Convolution, Fourier analysis, and distributions generated by Riesz bases

In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We develop the associated biorthogonal Fourier analysis and the theory of distributions, discuss properties of convolutions and give a number of examples.


Introduction
In this work we introduce a notion of a convolution generated by systems of elements of a Hilbert space H forming a Riesz basis.
Such collections often arise as systems of eigenfunctions of densely defined nonself-asjoint operators acting on H, and a suitable notion of convolution also leads to the development of the associated Fourier analysis. In the case of the eigenfunctions having no zeros the corresponding global theory of pseudo-differential operators has been recently developed in [25]. The assumption on eigenfunctions having no zeros has been subsequently removed in [26], and some applications of such analysis to the wave equation for the Landau Hamiltonian were carried out in [27,29], as well as for general operators with discrete spectrum in [28], and for nonlinear PDE in [30]. The analysis in these papers relied on the spectral properties of a fixed operator acting in H = L 2 (M) for a smooth manifold M with or without boundary.
In this note we aim at discussing an abstract point of view on convolutions when one is given only a Riesz basis in a Hilbert space, without making additional assumptions on an operator for which it may be a basis of eigenfunctions. Such an abstract point of view has a number of advantages, for example, the questions of whether the basis elements (for example in H = L 2 (M)) have zeros at some points, become irrelevant.
More specifically, let H be a separable Hilbert space, and denote by U := {u ξ | u ξ ∈ H} ξ ∈N and V := {v ξ | v ξ ∈ H} ξ ∈N collections of elements of H parametrised by a discrete set N. We assume that the system U is a Riesz basis of the space H and the system V is biorthogonal to U in H, i.e. we have the property that where δ ξη is the Kronecker delta, equal to 1 for ξ = η, and to 0 otherwise. Then from the classical Bari's work [5] (see also Gelfand [14]) it follows that the system V is also basis in H. The Riesz basis is characterised by the property that it is the image of an orthonormal basis in H under a linear invertible transformation. However, since our aim is to subsequently extend the present constructions in the future beyond the Riesz basis setting we will try not to make explicit use of this property. The results of this paper have been announced in [22]. The setting of Riesz bases has numerous applications to different problems, see e.g. [10,11], and in different settings and modifications, see e.g. [4,15,16], to mention only very few. Decomposition systems of different types and the subsequent function spaces is also an active area of research, see e.g. [7,8,17,23].
In this paper we define Uand V-convolutions in the following form: and for appropriate elements f, g, h, j ∈ H. These convolutions are clearly commutative and associative, and have a number of properties expected from convolutions, most importantly, they are mapped to the product by the naturally defined Fourier transforms associated to U and V. Without going too much into detail, let us briefly summarise the results of this paper: • • We discuss more general families of spaces l p U , 1 ≤ p ≤ ∞, on the Fourier transform side giving rise to further Fourier analysis in H. Namely, these spaces satisfy analogues of the usual duality and interpolation relations, as well as the Hausdorff-Young inequalities with the corresponding family of subspaces of H. Let us conclude the introduction by giving a concrete example of such convolution also relating it to the spectral analysis. Let us consider the operator L : H → H on the interval (0, 1) given by and let us equip this operator with boundary condition hy(0) = y(1) for some h > 0. The operator L is not self-adjoint on H = L 2 (0, 1) for h = 1. The spectral properties of L have been thoroughly investigated by e.g. Titchmarsh [31] and Cartwright [9]. In particular, it is known that the collections are the systems of eigenfunctions of L and L * , respectively, and form Riesz bases in H = L 2 (0, 1). In this case the abstract definition of convolution above can be shown (see Proposition 5.1) to yield the concrete expression which coincides with the usual convolution for h = 1, in which case also U = V is an orthonormal basis in H. Of course, in this example, the main interest for us is the case h = 1 corresponding to biorthogonal bases U and V in (1.3) and (1.4), respectively. In this paper, to avoid any confusion, we will be using the notation N 0 = N ∪ {0}.

Biorthogonal convolutions
In this section we describe the functional analytic setting for investigating convolutions (1.1) and (1.2). Let us take biorthogonal systems in a separable Hilbert space H, where N is a discrete set. We assume that U (and hence also V) is a Riesz basis in H, i.e. any element of H has a unique decomposition with respect of the elements of U. We note that the basis collections are uniformly bounded in H. Before we proceed with describing a version of the biorthogonal Fourier analysis, let us show that expressions in (1.1) and (1.2) are usually well-defined. Proposition 2.1 Let f U g and h V j be defined by (1.1) and (1.2), respectively, that is, Then there exists a constant M > 0 such that we have 3) for all f, g, h, j ∈ H.
The statement follows from the Cauchy-Schwarz inequality and the following fact: since systems of u ξ and of v ξ are Riesz bases in H, from [5,Theorem 9] we have that there are constants a, A, b, B > 0 such that for arbitrary g ∈ H we obtain (2.4) This amounts to simply stating that the Riesz basis collections form collections of frames in H. From the Riesz basis property it also follows that the families U and V are uniformly bounded in H, that is, Let us introduce Uand V-Fourier transforms by formulas and respectively, for all f, g ∈ H and for each ξ ∈ N. Here g * stands for the V-Fourier transform of the function g. Indeed, in general g * = g. Their inverses are given by and The Fourier transforms defined in (2.5) and (2.6) are the analysis operators, and, the inverse transforms (2.7) and (2.8) are the corresponding synthesis operators, see e.g. [23]. For more information, see e.g. [7,8,17] and references therein.
There is a straightforward relation between Uand V-convolutions, and the Fourier transforms: Therefore, the convolutions are commutative and associative.
Similarly, if K ( f, g) satisfies the property Proof Direct calculations yield Commutativity follows from the bijectivity of the U-Fourier transform, also implying the associativity. This can be also seen from the definition: Next, it is enough to prove that K is the U-convolution under the assumption (2.9). The similar property for V-convolutions under assumption (2.10) follows by simply replacing U by V in the part concerning U-convolutions.
Since for arbitrary f, g ∈ H and for K ( f, g) ∈ H the property (2.9) is valid then we can regain K ( f, g) from the inverse U-Fourier formula The last expression defines the U-convolution.

Biorthogonal Fourier analysis
From (2.4) we can conclude that the Uand V-Fourier coefficients of the elements of H belong to the space of square-summable sequences l 2 (N). However, we note that the Plancherel identity is also valid for suitably defined l 2 -spaces of Fourier coefficients, see [25, Proposition 6.1]. We explain it now in the present setting.
Indeed, the frame property in (2.4) can be improved to the exact Plancherel formula with a suitable choice of norms.

Plancherel formula
Let us denote by for arbitrary a, b ∈ l 2 U . The reason for this choice of the definition is the following formal calculation: which implies the Hilbert space properties of the space of sequences l 2 U . The norm of l 2 U is then given by the formula We note that individual terms in this sum may be complex-valued but the whole sum is real and non-negative.
Analogously, we introduce the Hilbert space for arbitrary a, b ∈ l 2 V . The norm of l 2 V is given by the formula for all a ∈ l 2 V . The spaces of sequences l 2 U and l 2 V are thus generated by biorthogonal systems {u ξ } ξ ∈N and {v ξ } ξ ∈N .
Since Riesz bases are equivalent to an orthonormal basis by an invertible linear transformation, we have the equality between the spaces l 2 U = l 2 V = l 2 (N) as sets; of course the special choice of their norms is the important ingredient in their definition.
Indeed, the reason for their definition in the above forms becomes clear again in view of the following Plancherel identity: respectively. In particular, we have The Parseval identity takes the form Furthermore, for any f ∈ H, we have f ∈ l 2 U , f * ∈ l 2 V , and Proof By the definition we get and Hence it follows that and To show Parseval's identity (3.4), using these properties and the biorthogonality of u ξ 's to v η 's, we can write proving (3.4). Taking f = g, we get proving the first equality in (3.5). Then, by checking that the proofs of (3.5) and of Theorem 3.1 are complete.

Hausdorff-Young inequality
Now, we introduce a set of Banach spaces {H p } 1≤ p≤∞ with the norms · p such that for all 1 ≤ p ≤ ∞, where 1 p + 1 q = 1. We assume that H 2 = H, and that H p are real interpolation properties in the following sense: and We also assume that U ⊂ H p and V ⊂ H p for all p ∈ [1, ∞].
is the Hilbert space of Hilbert-Schmidt operators on a Hilbert space K, then we can take H p = S 2 (K) ∩ S p (K), where S p (K) stands for the space of p-Schatten operators on K.
Below we introduce the p-Lebesgue versions of the spaces of Fourier coefficients. Here classical l p spaces on N are extended in a way so that we associate them to the given biorthogonal systems. and and, for p = ∞, Here, without loss of generality, we can assume that u ξ = 0 and v ξ = 0 for all ξ ∈ N, so that the above spaces are well-defined.
Analogously, we introduce spaces l p V = l p (V) as the spaces of all b : N → C such that From this we obtain the following property: where 0 < θ < 1 and p = 2 2−θ .
Using Theorem 3.3 and Corollary 3.4 we get the following Hausdorff-Young inequality.

Theorem 3.5 (Hausdorff-Young inequality)
Assume that 1 ≤ p ≤ 2 and 1 p + 1 p = 1. Then there exists a constant C p ≥ 1 such that for all f ∈ H p and a ∈ l p (U). Similarly, for all b ∈ l p (V) we obtain The last estimate gives the first inequality in (3.9) for p = 1. For the second inequality, using in view of the definition of l 1 (U), which gives (3.9) in the case p = 1. The proof is complete.
Let us establish the duality between spaces l p (U) and l q (V): Theorem 3.6 (Duality of l p (U) and l q (V)) Let 1 ≤ p < ∞ and 1 p + 1 q = 1. Then Proof The proof is standard. Meanwhile, we provide several details for clarity. The duality is given by for σ 1 ∈ l p (U) and σ 2 ∈ l q (V). Let 1 < p ≤ 2. Then, if σ 1 ∈ l p (U) and σ 2 ∈ l q (V), we obtain where that 2 ≤ q < ∞ and that 2 p − 1 = 1 − 2 q were used (last line). Now, let 2 < p < ∞. If σ 1 ∈ l p (U) and σ 2 ∈ l q (V), we get Put p = 1. Then we have The adjoint space cases could be proven in a similar way.

Rigged Hilbert spaces
In this section we will investigate a rigged structure of the Hilbert space H. Especially, we will construct a (Gelfand) triple ( , H, ) with the inclusion property where a role of will be played by the so-called 'spaces of test functions' C ∞ U , and C ∞ V, generated by the systems U and V, respectively, and by some sequence of complex numbers. For this aim, let us fix some sequence := {λ ξ } ξ ∈N of complex numbers such that the series for those f ∈ H for which the series converges in H. Then L is densely defined since Lu ξ = λ ξ u ξ for all ξ ∈ N, and U is a basis in H. We denote by Dom(L) the domain of the operator L, so that we have Span (U) ⊂ Dom(L) ⊂ H. We call L to be the operator associated to the pair (U, ). Operators defined as in (4.2) have been also studied in [3].
We note that this construction goes in the opposite direction to the investigations devoted to the development of the global theory of pseudo-differential operators associated to a fixed operator, as in the papers [12,13,[25][26][27], where one is given an operator L acting in H with the system of eigenfunctions U and eigenvalues . In this case we could 'control' only one parameter, i.e. the operator L. In the present (more abstract) point of view we have two parameters to control: the system U and the sequence of numbers .
In a similar way to Definition 4.1, we define the operator L * : H → H by for those g ∈ H for which it makes sense. Then L * is densely defined since L * v ξ = λ ξ v ξ and V is a basis in H, and Span (V) ⊂ Dom(L * ) ⊂ H. One readily checks that we have on their domains. We can now define the following notions: (i) the spaces of (U, )and (V, )-test functions are defined by where C k V, := {ψ ∈ H : |(ψ, u ξ )| ≤C(1 + |λ ξ |) −k for some constant C for all ξ ∈ N}.
The topology of these spaces is defined by a natural choice of seminorms. We can define spaces of (U, )and (V, )-distributions by D U , := (C ∞ V, ) and D V, := (C ∞ U , ) , as spaces of linear continuous functionals on C ∞ V, and C ∞ U , , respectively. We follow the conventions of rigged Hilbert spaces to denote this duality by extending the inner product on H for u, φ ∈ H, and similarly for the pair D V, := (C ∞ U , ) . (ii) the Uand V-Fourier transforms respectively, for arbitrary φ ∈ C ∞ U , , ψ ∈ C ∞ V, and for all ξ ∈ N, and hence by duality, these extend to D U , and D V, , respectively. Here we have where the space S(N) is defined in (4.7). Indeed, for w ∈ H we can calculate justifying definition (4.4). Similarly, we define The Fourier transforms of elements of D U , , D V, can be characterised by the property that, for example, for w ∈ D U , , there is N > 0 and C > 0 such that (iii) Uand V-convolutions can be extended by the same formula: for example, for all f ∈ D U , and g ∈ C ∞ U , . It is well-defined since the series converges in view of properties from (i) above and assumption (4.1). By commutativity that spaces for f and g can be swapped. Similarly, U , can be also described in terms of the operator L in (4.2). Namely, we have Summarising the above definitions and discussion, we record the basic properties of the Fourier transforms as follows: is given by (4.8) so that the Fourier inversion formula becomes

9)
Similarly, F V : C ∞ V, → S(N) is a bijective homeomorphism and its inverse is given by (4.10) so that the conjugate Fourier inversion formula becomes The proof is straightforward. Let us formulate the properties of the Uand V-convolutions: The convolutions are commutative and associative. If g ∈ C ∞ U , then for all f ∈ D U , we have

.12)
Proof Since the first part of the statement is proving in the same way as analogous one from Proposition 2.2, we will show only the property (4.12) which follows if we observe that for all k ∈ N 0 the series

Proposition 4.4 If L : H → H is associated to a pair (U, ) then we have
Proof The proof is valid since the equalities and are true for all ξ ∈ N.
As a small application, let us write the resolvent of the operator L in terms of the convolution. L : H → H be an operator associated to a pair (U, ). Then the resolvent of the operator L is given by the formula

Theorem 4.5 Let
where I is an identity operator in H and Proof Begin by calculating the following series where we used the continuity of the resolvent. Now the theorem is proved.

Examples
We give an example considered in [25] that can be also considered as an extension setting in an appropriate sense of the toroidal calculus studied in [24].
This leads to the setting of the classical Fourier analysis on the circle which can be viewed as the interval (0, 1) with periodic boundary conditions. The corresponding pseudo-differential calculus was consistently developed in [24] building on previous observations in the works by Agranovich [1,2] and others.
For h = 1, the operator O h is not self-adjoint. The spectral properties of O (1) h are well-known (see Titchmarsh [31] and Cartwright [9]), the spectrum of O (1) h is discrete and is given by λ j = −i ln h +2 jπ, j ∈ Z. The corresponding bi-orthogonal families of eigenfunctions of O (1) h and its adjoint are given by respectively. They form Riesz bases, and O (1) h is the operator associated to the pair U and = {λ j = −i ln h + 2 jπ } j∈Z .
Since N denoted an arbitrary discrete set before, all the previous constructions work with Z instead of N.
Formally, we can write Here integrals (5.1) and the last series should be understood in the sense of distributions. In the case h = 1, it can be shown that F(x, y, z) = δ(x − y − z), see [24]. For any h > 0, it can be shown that the U-convolution coincides with Kanguzhin's convolution that was studied in [19,21]:  h . The corresponding U-convolution can be written in the integral form: In particular, when h = 1, we obtain is the usual convolution on the circle.
Proof of Proposition 5.1 Let us denote Then we can calculate Consequently, by Theorem 2.2, we obtain that K ( f, g) = f * U g.

Further discussion
We note that in the case when we are given an We finally show that an L-convolution does not have to be a U-convolution for any choice of the set .
For this, let us consider an L-convolution associated to the so-called Ionkin operator considered in [18]. The Ionkin operator Y : H → H is the operator in H := L 2 (0, 1) generated by the differential expression for more details, see [18]. We consider the Y-convolution (Ionkin-Kanguzhin's convolution) given by the formula This is a Y-convolution in the sense of Definition 6.1, namely, it satisfies see [20]. For the collection it can be readily checked that the corresponding U-Fourier transform satisfies Therefore, by Theorem 2.2, the Y-convolution (Ionkin-Kanguzhin convolution) does not coincide with the U-convolution for any choice of numbers .