On Harmonic Hilbert Spaces on Compact Abelian Groups

Abstract

Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted $L^2$ spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach $^*$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group G to have the same spectrum as the $C^*$-algebra of continuous functions on G. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on $L^2(G)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

1 Introduction

The notion of a harmonic Hilbert space on the real line was introduced by Delvos [9], extending earlier work of Babuška [2, 3] and Práger [39] on periodic Hilbert spaces. Subsequently, Feichtinger, Pandey, and Werther [18] constructed harmonic Hilbert spaces on arbitrary locally compact abelian groups (LCAs), and characterized their properties using the theory of Wiener amalgam spaces [16]. In essence, a harmonic Hilbert space ${\mathcal {H}}_\lambda$ on an LCA G is the image under the Fourier transform of a weighted $L^2$ space on the dual group, ${\hat{G}}$, associated with a weight function $w \equiv \lambda ^{-1/2} : {\hat{G}} \rightarrow {\mathbb {R}}_+$, where $\lambda \in L^1({\hat{G}})$. Harmonic Hilbert spaces are reproducing kernel Hilbert spaces (RKHSs) of continuous functions, and have a number of useful properties for function approximation on LCAs [9, 10, 18]. In particular, using results in [15], it has been shown that if w is subadditive (up to a constant C),

$$\begin{aligned} w(\gamma + \gamma ') \le C (w(\gamma ) + w(\gamma ')), \quad \gamma ,\gamma ' \in {\hat{G}}, \end{aligned}$$

(1)

then ${\mathcal {H}}_\lambda$ is a Banach algebra under pointwise function multiplication [18].

In this paper, we study harmonic Hilbert spaces induced by subconvolutive functions $\lambda \in L^1({\hat{G}})$, satisfying

$$\begin{aligned} (\lambda * \lambda ) (\gamma ) \le C \lambda (\gamma ), \quad \gamma \in {\hat{G}}, \end{aligned}$$

(2)

along with strict positivity and symmetry conditions (see [15, Satz 3.6] and related ideas in earlier papers such as [12, 13]). A prototypical example is the class of subexponential weights on the dual group ${\hat{G}} = \mathbb Z^d$ of $G = {\mathbb {T}}^d$,

$$\begin{aligned} \lambda ^{-1}(\gamma ) = e^{\tau |\gamma |^p}, \quad \tau > 0, \quad p \in (0,1). \end{aligned}$$

(3)

One of our main results, Theorem 4, is that on a compact abelian group G every such space ${\mathcal {H}}_\lambda$ is a symmetric Banach $^*$-algebra under pointwise function multiplication and complex conjugation. In addition, in Theorem 6 we show that ${\mathcal {H}}_\lambda$ has the same spectrum (space of maximal ideals) as the $C^*$-algebra of continuous functions on G. We also study one-parameter families of RKHSs associated with self-adjoint Markov operators on $L^2(G)$. In this setting, we find that subconvolutivity of $\lambda$ is a necessary and sufficient condition for these spaces to have Banach algebra structure under pointwise function multiplication; see Theorem 11.

We will refer to RKHSs which are simultaneously Banach $^*$-algebras with respect to pointwise function multiplication and complex conjugation as reproducing kernel Hilbert algebras (RKHAs).

As an application of our results, we discuss connections between RKHAs and a class of Banach spaces ${\mathcal {A}}_w$ on a compact abelian group G induced from weights $w: {\hat{G}} \rightarrow {\mathbb {R}}_+$ such that ${\mathcal {A}}_w$ is a subspace of the Wiener algebra ${\mathcal {A}}(G)$ of continuous functions with absolutely convergent Fourier series. Spaces in this class have recently received attention in the context of function approximation in high-dimensional periodic domains, $G = {\mathbb {T}}^d$ with $d \gg 1$, where the weight function $w \equiv w_{s,r}$ is oftentimes from the class of dominating mixed smoothness,

$$\begin{aligned} w_{s,r}(\gamma ) = \prod _{i=1}^d (1 + |\gamma _i |^r)^{s/r}, \quad r,s \in (0, \infty ), \end{aligned}$$

(4)

with $\gamma = (\gamma _1, \ldots , \gamma _d) \in {\hat{G}} = \mathbb Z^d$ and a related definition for $r = \infty$ [31, 35]. Kolomoitsev, Lomako, and Tikhonov [31] refer to the spaces $\mathcal A_w({\mathbb {T}}^d)$ as weighted Wiener spaces. In the paper [35], Nguyen, Nguen, and Sickel call $\mathcal A_w({\mathbb {T}}^d)$ weighted Wiener algebras, though they do not claim that these spaces are actually algebras under pointwise multiplication. In Sect. 7, we consider cases where ${\mathcal {A}}_w$ and ${\mathcal {H}}_\lambda$ with $\lambda = w^{-2}$ are both Banach algebras associated with subconvolutive weights (which include the $w_{s,r}$ family for $s \ge 2$), using results from [5, 33] to verify the Banach algebra structure of ${\mathcal {A}}_w$. Following a suggestion of Feichtinger [17], we call the Banach algebras ${\mathcal {A}}_w$ Fourier–Wermer algebras owing to the fact that they are Fourier images of convolution algebras $L^1_w({\hat{G}})$ on the dual group ${\hat{G}}$ that were studied in the early paper of Wermer [48]. We establish embedding relationships between ${\mathcal {A}}_w$ and ${\mathcal {H}}_\lambda$ that should be useful in function approximation applications.

1.1 Related Work

The study of Banach algebras on LCAs associated with subconvolutive weights was initiated by Feichtinger [15]. That work established that $L^\infty _w$ is a Banach convolution algebra iff $w^{-1}$ is subconvolutive. It was also shown that if w is subadditive and $w^{-1}$ lies in $L^1$, then $w^{-1}$ is subconvolutive. Thus, subadditive weights with integrable inverses provide a useful route to constructing subconvolutive weights and associated $L^\infty _w$ convolution algebras. In general, however, subadditivity and subconvolutivity are independent notions. Indeed, the subexponential weights $w=\lambda ^{-1}$ from (3) are subconvolutive but not subadditive. Another relevant property of weight functions on LCAs is submultiplicativity,

$$\begin{aligned} w(\gamma +\gamma ') \le C w(\gamma )w(\gamma '), \end{aligned}$$

the main result here being that $L^1_w$ is a Banach convolution algebra iff w is submultiplicative [15].

Besides $L^1_w$ and $L^\infty _w$, sufficient conditions for $L^p_w$, $1< p < \infty$, to be Banach convolution algebras are also known. An early reference in that direction is the paper of Wermer [48], which studies $L^p_w$ convolution algebras on the real line, including the case $p=1$. More recently, Kuznetsova [33] has shown (using methods of proof for sequence spaces from [36]) that for a locally compact group, $L^p_w$ with $1< p < \infty$ is a convolution algebra if

$$\begin{aligned} (w^{-p} * w^{-p})(\gamma ) \le C w^{-p}(\gamma ). \end{aligned}$$

(5)

When $p=2$, the latter condition is equivalent to (2) with $\lambda = w^{-2}$. Related results for weighted convolution algebras on ${\mathbb {R}}^n$ have been obtained by Kerman and Sawyer [30].

See [28, 41] for surveys of weight functions in harmonic analysis. Closely related to weight function theory is the theory of Sobolev spaces and algebras on LCAs, e.g., [4, 6, 19, 25, 26].

In the paper [13], Essén studies the spectral properties of Banach convolution algebras on ${\mathbb {Z}}$ or ${\mathbb {R}}$ under subconvolutive weights with additional decay conditions. In [5], Brandenburg establishes sufficient conditions for the equivalence of the spectrum of unital commutative Banach algebra B and a subalgebra $S \subset B$. Using these conditions, it is shown that for a subadditive weight function $w \ge 1$ on an LCA, the space $L^1 \cap L^\infty _w$ (equipped with the norm $\Vert f \Vert _{L^1} + \Vert f \Vert _{L^\infty _w}$) is a Banach convolution algebra which has the same spectrum as the group convolution algebra $L^1$. The spectral properties of $L^p_w$ convolution algebras on LCAs were studied in [33], where it was shown that every such algebra is semisimple. In more recent work, Kuznetsova and Molitor-Braun [32] studied the representation theory for convolution Banach $^*$-algebras on locally compact, non-abelian groups, and established sufficient conditions for these algebras to be symmetric (i.e., the spectra of positive elements are subsets of the positive half-line), among other results.

A related notion of Banach algebras on locally compact groups with Hilbert space structure is the class of $H^*$-algebras proposed by Ambrose [1]. An $H^*$-algebra H is required to satisfy the identity

$$\begin{aligned} \langle fg, h \rangle _H = \langle g, f^*h \rangle _H = \langle f, h g^* \rangle _H, \quad \forall g,h\in H, \end{aligned}$$

(6)

which implies that for each $f\in H$, the operations of left and right multiplication by f are norm-preserving, $^*$-homomorphisms between H and B(H) (the $C^*$-algebra of bounded linear maps on H). A classical example of an $H^*$-algebra is the $L^2$ convolution algebra on a compact group. Note that the RKHAs and weighted convolution algebras on LCAs mentioned above do not, in general, satisfy (6).

In this paper, we focus on the setting of compact abelian groups, which allows us to approach the problem of constructing RKHAs using Mercer theory. In particular, in this setting the values $\lambda (\gamma )$ correspond to the eigenvalues of a compact integral operator on $L^2(G)$ associated with the reproducing kernel of ${\mathcal {H}}_\lambda$, and the characters $\gamma \in {\hat{G}}$, $\gamma : G \rightarrow S^1 \subset \mathbb {C}$, form a corresponding orthogonal eigenbasis. This structure allows us to deduce that $\mathcal H_\lambda$ is a Banach algebra if $\lambda \in L^1({\hat{G}})$ is subconvolutive, which is more general than the subconvolutivity implied by subadditivity of $\lambda ^{-1} = w^2$. Mercer theory also facilitates the characterization of the associated maximal ideal spaces using kernel integral operators.

1.2 Plan of the Paper

We introduce our notation in Sect. 2, and give an overview of relevant results from RKHS theory in Sect. 3. In Sect. 4, we describe the construction of RKHAs on compact abelian groups associated with subconvolutive functions. In Sect. 5, we study the spectra and state spaces of RKHAs (not necessarily associated with subconvolutive functions). In Sect. 6 we study one-parameter families of RKHAs associated with Markov semigroups. Sect. 7 discusses aspects of Fourier–Wermer algebras associated with subconvolutive weights and their embedding relationships with RKHAs. Appendix A contains the proof of an auxiliary result, Lemma 2, on integral operators associated with translation-invariant kernels on LCAs.

2 Notation and Preliminaries

2.1 Locally Compact Abelian Groups

Consider an LCA G equipped with a Haar measure $\mu$. In most instances, we will assume that G is compact, in which case $\mu$ will be normalized to a probability measure. We let $\hat{G}$ denote the dual group of G, i.e., the abelian group of continuous homomorphisms $\gamma : G \rightarrow S^1$, equipped with its dual measure, ${\hat{\mu }}$ [34, 42]. We identify each element of $\hat{G}$ with a continuous, complex-valued function on G, taking values in the unit circle $S^1 \subset {\mathbb {C}}$, and acting on ${\mathbb {C}}$ multiplicatively as a unitary character. The trivial character in ${\hat{G}}$ will be denoted by $0_{{\hat{G}}}$. When there is no risk of confusion with scalar multiplication of functions, the inverse of $\gamma \in {\hat{G}}$ will be denoted by $-\gamma$. We recall that if G is compact, ${\hat{G}}$ has a discrete topology and ${\hat{\mu }}$ is a weighted counting measure.

In what follows, $C_0(G)$ (resp. $C_0( \hat{G})$) will denote the Banach space of complex-valued, continuous functions on an LCA G (resp. its dual $\hat{G}$) vanishing at infinity, equipped with the uniform norm. Moreover, $\mathcal {F}: L^1(G) \rightarrow C_0(\hat{G})$ and $\hat{\mathcal {F}}: L^1(\hat{G})\rightarrow C_0(G)$ will denote the Fourier and inverse Fourier transforms, respectively, i.e.,

$$\begin{aligned} \mathcal {F}f(\gamma ) := \int _G f(x) \gamma (-x) \, d\mu (x), \quad \hat{\mathcal {F}}{\hat{f}}( x ) := \int _{\hat{G} } {\hat{f}}(\gamma ) \gamma (x) \, d{\hat{\mu }}(\gamma ). \end{aligned}$$

We also use $*$ and $^\star$ to denote the convolution (algebraic product) and antilinear involution operations on the group algebra $L^1(G)$, respectively, i.e.,

$$\begin{aligned} ( f * g )(x) = \int _G f(x - y) g(y) \, d\mu (y), \quad f^\star (x) = \overline{f(-x)}, \end{aligned}$$

and a similar notation for the corresponding operations on $L^1(\hat{G})$. When working with Hilbert spaces such as $L^2(G)$ or ${\mathcal {H}}_\lambda$, we will adopt the convention that the inner product is antilinear in the first argument, e.g., $\langle f, g \rangle _{L^2(G)} = \int _G {\bar{f}} g \, d\mu$.

Throughout the paper, $S^x$ will denote the (left) shift operator by group element $x \in G$, i.e., $S^x f( y ) = f( x + y )$ for any element $y \in G$ and function $f : G \rightarrow {\mathbb {C}}$. The collection $\{ S^x \}_{x \in G}$ forms a strongly continuous group of isometries on any of the spaces $L^p(G)$ with $1 \le p < \infty$ and $C_0(G)$. We also recall the standard property of Fourier transforms that

$$\begin{aligned} \mathcal {F}(S^x f)(\gamma ) = \gamma (x) ( \mathcal {F}f )( \gamma ), \quad \forall f \in L^1(G), \quad \forall x \in G, \quad \forall \gamma \in \hat{G}. \end{aligned}$$

(7)

Reusing notation, we shall let ${\mathcal {F}} : L^2(G) \rightarrow L^2( {\hat{G}})$ denote the unitary extension of the Fourier operator. Similarly, we will continue to use $*$ to denote the convolution operator on $L^2(G)$. Note that for any $f,g \in L^2(G)$, $f * g$ lies in $C_0(G)$ [21, Proposition 2.40], and we have

$$\begin{aligned} ( f * g )(x) = \langle {\hat{f}}^\star , S^x g \rangle _{L^2(G)}. \end{aligned}$$

(8)

We will let ${\hat{S}}^\gamma$ be the shift operator on the dual group ${\hat{G}}$, defined analogously to $S^x$ on G and satisfying the corresponding relations in (7) and (8).

We say that a function $f:G\rightarrow \mathbb {C}$ on an LCA is uniformly continuous if for every $\epsilon >0$ there is a neighborhood U of the identity element of G, such that for every $x\in G$ and $y\in x+U$, $\left| f(x) - f(y) \right| <\epsilon$. This notion is equivalent to uniform continuity of functions on uniform spaces [29, Definition 7.6].

2.2 Reproducing Kernel Hilbert Algebras

Let $\lambda$ be a positive-valued function in $L^1({\hat{G}})$. In this paper, our focus is on RKHSs on the group G with translation-invariant kernels $k: G \times G \rightarrow {\mathbb {C}}$ induced by $\lambda$, viz.

$$\begin{aligned} k(x,y) = l( x - y ), \quad l := \hat{ \mathcal {F}} \lambda \in C_0(G). \end{aligned}$$

(9)

By Bochner’s theorem for LCAs [42, Section 1.4.2], k is the reproducing kernel of an RKHS ${\mathcal {H}}_\lambda$ of continuous functions; that is,

$$\begin{aligned} f(x) = \langle k(x,\cdot ), f \rangle _{{\mathcal {H}}_\lambda }, \quad \forall f \in {\mathcal {H}}_\lambda , \quad \forall x \in G, \end{aligned}$$

which expresses the fact that pointwise evaluation functionals on ${\mathcal {H}}_\lambda$ are continuous. Note that ${\mathcal {H}}_\lambda$ is a harmonic Hilbert space associated with the weight function $w = \lambda ^{-1/2}$.

We will say that ${\mathcal {H}}_\lambda$ is a reproducing kernel Hilbert algebra (RKHA) if it is a Banach $^*$-algebra with respect to pointwise function multiplication and complex conjugation, i.e.,

$$\begin{aligned} \Vert fg\Vert _{{\mathcal {H}}_\lambda } \le C \Vert f\Vert _{{\mathcal {H}}_\lambda } \Vert g\Vert _{{\mathcal {H}}_\lambda }, \quad \Vert f^* \Vert _{{\mathcal {H}}_\lambda } \equiv \Vert {\bar{f}} \Vert _{{\mathcal {H}}_\lambda } = \Vert f \Vert _{{\mathcal {H}}_\lambda }, \quad \forall f, g \in {\mathcal {H}}_\lambda . \end{aligned}$$

(10)

If ${\mathcal {H}}_\lambda$ is unital, we will use the symbol $1_G$ to denote the unit of ${\mathcal {H}}_\lambda$, i.e., the function equal to 1 at every point in G. Note that we do not require that the norm of $1_G$ is equal to 1. We will also let $\sigma _\lambda (f)$ denote the spectrum of $f \in {\mathcal {H}}_\lambda$, i.e., the set of complex numbers z such that $f - z$ does not have a multiplicative inverse in ${\mathcal {H}}_\lambda$. We recall that ${\mathcal {H}}_\lambda$ is symmetric as a unital Banach $^*$-algebra if $\sigma _\lambda (f^*f)$ is a subset of $[0, \infty )$ for every element $f\in {\mathcal {H}}_\lambda$.

Remark 1

We have stated the Banach algebra condition $\Vert fg \Vert _{{\mathcal {H}}_\lambda } \le C \Vert f \Vert _{\mathcal H_\lambda } \Vert g \Vert _{{\mathcal {H}}_\lambda }$ allowing a general constant C, as opposed to the more conventional definition $\Vert fg \Vert _{{\mathcal {H}}_\lambda } \le \Vert f \Vert _{\mathcal H_\lambda } \Vert g \Vert _{{\mathcal {H}}_\lambda }$. This choice does not affect any of the results presented below, as C can be absorbed in a redefinition of the reproducing kernel k of ${\mathcal {H}}_\lambda$ to a scaled kernel ${\tilde{k}} := C^2 k$. The corresponding RKHA, $\tilde{{\mathcal {H}}}_\lambda$, has the same elements as ${\mathcal {H}}_\lambda$, and satisfies $\Vert fg \Vert _{\tilde{{\mathcal {H}}}_\lambda } \le \Vert f \Vert _{\tilde{{\mathcal {H}}}_\lambda } \Vert g \Vert _{\tilde{\mathcal H}_\lambda }$, so we can view $\Vert \cdot \Vert _{\tilde{\mathcal H}_\lambda }$ as an equivalent norm to $\Vert \cdot \Vert _{{\mathcal {H}}_\lambda }$. That being said, it should be kept in mind that a number of standard results on Banach algebras require appropriate modification when $C > 1$. For example, the fact that a state on a unital Banach $^*$-algebra with isometric involution has unit operator norm does not necessarily hold when $C > 1$. In the present work, it is natural to allow a general C, as this enables a direct identification of the reproducing kernels of certain RKHAs with Markov transition kernels without having to employ additional normalization factors.

3 Results from Reproducing Kernel Hilbert Space Theory

In this section, we collect results from RKHS theory that will be useful in the analysis that follows.

First, consider a locally compact Hausdorff space X. We use ${\mathcal {P}}(X)$ to denote the set of Borel probability measures on X and ${\mathbb {E}}_\nu (\cdot ) = \int _X (\cdot ) \, d\nu$ the expectation operator with respect to $\nu \in {\mathcal {P}}(X)$.

We recall that a positive-definite kernel k on X whose corresponding RKHS ${\mathcal {H}}$ is dense in $C_0(X)$ is known as $C_0$-universal [7], or C-universal if X is compact [47]. If the map $R : {\mathcal {P}}( X ) \rightarrow {\mathcal {H}}$ with

$$\begin{aligned} R(\nu ) = \int _X k(x,\cdot )\, d\nu (x) \end{aligned}$$

(11)

is well-defined and injective, the kernel k is called characteristic [23]. In that case, R is referred to as a kernel mean embedding of probability measures [44]. Moreover, for any $f\in {\mathcal {H}}$ and $\nu \in {\mathcal {P}}(X)$, we have

$$\begin{aligned} {\mathbb {E}}_\nu f = \langle R(\nu ), f \rangle _{{\mathcal {H}}}. \end{aligned}$$

Thus, we can evaluate expectation values of elements of ${\mathcal {H}}$ by means of Hilbert space inner products. For a characteristic kernel, the feature map $F : X \rightarrow {\mathcal {H}}$ with $F(x) = k(x,\cdot )$ is injective, and has linearly independent range [47].

On a compact Hausdorff space X, a C-universal kernel k is strictly positive-definite and characteristic [27, 46]. Moreover, the kernel mean embedding induced by k metrizes the weak-$^*$ topology of ${\mathcal {P}}(X)$ [45]. That is, a sequence of measures $\nu _j \in {\mathcal {P}}(X)$ converges to $\nu \in \mathcal P(X)$ weak-$^*$ sense iff $R(\nu _j)$ converges to $R(\nu )$ in the norm of ${\mathcal {H}}$. See [22] for a study of characteristic kernels on LCAs.

The following are standard results from Mercer theory [20, 37, 46], which we state without proof.

Lemma 1

Let X be a compact Hausdorff space, $\mu$ a finite Borel measure with full support in X, and $k : X \times X \rightarrow {\mathbb {C}}$ a positive-definite, continuous kernel with associated RKHS $\mathcal {H}$. Then, the following hold.

• (i) $\mathcal {H}$ is a subspace of C(X) , and the inclusion $\mathcal {H}\hookrightarrow C(X)$ is compact.

• (ii) $K : L^2(G) \rightarrow \mathcal {H}$ with $K f = \int _X k( \cdot , x ) f(x) \, d\mu (x)$ is a well-defined, compact integral operator with dense range.

• (iii) The adjoint $K^* : \mathcal {H}\rightarrow L^2(X)$ is equal to the restriction of the inclusion map $C(X) \hookrightarrow L^2(X)$ on $\mathcal {H}$; that is, $K^* f = f$, $\mu$-a.e.

• (iv) ${\mathcal {K}} := K^* K$ is a positive, self-adjoint Hilbert-Schmidt operator on $L^2(X)$, with eigenvalues $\lambda _0 \ge \lambda _1 \ge \cdots \searrow 0$ and a corresponding orthonormal basis $\{ \phi _0, \phi _1, \ldots \}$ of eigenfunctions. Moreover, if k is positive-valued, ${\mathcal {K}}$ is of trace class.

• (v) The set $\{ \psi _j = \lambda _j^{-1/2 } K \phi _j : \lambda _j > 0 \}$ is an orthonormal basis of $\mathcal {H}$, satisfying $K^* \psi _j = \lambda _j^{1/2} \phi _j$.

• (vi) The kernel admits the Mercer series expansion

  $$\begin{aligned} k(x,y) = \sum _{j : \lambda _j >0 } \overline{\psi _j(x)} \psi _j(y), \end{aligned}$$

  which converges uniformly for $(x,y) \in X \times X$.

• (vii) If the $\lambda _j$ are strictly positive, k is C-universal (and thus strictly positive-definite and characteristic).

Next, we state certain properties of integral operators associated with translation-invariant kernels on LCAs.

Lemma 2

Let $\lambda \in L^1({\hat{G}} )$ be an absolutely integrable, positive-valued function on the dual group ${\hat{G}}$ of an LCA G, with absolutely integrable Fourier transform $l=\hat{\mathcal F}\lambda \in L^1(G)$ and let $k : G \times G \rightarrow {\mathbb {C}}$ be the corresponding translation-invariant reproducing kernel from (9). Then:

• (i) k is uniformly continuous, and $\mathcal {H}_\lambda$ is a subspace of $C_0(G)$.

• (ii) The integral operator

  $$\begin{aligned} K : f \mapsto \int _G k( \cdot , x ) f(x) \, d\mu (x) \end{aligned}$$

  maps $L^\infty (G)$ into the Banach space of bounded functions on G. Moreover, K maps $L^2(G)$ and $L^1(G)$ into the space of uniformly continuous functions on G.

• (iii) For every $x \in G$ and $\gamma \in \hat{G}$,

  $$\begin{aligned} K \gamma (x) = \lambda (\gamma ) \gamma (x). \end{aligned}$$

Proof

See Appendix . $\square$

Recall that if G is compact, then $\hat{G}$ has a discrete topology. By Lemmas 1 and 2 (iii), when G is compact we can identify the eigenfunctions $\phi _j$ of the (compact) integral operator ${\mathcal {K}} : L^2(G) \rightarrow L^2(G)$ with the characters of G, $\phi _j \equiv \phi _\gamma \equiv \gamma$. Using $\Lambda \subseteq {\hat{G}}$ to denote the set $\{ \gamma \in \hat{G}: \lambda (\gamma ) > 0 \}$, and defining $\xi (\gamma ) := \sqrt{\lambda (\gamma )}$, the corresponding basis functions $\psi _j \equiv \psi _\gamma$ of ${\mathcal {H}}_\lambda$ are

$$\begin{aligned} \psi _{\gamma } := \frac{1}{ \xi (\gamma ) } K \gamma = \xi (\gamma ) \gamma , \quad \gamma \in \Lambda . \end{aligned}$$

(12)

As a result, the elements of $\mathcal {H}_\lambda$ can be explicitly characterized as

$$\begin{aligned} \mathcal {H}_\lambda = \left\{ f = \sum _{\gamma \in \Lambda } {\hat{f}}_\gamma \gamma = \sum _{\gamma \in \Lambda } {\hat{f}}_\gamma \psi _{\gamma }/ \xi (\gamma ): \; \sum _{\gamma \in \Lambda } |{\hat{f}}_\gamma |^2 / \lambda (\gamma ) < \infty \right\} . \end{aligned}$$

In the above, the coefficients ${\hat{f}}_\gamma$ coincide with the values of the Fourier transform of the continuous function $f \in \mathcal {H}_\lambda \subseteq C(G)$ on $\Lambda$, i.e., ${\hat{f}}_\gamma = {\mathcal {F}} f(\gamma )$. Moreover, the condition $\sum _{\gamma \in \Lambda } |{\hat{f}}_\gamma |^2 / \lambda (\gamma ) < \infty$ is equivalent to the statement that the function ${\hat{u}} : {\hat{G}} \rightarrow {\mathbb {C}}$ with

$$\begin{aligned} {\hat{u}}( \gamma ) = {\left\{ \begin{array}{ll} {\hat{f}}_\gamma / \xi (\gamma ), &{} \gamma \in \Lambda ,\\ 0, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

lies in $L^2({\hat{G}})$. Together, these facts imply:

Lemma 3

The following statements are equivalent:

• (i) f is an element of $\mathcal {H}_\lambda$.

• (ii) There exists ${\hat{u}} \in L^2({\hat{G}})$ with $\Vert f \Vert _{{\mathcal {H}}_\lambda } = \Vert {\hat{u}} \Vert _{L^2({\hat{G}})}$ such that ${\mathcal {F}} f = \xi {\hat{u}}$.

Moreover, ${\hat{u}}$ is unique, and can be explicitly constructed as ${\hat{u}} = \xi ^+ {\mathcal {F}} f$, where

$$\begin{aligned} \xi ^+(\gamma ) = {\left\{ \begin{array}{ll} 1/ \xi (\gamma ), &{} \gamma \in \Lambda , \\ 0, &{} \gamma \in {\hat{G}} \setminus \Lambda . \end{array}\right. } \end{aligned}$$

4 Reproducing Kernel Hilbert Algebras from Subconvolutive Functions

Unless otherwise stated, throughout this section we will assume that G is compact. One of our main results is the following.

Theorem 4

Suppose that $\lambda \in L^1({\hat{G}})$ is:

• (i) Strictly positive-valued, $\lambda (\gamma ) > 0$;

• (ii) Subconvolutive, $( \lambda * \lambda )(\gamma ) \le C \lambda (\gamma )$;

• (iii) Symmetric, $\lambda (-\gamma ) = \lambda (\gamma )$.

Then, ${\mathcal {H}}_\lambda$ is a unital, symmetric Banach $^*$-algebra with respect to pointwise multiplication and complex conjugation of functions, and lies dense in C(G).

Proof

See Sect. 4.1. $\square$

Remark 2

The fact that under Assumption (ii) of Theorem 4${\mathcal {H}}_\lambda$ is a Banach algebra can be readily deduced from [33, Theorem 1], which came to our attention after completion of this work. In particular, [33, Theorem 1] establishes that under Assumption (ii) $L^2_w({\hat{G}})$ with $w=\lambda ^{-1/2}$ is a convolution algebra, and taking Fourier transforms yields the Banach algebra property of ${\mathcal {H}}_\lambda$ under pointwise multiplication. See also [30, 32, 36] for related work. In Sect. 4.1, we include a proof tailored to the Hilbert space setting, which uses Cauchy-Schwarz inequalities and the representation of convolution as an $L^2({\hat{G}})$ inner product (rather than, e.g., Hölder inequalities employed in [32]).

As mentioned in Sect. 1.1, a useful way of constructing subconvolutive functions on ${\hat{G}}$ through positive-valued functions $\lambda \in L^1({\hat{G}})$ with subadditive inverses,

$$\begin{aligned} \lambda ^{-1}(\gamma + \gamma ') \le C ( \lambda ^{-1}(\gamma ) + \lambda ^{-1}(\gamma ')). \end{aligned}$$

By [15, Corollary 3.8], every such function $\lambda$ is subconvolutive. If, in addition, $\lambda$ is strictly positive-valued and symmetric, then by Theorem 4${\mathcal {H}}_\lambda$ is an RKHA. The subexponential weights $\lambda ^{-1}$ in (3) also satisfy the assumptions of Theorem 4, but in this case $\lambda ^{-1}$ is not subadditive. Another example mentioned in Sect. 1.1 is that of an LCA G (not necessarily compact) with a function $\lambda \in L^1({\hat{G}})$ such that $\lambda ^{-1/2}$ is subadditive. In that case, too, ${\mathcal {H}}_\lambda$ is a Banach algebra with respect to pointwise multiplication [18]. To make contact with that result in the compact case, we note the following fact.

Lemma 5

Suppose that G is compact and $\xi \in L^2({\hat{G}} )$ is positive-valued, self-adjoint with respect to convolution ($\xi ^\star = \xi )$, and subconvolutive. Then, $\lambda = \xi ^2 \in L^1({\hat{G}})$ is also subconvolutive.

Proof

Since G is compact, we have $L^1({\hat{G}} ) \subseteq L^2({\hat{G}})$ and $\Vert \cdot \Vert _{L^2({\hat{G}})} \le \Vert \cdot \Vert _{L^1({\hat{G}})}$ (since the dual measure ${\hat{\mu }}$ is a normalized counting measure). Thus, using (8) and the facts that $\xi ^\star = \xi$ and $\xi * \xi \le C \xi$, we get

$$\begin{aligned} \lambda * \lambda (\gamma )&= \langle \lambda , S^\gamma \lambda \rangle _{L^2({\hat{G}})} = \langle \xi ^2, S^\gamma \xi ^2 \rangle _{L^2({\hat{G}})} = \langle \xi S^\gamma \xi , \xi S^\gamma \xi \rangle _{L^2({\hat{G}})} = \Vert \xi S^\gamma \xi \Vert _{L^2({\hat{G}})}^2 \\&\le \Vert \xi S^\gamma \xi \Vert _{L^1({\hat{G}})}^2 = \langle \xi , S^\gamma \xi \rangle _{L^2({\hat{G}})}^2 = ( \xi * \xi (\gamma ) )^2 \\&\le C^2 \xi ^2(\gamma ) = C^2 \lambda (\gamma ). \end{aligned}$$

$\square$

If G is compact and $\xi \in L^1({\hat{G}}) \subseteq L^2({\hat{G}})$ has a subadditive inverse, then by [15, Corollary 3.9], $\xi$ is subconvolutive. As a result, by Lemma 5, $\lambda = \xi ^2$ is also subconvolutive. Thus, under the additional constraint $\lambda ^{1/2} \in L^1({\hat{G}})$, the subadditivity assumption on $\lambda ^{-1/2}$ underlying the construction of the Banach algebra ${\mathcal {H}}_\lambda$ in [18] implies the subconvolutivity assumption in Theorem 4. Note that the subexponential weights in (3) satisfy $\lambda ^{1/2} \in L^1({\hat{G}})$.

4.1 Proof of Theorem 4

Consider two elements $f,g \in \mathcal {H}_\lambda$. By Lemma 3, to show that the continuous function fg lies in $\mathcal {H}_\lambda$ it is enough to show that the function ${\hat{w}} : {\hat{G}} \rightarrow \mathbb C$ defined as ${\hat{w}} = \xi ^+ {\mathcal {F}}(fg)$ lies in $L^2({\hat{G}})$. To that end, letting ${\hat{u}} = \xi ^+ {\mathcal {F}} f$ and ${\hat{v}} = \xi ^+ {\mathcal {F}} g$ be the $L^2({\hat{G}})$ representatives of f and g from Lemma 3, we obtain

$$\begin{aligned} {\mathcal {F}}(fg)(\gamma ) = ( {\mathcal {F}} f * {\mathcal {F}} g )(\gamma ) = ( ( \xi {\hat{u}} ) * ( \xi {\hat{v}} ) )( \gamma ) = \langle ( \xi {\hat{u}} )^{\star }, {\hat{S}}^\gamma ( \xi {\hat{v}}) \rangle _{L^2({\hat{G}})}. \end{aligned}$$

Then, using standard properties of shift operators and $L^2$ inner products, as well as the fact that $\xi$ is real and $L^1({\hat{G}})$-self-adjoint, we get

$$\begin{aligned} |{\mathcal {F}}(fg)(\gamma ) |^2&= |\langle ( \xi {\hat{u}} )^{\star }, {\hat{S}}^\gamma ( \xi {\hat{v}}) \rangle _{L^2({\hat{G}})} |^2 = |\langle \xi {\hat{u}}^\star , ( {\hat{S}}^\gamma \xi ) ( {\hat{S}}^\gamma {\hat{v}}) \rangle _{L^2({\hat{G}})} |^2 \\&= |\langle \xi S^\gamma \xi , \overline{{\hat{u}}^*} {\hat{S}}^\gamma {\hat{v}} \rangle _{L^2({\hat{G}})} |^2 \le \langle \xi S^\gamma \xi , \xi S^\gamma \xi \rangle _{L^2({\hat{G}})} \langle \overline{{\hat{u}}^*} {\hat{S}}^\gamma v, \overline{{\hat{u}}^*} {\hat{S}}^\gamma v \rangle _{L^2({\hat{G}})} \\&= \langle \lambda , {\hat{S}}^\gamma \lambda \rangle _{L^2({\hat{G}})} \langle |{\hat{u}}^\star |^2, S^\gamma |{\hat{v}} |^2 \rangle _{L^2({\hat{G}})} = [ ( \lambda * \lambda )(\gamma ) ] [ ( |{\hat{u}}^\star |^2 * |{\hat{v}} |^2 )( \gamma ) ] \\&\le C \lambda (\gamma ) [ ( |{\hat{u}}^\star |^2 * |{\hat{v}} |^2 )( \gamma ) ], \end{aligned}$$

where we used the subconvolutivity of $\lambda$ to arrive at the last line. Thus, since $\lambda$ is strictly positive-valued, we have $\xi ^+(\gamma ) = 1 / \sqrt{\lambda }(\gamma )$, and $|{\hat{w}}(\gamma ) |^2 \le C ( |{\hat{u}}^\star |^2 * |{\hat{v}} |^2 )( \gamma )$. Therefore,

$$\begin{aligned} \Vert {\hat{w}} \Vert _{L^2({\hat{G}})} \le C \Vert |{\hat{u}}^\star |^2 * |{\hat{v}} |^2 \Vert _{L^1({\hat{G}})}, \end{aligned}$$

and it follows that ${\hat{w}}$ lies in $L^2({\hat{G}})$ since $|{\hat{u}}^\star |^2 * |{\hat{v}} |^2$ is the convolution of the $L^1({\hat{G}})$ elements $|{\hat{u}}^\star |^2$ and $|{\hat{v}} |^2$.

We thus conclude that ${\mathcal {H}}_\lambda$ is a Banach algebra with respect to pointwise function multiplication. The fact that ${\mathcal {H}}_\lambda$ is a dense subspace of C(G) follows from the strict positivity of $\lambda$ in conjunction with Lemma 1(vii).

Next, we verify that $f^* (x) = \overline{f(x)}$ is an isometric, antilinear involution on $\mathcal {H}_\lambda$. Since $\lambda (\gamma ) = \lambda (-\gamma )$ for every $\gamma \in {\hat{G}}$, the orthonormal basis elements $\psi _{\gamma }$ from (12) satisfy

$$\begin{aligned} \overline{ \psi _{\gamma }(x) } = \overline{\lambda ^{1/2}(\gamma ) \gamma (x)} = \lambda ^{1/2}(-\gamma ) \overline{\gamma (x)} = \lambda ^{1/2}(-\gamma ) \gamma ^{-1}(x) = \psi _{-\gamma }(x), \end{aligned}$$

so $\Vert \psi ^*_{\gamma } \Vert = \Vert \psi _{-\gamma } \Vert _{{\mathcal {H}}_\lambda } = 1$. Therefore, $^*$ preserves the norm of orthonormal basis vectors of $\mathcal {H}_\lambda$. Moreover, it is clearly antilinear and involutive, so ${\mathcal {H}}_\lambda$ is a Banach $^*$-algebra satisfying (10).

The RKHA $\mathcal {H}_\lambda$ is also unital and satisfies $\Vert 1_G \Vert _{{\mathcal {H}}_\lambda } = 1$ since the unit basis vector $\psi _0$ is equal to the trivial character in ${\hat{G}}$, and thus everywhere equal to 1 on G.

Finally, the symmetry of ${\mathcal {H}}_\lambda$ follows from Corollary 8 in Sect. 5 below.$\square$

5 Spectra and States of Reproducing Kernel Hilbert Algebras

In general, an RKHA ${\mathcal {H}}_\lambda$ on a compact abelian group G does not satisfy the $C^*$ identity, $\Vert f^* f \Vert _{C(G)} = \Vert f \Vert ^2_{C(G)}$, holding for the $C^*$-algebra of continuous functions on G under pointwise multiplication and complex conjugation, nor does it satisfy the $H^*$-identity in (6) enjoyed by the $L^2(G)$ convolution algebra. Failure to meet, in particular, the last property means that the regular representation of $\mathcal H_\lambda$ into $B({\mathcal {H}}_\lambda )$ is not a $^*$-representation.

Yet, by virtue of their RKHS structure, RKHAs possess continuous evaluation functionals $\delta _x : {\mathcal {H}}_\lambda \rightarrow \mathbb C$ at every $x \in G$,

$$\begin{aligned} \delta _x f = f(x) = \langle k( x, \cdot ), f \rangle _{{\mathcal {H}}_\lambda }, \quad \Vert \delta _x \Vert _{{\mathcal {H}}'_\lambda } = \sqrt{k(x,x)}, \end{aligned}$$

(13)

satisfying

$$\begin{aligned} \delta _x(fg) = (\delta _x f ) ( \delta _x g ), \quad \delta _x f^* = \overline{\delta _x f}, \quad \forall f, g \in {\mathcal {H}}_\lambda , \end{aligned}$$

where $\Vert \cdot \Vert _{{\mathcal {H}}'_\lambda }$ is the operator norm of functionals in the dual space ${\mathcal {H}}'_\lambda$. Every nonzero evaluation functional $\delta _x$ is an element of the spectrum of ${\mathcal {H}}_\lambda$, i.e., the set of nonzero homomorphisms of ${\mathcal {H}}_\lambda$ into ${\mathbb {C}}$, denoted by $\sigma ( {\mathcal {H}}_\lambda )$. In addition, as we will see below, under appropriate conditions on the kernel, the $\delta _x$ provide an abundance of states on ${\mathcal {H}}_\lambda$, and also induce a set of states on the non-abelian $C^*$-algebra $B(\mathcal H_\lambda )$.

Recall now that for a compact Hausdorff space G, the spectrum of the $C^*$-algebra C(G) consists precisely of the evaluation functionals $\delta _x$ at every $x \in G$ [11]. Moreover, the map $\beta : G \rightarrow \sigma (C(G))$ with $\beta (x) = \delta _x$ and the Gelfand transform $\Gamma : C(G) \rightarrow C(\sigma (C(G)))$ with $( \Gamma f )(\delta _x) = f(x)$ are homeomorphisms with respect to the weak-$^*$ topology of $\sigma (C(G))$.

The following theorem characterizes analogously the spectra of RKHAs on compact abelian groups and the associated Gelfand transforms.

Theorem 6

Let ${\mathcal {H}}_\lambda$ be an RKHA on a compact abelian group associated with a strictly positive function $\lambda \in L^1({\hat{G}})$. Then, the following hold.

• (i) The map $\beta _\lambda : G \rightarrow \sigma ({\mathcal {H}}_\lambda )$ with $\beta _\lambda (x) = \delta _x$ is a homeomorphism with respect to the weak-$^*$ topology on $\sigma (\mathcal {H}_\lambda )$ inherited as a subset of $\mathcal {H}'_\lambda$.

• (ii) Under the identification $G \simeq \sigma (C(G))$ induced by $\beta$, the Gelfand transform $\Gamma _\lambda : {\mathcal {H}}_\lambda \rightarrow C(\sigma ({\mathcal {H}}_\lambda ))$ with $( \Gamma _\lambda f )(\delta _x) = f(x)$ coincides with the inclusion map $\iota : {\mathcal {H}}_\lambda \hookrightarrow C(G)$. In particular, the operator norm of $\Gamma _\lambda$ is equal to $\sqrt{l(0_G)}$.

Proof

See Sect. 5.1. $\square$

Theorem 6 establishes that ${\mathcal {H}}_\lambda$ has the same spectrum as C(G). Analogous results were found in the paper [5] for convolution algebras on LCAs. In particular, a class of convolution algebras associated with subadditive weight functions was shown to have the same spectrum as the group convolution algebra $L^1(G)$. Theorem 6 addresses the case of algebras with respect to pointwise function multiplication that are simultaneously RKHSs. In particular, our method of proof in Sect. 5.1 makes explicit use of the RKHSs structure of ${\mathcal {H}}_\lambda$.

The following are corollaries of Theorem 6.

Corollary 7

   

• (i) Every non-vanishing function $f \in {\mathcal {H}}_\lambda$ has a multiplicative inverse in ${\mathcal {H}}_\lambda$.

• (ii) Every strictly positive function $f \in {\mathcal {H}}_\lambda$ has a square root in ${\mathcal {H}}_\lambda$, i.e., there exists a (strictly positive) $g \in {\mathcal {H}}_\lambda$ such that $f = g^2$.

• (iii) The spectrum $\sigma _\lambda (f)$ of any $f\in {\mathcal {H}}_\lambda$ is equal to the range of f.

Proof

If f is non-vanishing, then $\delta _x f \ne 0$ for any $x \in G$, and thus by Theorem 6(i) f does not lie in any maximal ideal of ${\mathcal {H}}_\lambda$. As a result, f is invertible. This proves Claim (i).

Turning to Claim (iii), let $f \in {\mathcal {H}}_\lambda$ be arbitrary. It is clear that ${{\,\textrm{ran}\,}}f \subseteq \sigma _\lambda (f)$ (since $f-z$ has a zero whenever $z \in {{\,\textrm{ran}\,}}f$, and thus cannot have a multiplicative inverse). If $z \in \sigma _\lambda (f)$ and $z \notin {{\,\textrm{ran}\,}}f$, then $f - z$ is a nowhere-vanishing non-invertible element of ${\mathcal {H}}_\lambda$, which contradicts Claim (i). Thus, we have $\sigma _\lambda (f) \subseteq {{\,\textrm{ran}\,}}f$, and we conclude that $\sigma _\lambda (f) = {{\,\textrm{ran}\,}}f$.

Finally, to verify Claim (ii), we recall that every element of a unital Banach-$^*$ algebra with strictly positive spectrum has a square root, which can be chosen to also have strictly positive spectrum; e.g., [43, §10.30]. Since, by Claim (iii), $f>0$ has $\sigma _\lambda (f) = {{\,\textrm{ran}\,}}f \subset (0,\infty )$, it follows that there exists $g \in {\mathcal {H}}_\lambda$ with $\sigma _\lambda (g) \in (0,\infty )$ such that $f = g^2$. Again by Claim (iii), ${{\,\textrm{ran}\,}}g = \sigma _\lambda (g)$, and thus g is strictly positive. $\square$

Corollary 8

The RKHA ${\mathcal {H}}_\lambda$ is (i) semisimple; and (ii) symmetric.

Proof

Claim (i) follows from the fact that the Gelfand transform $\Gamma _\lambda$ has trivial kernel (by Theorem 6(ii)). For Claim (ii) we use Corollary 7(iii) in conjunction with the fact that $f^* f \equiv {\bar{f}} f \ge 0$ to conclude that $\sigma _\lambda (f^* f) = {{\,\textrm{ran}\,}}(f^* f) \subseteq [0, \infty )$. $\square$

Next, we consider the state space, ${\mathcal {S}}(\mathcal H_\lambda )$, of a unital RKHA ${\mathcal {H}}_\lambda$, i.e., the set of (automatically continuous) positive functionals $\varphi : \mathcal H_\lambda \rightarrow {\mathbb {C}}$, normalized such that $\varphi (1_G) = 1$. By (13), for a unital RKHA ${\mathcal {H}}_\lambda$ with reproducing kernel k, each nonzero evaluation functional is a state with operator norm equal to $\sqrt{k(x,x)}$. It should be noted that because we allow continuity constants C different from 1 in our definition of Banach algebras in (10), the elements of ${\mathcal {S}}({\mathcal {H}}_\lambda )$ need not have unit operator norm (which would be the case if $C =1$).

Suppose now that the evaluation functional $\delta _x$ at every $x\in G$ is nonzero (a condition that holds iff $\lambda > 0$). Then, viewing $\delta _x$ as a Dirac probability measure in $\mathcal P(G)$ leads to the identity

$$\begin{aligned} \delta _x = \langle R(\delta _x), \cdot \rangle _{{\mathcal {H}}_\lambda }, \quad \forall x \in G, \end{aligned}$$

(14)

where $R : {\mathcal {P}}(G) \rightarrow {\mathcal {H}}_\lambda$ is the kernel mean embedding of probability measures defined in (11). By continuity of the feature map $x \mapsto F(x) \equiv k(x,\cdot )$ as a map from G into ${\mathcal {H}}_\lambda$, (14) extends to a map $P : {\mathcal {P}}(G) \rightarrow {\mathcal {S}}({\mathcal {H}}_\lambda )$ such that

$$\begin{aligned} ( P \nu ) f = \int _G \delta _x f \, d\nu (x) = \langle R( \nu ), f \rangle _{{\mathcal {H}}_\lambda }. \end{aligned}$$

(15)

Similarly, to each Dirac probability measure $\delta _x \in \mathcal P(G)$, we can assign a state $\rho _x \in {\mathcal {S}}(B(\mathcal H_\lambda ))$ of the $C^*$-algebra $B({\mathcal {H}}_\lambda )$ given by

$$\begin{aligned} \rho _x = {{\,\textrm{tr}\,}}( \Pi _x \cdot ), \end{aligned}$$

(16)

where $\Pi _x : {\mathcal {H}}_\lambda \rightarrow {\mathcal {H}}_\lambda$ is the rank-1 projection operator

$$\begin{aligned} \Pi _x f = \frac{\langle k(x,\cdot ), f\rangle _{{\mathcal {H}}_\lambda }k(x,\cdot )}{k(x,x)} = \frac{f(x) k(x,\cdot )}{k(x,x)}. \end{aligned}$$

The assignment $\delta _x \mapsto \rho _x$ in (16) extends to a map $Q : {\mathcal {P}}(G) \rightarrow {\mathcal {S}}(B({\mathcal {H}}_\lambda ))$ with

$$\begin{aligned} (Q \nu ) A = \int _G \rho _x A \, d\nu (x). \end{aligned}$$

Intuitively, we can think of states of the non-abelian $C^*$-algebra $B({\mathcal {H}}_\lambda )$ in the range of Q as “classical” states induced by Borel probability measures on G (which are states of the abelian Banach $^*$-algebra ${\mathcal {H}}_\lambda$). Letting $\pi : {\mathcal {H}}_\lambda \rightarrow B({\mathcal {H}}_\lambda )$ denote the regular representation of ${\mathcal {H}}_\lambda$ with $\pi (f) g = fg$, the following proposition justifies the interpretation of states in ${{\,\textrm{ran}\,}}Q$ as classical states, in the sense of acting consistently on regular representatives of ${\mathcal {H}}_\lambda$ with expectation operators.

Proposition 9

With notation as above and under the assumptions of Theorem 6, the following hold.

• (i) The maps P and Q are injective and weak-$^*$ continuous.

• (ii) For every $\nu \in {\mathcal {P}}(G)$ and $f \in {\mathcal {H}}_\lambda$ the compatibility relations

  $$\begin{aligned} {\mathbb {E}}_\nu f = P(\nu )( f ) = Q(\nu )( \pi ( f ) ) \end{aligned}$$

  hold. In particular, we have $Q(\nu )(\pi (f^*)) = \overline{Q(\nu )( \pi (f))}$, even though $\pi$ need not be a $^*$-homomorphism.

Proof

Since G is compact and ${\mathcal {H}}_\lambda$ is a dense subspace of C(G), the reproducing kernel of ${\mathcal {H}}_\lambda$ is C-universal and thus characteristic. It follows that R is injective, and therefore so is P since $R(\nu )$ is the Riesz representative of $P(\nu )$ according to (15).

For every $\nu \in {\mathcal {P}}(G)$ and $f \in {\mathcal {H}}$ we get

$$\begin{aligned} Q(\nu )(\pi (f))&= \int _G {{\,\textrm{tr}\,}}(\Pi _x \pi (f)) \, d\nu (x) \\&= \int _G \sum _{\gamma \in {\hat{G}}} \frac{f(x) \langle \psi _\gamma , k(x,\cdot )\rangle _{{\mathcal {H}}} \langle k(x,\cdot ), \psi _\gamma \rangle _{{\mathcal {H}}}}{k(x,x)} \, d\nu (x) \\&= \int _G f(x) \frac{ \sum _{\gamma \in {\hat{G}}} \overline{\psi _\gamma (x)} \psi _\gamma (x) }{ k(x,x) } \, d\nu (x) \\&= \int _G f(x) \, d\nu (x) = {\mathbb {E}}_\nu f = P(\nu )(f), \end{aligned}$$

where $\{ \psi _{\gamma } = \sqrt{\lambda (\gamma )} \gamma : \gamma \in {\hat{G}} \}$ is the orthonormal basis of ${\mathcal {H}}_\lambda$ from (12). The above proves the compatibility relations in Claim (ii), and also implies that Q is injective by the injectivity of P. We also have

$$\begin{aligned} Q(\nu )(\pi (f^*)) = {\mathbb {E}}_\nu f^* = \overline{ E_\nu f} = \overline{Q(\nu )(\pi (f))}, \end{aligned}$$

verifying the $^*$-compatibility relation in Claim (ii).

Since ${\mathcal {H}}_\lambda$ is a subspace of C(G), the weak-$^*$ continuity of P follows directly from the fact that $P_\nu f = {\mathbb {E}}_\nu f$. Similarly, to deduce weak-$^*$ continuity of Q, note that for any $A \in B({\mathcal {H}}_\lambda )$, we have

$$\begin{aligned} (Q\nu ) A = {\mathbb {E}}_\nu f_A, \end{aligned}$$

where the function $f_A : x \mapsto \rho _x A$ is continuous. $\square$

The map $G\ni x \mapsto \Pi _x$ can be interpreted in an RKHS context as an operator-valued feature map. This feature map along with the corresponding embedding Q of probability measures on G generalize the standard RKHS feature maps and kernel mean embeddings of probability measures to the operator-valued setting of $B({\mathcal {H}}_\lambda )$. In other work [24], we have found these constructions to be useful in the context of quantum computation.

5.1 Proof of Theorem 6

We begin with the following observation about maximal ideals of unital RKHAs.

Lemma 10

With the assumptions of Theorem 6, every maximal ideal I in ${\mathcal {H}}_\lambda$ is orthogonal to the unit $1_G$.

Proof

Let $1_G = u + v$ with $u \in I$ and $v \in I^\perp$. Since I is a proper, closed subspace of ${\mathcal {H}}_\lambda$, the unit $1_G$ does not lie in I, and v is nonzero, i.e., $0 < \Vert v \Vert _{{\mathcal {H}}_\lambda } \le 1$. We claim that, in fact, $\Vert v \Vert _{{\mathcal {H}}_\lambda } = 1$.

To verify this, by rescaling the kernel of ${\mathcal {H}}_\lambda$, we assume without loss of generality that the multiplicative constant C in (10) is equal to 1 (see Remark 1).

Next, following standard techniques for unital Banach algebras, we equip ${\mathcal {H}}_\lambda$ with an equivalent norm, , induced from the operator norm of $B(\mathcal H_\lambda )$ and the regular representation $\pi$, viz.

This norm is a Banach algebra norm satisfying

In particular, , so if

then and $u = 1- v$ is invertible.

The latter, implies that I contains an invertible element, contradicting the fact that it is a maximal, and thus proper, ideal in ${\mathcal {H}}$. It follows that $\Vert v \Vert _{\mathcal H_\lambda } = 1$ and $u = 0$, proving that $1_G$ lies in $I^\perp$. $\square$

We now continue with the proof of Theorem 6.

5.2 Claim (i)

For every $f\in \mathcal {H}_\lambda$, $x \in G$, and net $(x_i)$ converging to x, we have

$$\begin{aligned} \beta _{\lambda }(x_i)(f ) = f (x_i) \rightarrow f (x) = \beta _{\lambda }(x)(f ), \end{aligned}$$

so $\beta _{\lambda }$ is weak-$^*$ continuous. In addition, $\beta _{\lambda }$ is injective since $\beta _\lambda (x) = \delta _x = \langle k(x,\cdot ), \cdot \rangle _{{\mathcal {H}}_\lambda }$, and $k(x,\cdot )$ is the image of x under the feature map $F : G \rightarrow {\mathcal {H}}_\lambda$ (which is injective since k is characteristic). Therefore, since G and $\sigma ({\mathcal {H}}_\lambda )$ are compact Hausdorff spaces, to show that $\beta _\lambda$ is a homeomorphism it suffices to show that it is surjective.

To prove the latter by contradiction, suppose that there exists $\psi \in \sigma ({\mathcal {H}}_\lambda ) \setminus {{\,\textrm{ran}\,}}\beta _{\mathcal H_\lambda }$. Then, $I := \ker \psi$ is a maximal ideal in ${\mathcal {H}}_\lambda$, which is distinct from $\ker \delta _x$ for all $x \in G$. We claim that I is a dense subspace of C(G) .

To verify this claim, observe first that the closure ${\bar{I}}$ of I in C(G) is an ideal. Indeed, if that were not the case, there would exist $f \in C(G)$ and $g \in {\bar{I}}$ such that $f g \notin {\bar{I}}$. But since ${\mathcal {H}}_\lambda$ is a dense subspace of C(G), there exists a sequence $f_n \in {\mathcal {H}}_\lambda$ converging to f in C(G) norm, and similarly there exists $g_n \in I$ such that $g_n \rightarrow g$ in C(G). Defining $h_n = f_n g_n$, it follows that $h_n$ is a sequence in I (since $f_n \in {\mathcal {H}}_\lambda$, $g_n \in I$, and I is an ideal in ${\mathcal {H}}_\lambda$) with a C(G)-norm limit $h \in {\bar{I}}$. The latter is equal to fg, contradicting the assertion that $fg \notin {\bar{I}}$.

Now suppose that ${\bar{I}}$ were contained in a maximal ideal in C(G). Since the maximal ideal space of C(G) is in bijective correspondence with the spectrum $\sigma (C(G))$, there would exist an $x \in G$ such that ${\bar{I}} \subseteq \ker \delta _x$ (with $\delta _x$ understood here as an evaluation functional on C(G)), contradicting the fact that I is distinct from the kernels of all evaluation functionals on ${\mathcal {H}}_\lambda$. We therefore conclude that ${\bar{I}}$ is an ideal distinct from any maximal ideal of C(G) , so it must be equal to the whole space C(G). We have thus verified that I is dense in C(G).

Next, since I is a maximal ideal in ${\mathcal {H}}_\lambda$, every $f \in I$ is ${\mathcal {H}}_\lambda$-orthogonal to $1_G$ by Lemma 10. Moreover, since $c := \lambda (0_{{\hat{G}}}) > 0$, the integral operator $K: L^2(G) \rightarrow {\mathcal {H}}_\lambda$ associated with k satisfies $c^{-1} K 1_G = 1_G$, and we get

$$\begin{aligned} 0 = \langle 1_G, f \rangle _{{\mathcal {H}}_\lambda } = c \langle 1_G, f \rangle _{{\mathcal {H}}_\lambda } = \langle K 1_G, f \rangle _{{\mathcal {H}}_\lambda } = \langle 1_G, K^* f \rangle _{L^2(G)}. \end{aligned}$$

As a result,

$$\begin{aligned} \Vert 1_G - f \Vert ^2_{C(G)} \ge \Vert 1_G - K^* f \Vert _{L^2(G)}^2 = 1 + \Vert K^* f \Vert ^2_{L^2(G)} \ge 1, \end{aligned}$$

which contradicts the assertion that I is dense in C(G), proving Claim (i).

5.3 Claim (ii)

The fact that $\Gamma _\lambda$ coincides with the inclusion map $\iota$ follows directly from the definition of the former and the fact that ${\mathcal {H}}_\lambda$ is a subspace of C(G), viz.,

$$\begin{aligned} ( \Gamma _\lambda f )(\delta _x) = f(x) = (\iota f )( x ). \end{aligned}$$

To verify the claim on the operator norm of $\Gamma _\lambda$, we use the reproducing property of ${\mathcal {H}}_\lambda$ to get

$$\begin{aligned} |( \Gamma _\lambda f )( \delta _x ) |&= |f(x) |= |\langle k(x,\cdot ), f \rangle _{{\mathcal {H}}_\lambda } \Vert \\&\le \Vert k(x,\cdot ) \Vert _{{\mathcal {H}}_\lambda } \Vert f \Vert _{{\mathcal {H}}_\lambda } \\&= \sqrt{ k(x,x ) } \Vert f \Vert _{{\mathcal {H}}_\lambda } =\sqrt{l(0_{G})} \Vert f \Vert _{{\mathcal {H}}_\lambda }. \end{aligned}$$

Setting f to the unit vector $f = k(x,\cdot ) / \Vert k(x,\cdot ) \Vert _{{\mathcal {H}}_\lambda }$ (which is well defined since $\delta _x$ is nonzero at any $x \in G$) then saturates the inequality, proving the claim and the theorem. $\square$

6 Reproducing Kernel Hilbert Algebras and Markov Semigroups

In this section, we study 1-parameter families of RKHAs associated with ergodic Markov semigroups on $L^2(G)$. We assume throughout that G is compact and the Haar measure $\mu$ is normalized to a probability measure.

Deferring additional details on the relevant theory to one of the many references in the literature, e.g., [40], we recall that a strongly continuous semigroup $\{ M_\tau \}_{\tau \ge 0 }$ of operators on $L^2(G)$ is a Markov semigroup if for every $\tau \ge 0$, $M_\tau$ is positivity-preserving (i.e., $M_\tau f \ge 0$, $\mu$-a.e., whenever $f \ge 0$, $\mu$-a.e.), $M_\tau 1_G = 1_G$, and $\int _G M_\tau f \, d\mu = \int _G f \, d\mu$ for all $f \in L^2(G)$. Moreover, $\{ M_\tau \}_{\tau \ge 0 }$ is said to be ergodic if $M_\tau f = f$ for all $\pi$ implies that f is constant $\mu -a.e$.

With these definitions, consider a family $\{ \lambda _\tau \in L^1({\hat{G}}) \}_{\tau > 0}$ of functions on the dual group satisfying the conditions

$$\begin{aligned}{} & {} \lambda _\tau (0_{{\hat{G}}}) = 1, \quad \forall \tau> 0, \nonumber \\{} & {} 0< \lambda _\tau ( \gamma ) < 1, \quad \lambda _{\tau }(\gamma ) \lambda _{\tau '}(\gamma ) = \lambda _{\tau +\tau '}(\gamma ), \quad \forall \tau ,\tau '>0, \quad \forall \gamma \in {\hat{G}} \setminus \{ 0_{{\hat{G}}} \}, \nonumber \\{} & {} \lambda _\tau (\gamma ) = \lambda _\tau (-\gamma ), \quad \forall \tau > 0, \quad \forall \gamma \in {\hat{G}}. \end{aligned}$$

(17)

We let $l_\tau = \hat{{\mathcal {F}}} \lambda _\tau$ and $k_\tau (x,y) = l_\tau (x-y)$ be the kernel functions defined as in (9), and ${\mathcal {H}}_\tau \equiv {\mathcal {H}}_{\lambda _\tau }$ the corresponding RKHSs. We also let ${\mathcal {K}}_\tau : L^2(G) \rightarrow L^2(G)$ be the corresponding kernel integral operators on $L^2(G)$ from Lemma 1. Note that $l_\tau$ and $k_\tau$ are real since $\lambda _\tau$ is real and symmetric (i.e., $\lambda _\tau ^\star = \lambda _\tau$).

By (17), for each $\gamma \in {\hat{G}}$, $\tau \mapsto \lambda _\tau (\gamma )$ is a continuous function, increasing monotonically to 1 as $\tau \rightarrow 0^+$. Consequently, the operators ${\mathcal {K}}_\tau$ converge pointwise to the identity on $L^2(G)$, i.e., $\lim _{\tau \rightarrow 0^+} K_\tau f = f$ for all $f \in L^2(G)$. Moreover, $\Vert {\mathcal {K}}_\tau \Vert = \lambda _\tau (0_{{\hat{G}}}) = 1$, so $\{ {\mathcal {K}}_\tau \}_{\tau > 0 } \cup {\mathcal {K}}_0$ with ${\mathcal {K}}_0 := {{\,\textrm{Id}\,}}$ is a strongly continuous contraction semigroup, consisting of self-adjoint compact operators.

By the Hille-Yosida theorem, there exists a positive, self-adjoint operator ${\mathcal {D}}$ such that, for all $\tau \ge 0$, ${\mathcal {K}}_\tau = e^{-\tau {\mathcal {D}}}$. This operator is diagonal in the character basis of $L^2(G)$, i.e., ${\mathcal {D}} \gamma = \eta (\gamma ) \gamma$, where $\eta (\gamma ) = - \tau ^{-1} \log \lambda _\tau$ for any $\tau > 0$. In particular, ${\mathcal {D}}$ has a simple eigenvalue $\eta (0_{{\hat{G}}}) = 0$ corresponding to the constant eigenfunction $1_G$. It then follows from results on Markov semigroups (e.g., [8, Chapter 14, Theorem 2]) that $-{\mathcal {D}}$ is the infinitesimal generator of an ergodic Markov semigroup, $\{ e^{-\tau {\mathcal {D}}} \}_{\tau \ge 0 }$.

By construction, the operators ${\mathcal {K}}_\tau$ are identical to $e^{-\tau {\mathcal {D}}}$, which implies that for $\tau > 0$, $k_\tau (x,\cdot )$ is a transition probability density with respect to Haar measure. That is, we have

$$\begin{aligned} k_\tau (x,\cdot ) \ge 0, \quad \int _G k_\tau ( x, \cdot ) \, d \mu = 1, \quad \forall \tau > 0, \quad \forall x \in G. \end{aligned}$$

The following theorem provides necessary and sufficient conditions for the spaces ${\mathcal {H}}_\tau$ to have RKHA structure.

Theorem 11

Suppose that the functions $\lambda _\tau \in L^1({\hat{G}})$ satisfy the Markov properties in (17). Then, the corresponding RKHSs ${\mathcal {H}}_\tau$ are RKHAs iff the $\lambda _\tau$ are subconvolutive for each $\tau > 0$, i.e.,

$$\begin{aligned} ( \lambda _\tau * \lambda _\tau )(\gamma ) \le C_\tau \lambda _\tau (\gamma ). \end{aligned}$$

Proof

See Sect. 6.1. $\square$

The subexponential functions from (3),

$$\begin{aligned} \lambda _\tau (\gamma ) = e^{-\tau |\gamma |^p}, \end{aligned}$$

are a concrete example satisfying the assumptions of Theorem 11 for the d-torus, $G = {\mathbb {T}}^d$, ${\hat{G}} = {\mathbb {Z}}^d$. In the case of the circle, ${\mathbb {T}}^1$, the Markov generator ${\mathcal {D}}$ is a fractional diffusion operator given by the p/2-th power of the Laplacian, ${\mathcal {D}} = \Delta ^{p/2}$.

6.1 Proof of Theorem 11

The “if” part of the theorem follows directly from Theorem 4. To prove the “only if” part, suppose that $\{ {\mathcal {H}}_\tau \}_{\tau > 0 }$ is a 1-parameter family of RKHAs associated with the functions $\{ \lambda _\tau \}_{\tau >0}$ satisfying (17). We show that the $\lambda _\tau$ are subconvolutive. To that end, letting $\xi _\tau = \lambda _\tau ^{1/2} = \lambda _{\tau /2}$, we make use of the following result.

Lemma 12

If ${\mathcal {H}}_\tau$ is a Banach algebra under pointwise multiplication of functions, then for every ${\hat{u}}, {\hat{v}} \in L^2({\hat{G}})$ there exists (a unique) ${\hat{w}} \in L^2({\hat{G}})$ such that

$$\begin{aligned} \xi _\tau {\hat{w}} = (\xi _\tau {\hat{u}}) * (\xi _\tau {\hat{v}}). \end{aligned}$$

Proof

By Lemma 3, there exist (unique) $f, g \in \mathcal H_\tau$ such that ${\mathcal {F}} f = \xi _\tau {\hat{u}}$ and ${\mathcal {F}} g = \xi _\tau {\hat{v}}$. Moreover, fg lies in ${\mathcal {H}}_\tau$, so again by Lemma 3 there exists ${\hat{w}} \in L^2({\hat{G}})$ such that ${\mathcal {F}}(fg) = \xi _\tau {\hat{w}}$. The claim follows from the fact that ${\mathcal {F}}(fg) = ({\mathcal {F}} f) * ({\mathcal {F}} g)$. $\square$

Since G is compact, for any $\epsilon > 0$ we have $\xi _\epsilon \in L^1({\hat{G}}) \subseteq L^2({\hat{G}})$, so setting ${\hat{u}} = {\hat{v}} = \xi _\epsilon$ in Lemma 12, it follows that there exist ${\hat{w}}_\epsilon \in L^2({\hat{G}})$ such that

$$\begin{aligned} \xi _\tau {\hat{w}}_{\epsilon } = \xi _{\tau +\epsilon } * \xi _{\tau +\epsilon }. \end{aligned}$$

In particular, we have

$$\begin{aligned} \hat{{\mathcal {F}}}(\xi _{\tau +\epsilon } * \xi _{\tau +\epsilon }) = (\hat{{\mathcal {F}}} \xi _{\tau +\epsilon }) (\hat{{\mathcal {F}}} \xi _{\tau +\epsilon }) = l_{(\tau +\epsilon )/2}^2. \end{aligned}$$

Note now that for each $\gamma \in {\hat{G}}$, $\epsilon \mapsto (\xi _{\tau +\epsilon } * \xi _{\tau +\epsilon })(\gamma )$ is a continuous function that increases monotonically as $\epsilon \rightarrow 0^+$ to $(\xi _{\tau } * \xi _{\tau })(\gamma )$. As a result, the family $\{ {\hat{w}}_\epsilon \}_{\epsilon > 0 }$ is bounded in $L^2({\hat{G}})$ norm and thus in $L^\infty ({\hat{G}})$ norm (since $\Vert \cdot \Vert _{L^\infty ({\hat{G}})} \le \Vert \cdot \Vert _{L^2({\hat{G}})}$ by compactness of G) by $\Vert \xi _\tau * \xi _\tau \Vert _{L^2({\hat{G}})}$.

By the above, for every $\tau > 0$, there exists a constant $\tilde{C}_\tau$ such that for every $\epsilon > 0$ and $\gamma \in {\hat{G}}$ we have

$$\begin{aligned} {\hat{w}}_\epsilon ( \gamma ) = \frac{(\xi _{\tau +\epsilon } * \xi _{\tau +\epsilon })(\gamma )}{\xi _\tau (\gamma )} \le {\tilde{C}}_\tau . \end{aligned}$$

Taking the limit $\epsilon \rightarrow 0^+$, we obtain

$$\begin{aligned} (\xi _\tau * \xi _\tau )( \gamma ) \le {\tilde{C}}_\tau \xi _\tau (\gamma ), \end{aligned}$$

and since $\xi _\tau = \lambda _{\tau /2}$ and $\tau$ was arbitrary, we conclude that

$$\begin{aligned} (\lambda _\tau * \lambda _\tau )(\gamma ) \le C_\tau \lambda _\tau (\gamma ), \end{aligned}$$

where $C_\tau = {\tilde{C}}_{2\tau }$. This verifies the subconvolutive property of the $\lambda _\tau$. $\square$

7 Fourier–Wermer Algebras

Motivated by applications to high-dimensional function approximation, we end the paper with a discussion on the inclusion relationships between RKHAs and Fourier–Wermer algebras on compact abelian groups associated with subconvolutive weights.

Given a positive weight $w: {\hat{G}} \rightarrow {\mathbb {R}}_+$, consider the space

$$\begin{aligned} {\mathcal {A}}_w = \left\{ f \in L^1(G): \sum _{\gamma \in {\hat{G}}} w(\gamma )|{\mathcal {F}} f(\gamma ) |< \infty \right\} , \end{aligned}$$

equipped with the norm $\Vert f \Vert _{{\mathcal {A}}_w} = \sum _{\gamma \in {\hat{G}}} w(\gamma ) |{\mathcal {F}} f(\gamma ) |$. As mentioned in Sect. 1, in the case $G = {\mathbb {T}}^d$, spaces in the class ${\mathcal {A}}_w$ are Fourier–Wermer algebras that have been employed in high-dimensional ($d \gg 1$) function approximation methods [31, 35]. In more detail, assuming (as we will henceforth do) that w is bounded away from 0, every space ${\mathcal {A}}_w$ embeds continuously into the Wiener algebra ${\mathcal {A}}(G)$, i.e., the Banach algebra of functions on G with absolutely convergent Fourier series,

$$\begin{aligned} {\mathcal {A}}(G) = \left\{ f \in L^1(G): \sum _{\gamma \in {\mathbb {Z}}^d} |{\mathcal {F}} f(\gamma ) |< \infty \right\} , \quad \Vert f \Vert _{{\mathcal {A}}({\mathbb {T}}^d)} := \Vert {\mathcal {F}} f \Vert _{L^1({\mathbb {Z}}^d)}. \end{aligned}$$

In particular, ${\mathcal {A}}_w$ may be identified with the image of $L^1_w({\hat{G}})$ under the inverse Fourier operator, ${\mathcal {A}}_w = \hat{{\mathcal {F}}} (L^1_w({\hat{G}}))$, and can thus be understood as a Banach space of continuous functions whose regularity depends on the weight function w. Here, and in what follows, we let $L^p_w({\hat{G}})$, $p \in (1,\infty )$, be the Banach space on ${\hat{G}}$ equipped with the norm $\Vert {\hat{f}} \Vert _{L^p_w({\hat{G}})} = ( \sum _{\gamma \in {\hat{G}}} ( w(\gamma ) |{\hat{f}}(\gamma )|)^p )^{1/p}$.

An important problem in numerical analysis is the approximation of functions f in an input space X such as ${\mathcal {A}}_w(\mathbb T^d)$ by elements $f_n$ in subspaces of finite dimension, such that for a given n the residual $f - f_n$ has low norm, uniformly over X, with respect to a target space Y into which X is continuously embedded (e.g., $L^\infty ({\mathbb {T}}^d)$, $L^2(\mathbb T^d)$, or ${\mathcal {A}}({\mathbb {T}}^d)$). Typically, the error of such approximations is measured using s-numbers for the embedding $\iota : X \hookrightarrow Y$ [38], which can be thought of as generalizations of the singular values of $\iota$ when X and Y are Hilbert spaces and $\iota$ is compact. In high-dimensional applications, of particular interest is the dependence of the s-numbers on $d\gg 1$. Intuitively, one seeks to take advantage of the regularity properties of X to alleviate the “curse of dimension” suffered by finite-rank approximation of arbitrary elements of the target space Y.

The recent paper [35] has shown that for weights $w = w_{s,r}$ in the class of dominating mixed smoothness (4) the optimal approximation error from $X = {\mathcal {A}}_w({\mathbb {T}}^d)$ scales as $n^{-s} (\log n)^{s(d-1)}$ when the output space is $Y = {\mathcal {A}}({\mathbb {T}}^d)$. However, the question of whether $\mathcal A_w({\mathbb {T}}^d)$ has Banach algebra structure is left open. An affirmative answer to that question would present additional opportunities to build approximation schemes that leverage algebra structure; see, e.g., [14] for an example in the setting of harmonic Hilbert spaces.

7.1 Algebra Structure of ${\mathcal {A}}_w$

Recall the subconvolutivity condition (5) that implies [33] that $L^p_w({\hat{G}})$ is a convolution algebra on the dual group. Requiring that this condition holds for $p=1$,

$$\begin{aligned} (w^{-1} * w^{-1})(\gamma ) \le C w^{-1}(\gamma ), \quad \forall \gamma ,\gamma ' \in {\hat{G}}, \end{aligned}$$

(18)

and using the inverse Fourier operator to pass to the primal group G, we can deduce that ${\mathcal {A}}_w = \hat{{\mathcal {F}}}(L^1_w({\hat{G}}))$ is a Banach algebra under pointwise multiplication. This algebra is a dense subalgebra of the Wiener algebra ${\mathcal {A}}(G)$. Moreover, the spectrum of $L^1_w({\hat{G}})$ contains a homeomorphic image of G [33, Theorem 4], which implies that $\sigma ({\mathcal {A}}_w)$ has the same property.

Note that the weights $w_{s,r}$ satisfy (18) for any $s \ge 2$ and $r > 0$. Indeed, in dimension $d=1$ the weight ${\tilde{w}}_{s,r}(\gamma ) := (1 + |\gamma |^r)^{s/r}$, $\gamma \in {\mathbb {Z}}$, is subadditive and ${\tilde{w}}_{s,r}^{-1}$ lies in $L^1({\mathbb {Z}})$, so $w_{s,r}^{-1}$ is subconvolutive (see [15]). Since, in any dimension $d\in {\mathbb {N}}$, $w_{s,r}^{-1}$ is built up as the product $w^{-1}_{s,r}(\gamma ) = \prod _{i=1}^d {\tilde{w}}_{s,r}^{-1}(\gamma _i)$ with $\gamma = (\gamma _1, \ldots , \gamma _d)$, it follows that $w_{s,r}$ satisfies (18), so ${\mathcal {A}}_{w_{r,s}}$ is a Banach algebra on ${\mathbb {T}}^d$.

For completeness, we note that the algebra property of $\mathcal A_w$ also holds for subadditive weights (without requiring that $w^{-1}$ lies in $L^1({\hat{G}})$). In this case, we can additionally deduce that the spectrum $\sigma ({\mathcal {A}}_w)$ is homeomorphic to G.

Recall Theorems 3 and 4 in [5], which collectively imply that if ${\hat{G}}$ is discrete (which is the case here since G is compact), and $w : {\hat{G}} \rightarrow {\mathbb {R}}$ satisfies $w \ge 1$ and (1), then $L^1({\hat{G}}) \cap L^\infty _w({\hat{G}})$ is a Banach convolution algebra with a homeomorphic spectrum to G. It has been pointed out to us by Feichtinger [17] that this result readily generalizes to ${\mathcal {L}}_w^p({\hat{G}}) := L^1({\hat{G}}) \cap L^p_w({\hat{G}})$ with $p \in [1,\infty ]$ and the norm $\Vert f \Vert _{{\mathcal {L}}^p_w({\hat{G}})} := \Vert f \Vert _{L^1({\hat{G}})} + \Vert f \Vert _{L^p_w({\hat{G}})}$. Indeed, for a discrete group ${\hat{G}}$, the subadditivity property (1) implies

$$\begin{aligned} w(\gamma ) \le C (w(\gamma ') + w(\gamma -\gamma ')), \quad \forall \gamma ,\gamma ' \in {\hat{G}}, \end{aligned}$$

leading to the pointwise estimate

$$\begin{aligned} |w(f*g)|(\gamma )&\le C\int _{{\hat{G}}}(w(\gamma ') + w(\gamma -\gamma ')) |f(\gamma ') g(\gamma -\gamma ') |\, d\gamma '\\&\le C\left( (|w f |* |g |)(\gamma ) + (|f|* |wg |)\right) (\gamma ), \end{aligned}$$

which holds for every $f,g \in {\mathcal {L}}^p_w({\hat{G}})$ and $\gamma \in {\hat{G}}$. From the above, we get

$$\begin{aligned} \nonumber \Vert f * g \Vert _{L^p_w({\hat{G}})}&\le C \left\Vert |wf |* |g |+ |f |* |w g |\right\Vert _{L^p({\hat{G}})} \\ \nonumber&\le C \left( \Vert wf \Vert _{L^p({\hat{G}})} \Vert g \Vert _{L^\infty ({\hat{G}})} + \Vert f \Vert _{L^\infty ({\hat{G}})} \Vert w g \Vert _{L^p({\hat{G}})} \right) \\&\le C \left( \Vert f \Vert _{L^p_w({\hat{G}})} \Vert g \Vert _{L^1({\hat{G}})} + \Vert f \Vert _{L^1({\hat{G}})} \Vert g \Vert _{L^p_w({\hat{G}})} \right) , \end{aligned}$$

(19)

and thus $\Vert f * g \Vert _{L^p_w({\hat{G}})} \le 2 C \Vert f \Vert _{{\mathcal {L}}^p_w({\hat{G}})}\Vert g \Vert _{{\mathcal {L}}^p_w({\hat{G}})}$. Moreover, since $\Vert f * g \Vert _{L^1({\hat{G}})} \le \Vert f \Vert _{L^1({\hat{G}})} \Vert g \Vert _{L^1({\hat{G}})}$, we have,

$$\begin{aligned} \Vert f * g \Vert _{{\mathcal {L}}^p_w({\hat{G}})} \le (2C + 1) \Vert f \Vert _{{\mathcal {L}}^p_w({\hat{G}})}\Vert g \Vert _{{\mathcal {L}}^p_w({\hat{G}})}, \end{aligned}$$

and we conclude that ${\mathcal {L}}^p_w({\hat{G}})$ is a Banach convolution algebra (cf. [5, Theorem 3]).

Next, it follows from (19) that for any $n\in {\mathbb {N}}$

$$\begin{aligned} \Vert f^{*2n} \Vert _{L^p_w({\hat{G}})} \le 2 C \Vert f^{*n} \Vert _{L^1({\hat{G}})} \Vert f^{*n} \Vert _{L^p_w({\hat{G}})} \le 2 C \Vert f^{*n} \Vert _{L^1({\hat{G}})} \Vert f^{*n} \Vert _{{\mathcal {L}}^p_w({\hat{G}})}. \end{aligned}$$

(20)

By [5, Theorem 2], the above is a sufficient condition for ${\mathcal {L}}^p_w({\hat{G}})$ and $L^1({\hat{G}})$ to have equal spectra,

$$\begin{aligned} \sigma ({\mathcal {L}}^p_w({\hat{G}})) = \sigma (L^1({\hat{G}})). \end{aligned}$$

(21)

Therefore, setting $p=1$ in (21) (in which case ${\mathcal {L}}^1_w({\hat{G}}) = L^1_w({\hat{G}})$ since $w\ge 1$), and passing to the primal group by Fourier transforms, we obtain

$$\begin{aligned} \sigma ({\mathcal {A}}_w) = \sigma (\hat{{\mathcal {F}}}L^1_w({\hat{G}})) = \sigma (\hat{{\mathcal {F}}}L^1({\hat{G}})) = \sigma (C(G)). \end{aligned}$$

(22)

Thus, we can conclude that the Fourier–Wermer algebras $\mathcal A_w$ associated with subadditive weights have the same spectra as C(G), analogously to the result in Theorem 6 for RKHAS ${\mathcal {H}}_{\lambda }$ associated with subconvolutive weights.

7.2 Embedding Relationships with RKHAs

We now consider the case where w and $\lambda = w^{-2}$ satisfy (18) and the assumptions of Theorem 4 , respectively, so that ${\mathcal {A}}_w$ and ${\mathcal {H}}_\lambda$ are both Banach algebras. This will be the case, for instance, for $w = w_{s,r}$ with $s \ge 2$, as well as the weights $w = \lambda _\tau ^{-1/2}$ obtained from Markov semigroups as in Sect. 6 (which include the subexponential weights $w = \lambda ^{-1/2}$ from (3) as a special case). Inclusion relationships between ${\mathcal {A}}_w$ and ${\mathcal {H}}_\lambda$ are particularly interesting when both of these spaces are algebras, as they induce algebra representations via the corresponding multiplication operators.

First, it is straightforward to deduce that ${\mathcal {A}}_w$ embeds continuously into ${\mathcal {H}}_\lambda$, and the operator norm of the embedding is equal to 1 [35]. Indeed, defining the linear operators $A: {\mathcal {A}}_w \rightarrow L^1({\hat{G}})$ and ${\tilde{B}}: {\mathcal {H}}_\lambda \rightarrow L^2({\hat{G}})$ such that

$$\begin{aligned} A f(\gamma ) = w(\gamma ) {\mathcal {F}} f(\gamma ), \quad {\tilde{B}} {\hat{f}}(x) = \sum _{\gamma \in {\hat{G}}} w^{-1}(\gamma ) \gamma (x) {\hat{f}}(\gamma ), \end{aligned}$$

it follows by direct calculation that the following diagram commutes,

where ${{\,\textrm{Id}\,}}$ is the identity map on functions and $\iota$ is the inclusion map. One also verifies that A and ${\tilde{B}}$ have unit operator norm (in fact, ${\tilde{B}}$ is unitary), and since $\Vert 1_G \Vert _{{\mathcal {A}}_w} = \Vert 1_G \Vert _{{\mathcal {H}}_\lambda } = 1$, the embedding ${\mathcal {A}}_w \hookrightarrow {\mathcal {H}}_\lambda$ has unit operator norm. It then follows that the map $\pi : \mathcal A_w \rightarrow B({\mathcal {H}}_\lambda )$, where $\pi f$ is the multiplication operator by f, provides a strongly continuous, faithful representation of ${\mathcal {A}}_w$ into $B({\mathcal {H}}_\lambda )$.

Next, turning to embeddings of the opposite direction, in general we cannot expect a continuous embedding of ${\mathcal {H}}_\lambda$ into ${\mathcal {A}}_w$ since, unless G is finite, $L^1({\hat{G}})$ is a strict subspace of $L^2({\hat{G}})$. Nevertheless, under appropriate assumptions, giving up a small amount of regularity $\epsilon$ is sufficient to obtain a continuous embedding of $\mathcal H_{\lambda ^{1+\epsilon }}$ into ${\mathcal {A}}_w$. Here, we assume that the weight w is chosen such that $\lambda ^\epsilon = w^{-2\epsilon }$ satisfies the assumptions of Theorem 4 for any $\epsilon >0$. This will hold, for instance, for the “semigroup” weights $w = \lambda _\tau ^{-1/2}$ from Sect. 6 but not the $w_{s,r}$ family (4).

For any $\epsilon >0$ let us define the linear map $D_\epsilon : L^2({\hat{G}}) \rightarrow L^1({\hat{G}})$ such that

$$\begin{aligned} D_\epsilon {\hat{f}} = w^{-\epsilon } {\hat{f}}; \end{aligned}$$

this map has operator norm $\Vert D_\epsilon \Vert \le \Vert w^{-\epsilon } \Vert _{L^1({\hat{G}})}$. Defining also ${\tilde{A}} : L^1({\hat{G}}) \rightarrow {\mathcal {A}}_w$ and $B_\epsilon : \mathcal H_{\lambda ^{1+\epsilon }} \rightarrow L^2({\hat{G}})$ as

$$\begin{aligned} {\tilde{A}} {\hat{f}}(x) = \sum _{\gamma \in {\hat{G}}} w^{-1}(\gamma ) \gamma (x) \xi (\gamma ), \quad B_\epsilon f(\gamma ) = w^{1+\epsilon }(\gamma ) {\mathcal {F}} f(\gamma ), \end{aligned}$$

where $\Vert {\tilde{A}} \Vert = \Vert B_\epsilon \Vert = 1$ and $B_\epsilon$ is unitary, leads to the following commutative diagram:

We thus conclude that the embedding $\mathcal H_{\lambda ^{1+\epsilon }} \hookrightarrow {\mathcal {A}}_w$ is continuous for any $\epsilon > 0$.

Appendix A. Proof of Lemma 2

1.1 Claim (i)

To show that k is uniformly continuous, note first that the kernel shape function l lies in $C_0(G)$, and is thus uniformly continuous. As a result, for every $\epsilon > 0$ and $(x,x') \in G \times G$ there exists a neighborhood U of the identity element of G such that

$$\begin{aligned} |l( x - x' ) - l(z) |< \epsilon , \quad \forall z \in ( x - x' ) + U. \end{aligned}$$

Therefore, defining the open neighborhood $V = \{ (y,y') \in G \times G : y - y' \in U \}$ of the identity of $G \times G$, we get

$$\begin{aligned} |l( x - x' ) - l( y - y' ) |= |k(x,x') - k(y,y') |< \epsilon , \quad \forall (y,y') \in (x,x') + V, \end{aligned}$$

which proves that k is uniformly continuous.

To show that $\mathcal {H}_\lambda$ is a subspace of $C_0(G)$, note that every $f \in \mathcal {H}_\lambda$ is the $\mathcal {H}_\lambda$-norm limit of finite linear combinations of kernel sections of the form $f_n = \sum _{j=0}^{n-1} c_j k( x_j, \cdot )$, where $k( x_j, \cdot )$ lies in $C_0(G)$. Moreover, proceeding similarly to [20, Lemma 2.1], we have

$$\begin{aligned} |f(x)| = \left| \langle k(x, \cdot ), f \rangle _{\mathcal {H}_\lambda } \right| \le \left\| k(x, \cdot ) \right\| _{\mathcal {H}_\lambda } \left\| f \right\| _{\mathcal {H}_\lambda } = \sqrt{k(x,x)} \left\| f \right\| _{\mathcal {H}_\lambda } \le \Vert l \Vert _{C_0(G)}^{1/2} \Vert f \Vert _{\mathcal {H}_\lambda }, \end{aligned}$$

and thus

$$\begin{aligned} \left\| f \right\| _{\sup } \le \left\| l \right\| _{C_0(G)}^{1/2} \left\| f \right\| _{\mathcal {H}_\lambda }. \end{aligned}$$

The above implies that the Cauchy sequence $f_n \in \mathcal {H}_\lambda \cap C_0(G)$ converging to $f \in \mathcal {H}_\lambda$ is also Cauchy with respect to $C_0(G)$ norm, so f lies in $C_0(G)$. This proves Claim (i).

1.2 Claim (ii)

Since $\lambda \in {L}^1(G)$ and $l = \hat{{\mathcal {F}}} \lambda \in L^1(G)$, we have $l\in L^1 (G)\cap C_0 (G)$, which implies that for every $x \in G$, $k( x, \cdot ) = S^x l$ lies in $L^1(G) \cap C_0(G)$ Therefore, for every $f\in L^\infty (G)$, $K f(x) = \int _G k(x,\cdot ) f\, d\mu$ exists for every $x\in G$, and we have

$$\begin{aligned} |K f(x) |= \left|\int _G k(x,\cdot ) f \, d\mu \right|\le \Vert S^x l \Vert _{L^1(G)} \Vert f \Vert _{L^\infty (G)} = \Vert l \Vert _{L^1(G)} \Vert f \Vert _{L^\infty (G)}. \end{aligned}$$

This shows that K is well-defined as a bounded linear map from $L^\infty (G)$ to the space of bounded functions on G, as claimed.

Next, we have $\Vert l \Vert ^2_{L^2(G)} \le \Vert l \Vert _{C_0(G)} \Vert l \Vert _{L^1(G)}$, which implies that l lies in $L^2(G)$. Thus, for every $f \in L^2(G)$ we can express Kf(x) as the inner product

$$\begin{aligned} K f( x ) = \langle k(x,\cdot ), f \rangle _{L^2(G)} = \langle S^x l, f \rangle _{L^2(G)}. \end{aligned}$$

Therefore, for any $x, y \in G$, we obtain

$$\begin{aligned} |K f( x) - K f( y ) |^2&\le \Vert S^x l - S^y l \Vert _{C_0(G)} \Vert S^x l - S^y l \Vert _{L^1(G)} \Vert f \Vert _{L^2(G)}^2 \\&\le 2 \Vert S^x l - S^y l \Vert _{C_0(G)} \Vert l \Vert _{L^1(G)} \Vert f \Vert _{L^2(G)}^2, \end{aligned}$$

The uniform continuity of Kf then follows by the strong continuity of $S^x$, using a neighborhood of the identity of $G \times G$ analogous to V in the proof of Claim (i).

Finally, the uniform continuity of Kf for $f \in L^1(G)$ follows from a similar argument using the bound

$$\begin{aligned} |K f( x) - K f( y ) |\le \Vert k( x, \cdot ) - k(y, \cdot ) \Vert _{C_0(G)} \Vert f \Vert _{L^1(G)}. \end{aligned}$$

This completes the proof of Claim (ii).

1.3 Claim (iii)

The claim is a direct consequence of (7) and the definition of k in (9), viz.

$$\begin{aligned} K \gamma ( x )&= \int _G k( x, y ) \gamma ( y ) \, d\mu (y)\\&= \int _G S^x l(-y ) \gamma ( y ) \, d\mu (y) = \int _G S^x l( y ) \gamma ( -y ) \, d\mu (y) \\&= \mathcal {F}(S^x l)(\gamma ) = \gamma (x) ( \mathcal {F}l )(\gamma ) = \gamma (x) \lambda (\gamma ). \end{aligned}$$