Uniqueness of minimal Fourier-type extensions in $L_1$-spaces

Abstract

We give a characterization of uniqueness of finite rank Fourier-type minimal extensions in $L_1$-norm. This generalizes the main result obtained by Lewicki (Proceedings of the Fifth International Conference on Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, pp. 337–345, 1998) to the case of $n$-circular sets in $\mathbb{C }^n$.

1 Introduction

We start with some notation which will be used in this paper. By $S_X(x,r)$ ($S_X$ if $x=0$ and $r=1$) we denote a sphere in a Banach space $X$ with the center $x$ and the radius $r$, by $\text{ ext}(S_X)$ the set of extreme points of $S_X$ and the symbol $X^*$ stands for a dual space of $X$. An element $x\in X$ is called a norming point for $f\in X^*$ if $x\in S_X$ and $f(x)=\Vert x\Vert$. For $z\in \mathbb{C }$, $\text{ sgn}{z}=\overline{z}/|z|$ for $z\ne 0$ and $0$ for $z=0$.

Let $Y$ be a linear subspace of $X$ and let $\mathcal L (X,Y)$ ($\mathcal L (X)$ if $X=Y$) be the space of all linear, continuous operators from $X$ into $Y$. Given $A\in \mathcal L (Y)$, an operator $P\in \mathcal L (X,Y)$ is called an extension of $A$ (or a projection in the case of $A=\text{ Id}_Y$) if $P|_Y=A$. The set of all extensions of $A$ will be denoted by $\mathcal P _A(X,Y)$. An extension $P_0\in \mathcal P _A(X,Y)$ is minimal if

$$\begin{aligned} \Vert P_0\Vert =\lambda _{A}(X,Y)=\inf \{\Vert P\Vert :P\in \mathcal P _{A}(X,Y)\}, \end{aligned}$$

(1)

We write briefly $\mathcal P (X,Y)$ and $\lambda (Y,X)$ instead of $\mathcal P _{\text{ Id}_Y}(X,Y)$ and $\lambda _{\text{ Id}_Y}(Y,X)$ respectively. Basic results on minimal projections and extensions can be found in [1, 7, 9, 12, 13, 15–18, 20, 22, 24]. Set

$$\begin{aligned} \mathcal L _Y(X,Y) = \{L\in \mathcal L (X,Y): L|_{Y}=0\}. \end{aligned}$$

(2)

Let ${\pi }_n$ denote the space of all trigonometric polynomials of degree $\leqslant n$ and let $\mathcal C _0(2\pi )$ be the space of all continuous, real valued, $2\pi$-periodic functions. The classical Fourier projection from $\mathcal C _0(2\pi )$ onto ${\pi }_n$ is defined by a formula

$$\begin{aligned} (F_nf)(t)=(f*D_n)(t)=(1/2\pi )\int _{0}^{2\pi }f(s)D_n(t-s)\,ds, \end{aligned}$$

(3)

where $D_n(t)=\sum _{j=-n}^{n}e^{ijt}$. It is well-known that $F_n$ is the unique operator of minimal norm in the space $\mathcal P (\mathcal C _0(2\pi ),{\pi }_n)$ [10, 23]. Moreover, $\Vert F_n\Vert =\lambda (\pi _n,L_p[0,2\pi ])$ for $1\leqslant p\leqslant \infty$, which follows from Rudin Theorem [8], however, in general, it is an open question if $F_n$ is the unique minimal projection from $L_p[0,2\pi ]$ onto $\pi _n$ for $p\ne 1,2,+\infty$. Partial results concerning subject can be found in [26] and [27].

In this paper we study the problem of the unique minimality of the Fourier-type extensions in the space $L_1$. More precisely, let $M$ be a set, $\Sigma$-$\sigma$-algebra of subsets of $M$, $\nu$ a positive measure on $\Sigma$ such that $(M,\Sigma , \nu )$ is a complete measure space. By $L_1(M,\Sigma , \nu )$ denote a space of complex-valued, $\nu$-measurable functions on $M$ satisfying a condition

$$\begin{aligned} \Vert f\Vert _1=\int _{M}|f(z)|d\nu (z)<+\infty . \end{aligned}$$

To the end of this paper we assume that $(L_{1}(M,\Sigma , \nu ))^*=L_{\infty }(M,\Sigma , \nu )$, which is satisfied, for example, if $\nu$ is $\sigma$-finite.

Definition 1

It is said that $w\in V\subset L_{\infty }(M,\Sigma , \nu )$, $w\ne 0$ is determined by its roots in the space $V$ if and only if for any $g\in V$ the condition $g/w\in L_{\infty }(M,\Sigma , \nu )$ implies that $g=cw$ for some $c\in \mathbb{C }$.

Definition 2

A subspace $V\subset L_1(M,\Sigma , \nu )$ is called smooth if and only if each member of $\ V{\setminus }\{0\}$ is almost everywhere different from $0$.

Take $V$ a smooth, finite-dimensional subspace of $L_1(M,\Sigma , \nu )$ with a basis $\{v_1,\ldots ,v_k\}$ and fix an operator $A\in \mathcal L (V)$. Observe that any $P\in \mathcal P _A(L_1(M,\Sigma , \nu ),V)$ has a form

$$\begin{aligned} Pf=\sum _{j=1}^{k}\widehat{u_j}(f)v_j, \ \ \widehat{u_j}(v_i)=a_{j,i}, i,j=1,\ldots ,k, \text{ where} \end{aligned}$$

(4)

$[a_{i,j}]_{i,j=1}^{k}$ is a matrix of the operator $A$ in the basis $\{v_1,\ldots ,v_k\}$ and $\widehat{u}\in (L_1(M,\Sigma , \nu ))^*$ denotes a functional associated with $u\in L_{\infty }(M,\Sigma , \nu )$ by

$$\begin{aligned} \widehat{u}(f)=\int _{M}f(z)u(z)\,d\nu (z). \end{aligned}$$

(5)

Let $P\in \mathcal P _A(L_1(M,\Sigma , \nu ),V)$ be given by (4) and $z, w\in M$. Define

$$\begin{aligned} x^{P}_{z}(w)&= \sum _{j=1}^{k}u_{j}(z)v_j(w),\nonumber \\ V_j(z)&= \int _{M}v_j(w)\text{ sgn}(x^{P}_{z}(w))\,d\nu (w),\\ P_z&= \int _{M}|x^{P}_{z}(w)|\,d\nu (w).\nonumber \end{aligned}$$

(6)

A map $z\rightarrow P_z$ is called the Lebesgue function of the operator $P$. It is well known that

$$\begin{aligned} \Vert P\Vert =\text{ ess}\sup _{z\in M} P_z \text{(see} \text{[14,} \text{ Lem.} \text{2])}. \end{aligned}$$

(7)

Lemma 3

Assume that $P_0\in \mathcal P _{A}(L_1(M,\Sigma , \nu ),V)$ and the Lebesgue function of the operator $P_0$ is constant on $M$ $(\nu ~a.a.)$ Let $P_1, P_2\in \mathcal P _{A}(L_1(M,\Sigma , \nu ),V)$, $\Vert P_1\Vert =\Vert P_2\Vert =\Vert P_0\Vert$ and $P_0=(P_1+P_2)/2$. Then the Lebesgue functions of the operators $P_j: j=1,2$ are constant on $M$ and for $\nu$ a.a. $z\in M$,

$$\begin{aligned} \text{ sgn}(x^{P_1}_{z})=\text{ sgn}(x^{P_2}_{z})=\text{ sgn}(x^{P_0}_{z}). \end{aligned}$$

Proof

Let

$$\begin{aligned} P_j(f)=\sum _{i=1}^{k}\widehat{u_{ji}}(f)v_i: \ j=1,2 \end{aligned}$$

for some $u_{ji}\in L_{\infty }(M,\Sigma , \nu ): j=1,2, i=1,\ldots ,k$ [see (5)]. Then

$$\begin{aligned} P_0(f)=\frac{P_1(f)+P_2(f)}{2}=\sum _{i=1}^{k}\frac{1}{2} (\widehat{u_{1i}}(f)+\widehat{u_{2i}}(f))v_i. \end{aligned}$$

For $\nu$ a.a. $z\in M$,

$$\begin{aligned} \Vert P_0\Vert&= (P_0)_z=\int _{M}\Big |\sum _{i=1}^{k} \frac{1}{2}[u_{1i}(z)+u_{2i}(z)]v_i(w)\Big |d\nu (w)\\&\leqslant \frac{1}{2}\int _{M}\Big |\sum _{i=1}^{k}u_{1i}(z)v_i(w)\Big |\,d \nu (w)+\frac{1}{2}\int _{M}\Big |\sum _{i=1}^{k}u_{2i}(z)v_i(w)\Big |d\nu (w)\\&= \frac{1}{2}(P_1)_z+\frac{1}{2}(P_2)_z\leqslant \frac{1}{2}(\Vert P_1\Vert +\Vert P_2\Vert )= \Vert P_0\Vert , \end{aligned}$$

so in the above inequalities we get equalities. In particular, for $\nu$ a.a. $z,w\in M$,

$$\begin{aligned} \Big |\sum _{i=1}^{k}[u_{1i}(z)+u_{2i}(z)]v_i(w)\Big |=\Big |\sum _{i=1}^{k}u_{1i}(z)v_i(w)\Big |+\Big | \sum _{i=1}^{k}u_{2i}(z)v_i(w)\Big |, \end{aligned}$$

or equivalently

$$\begin{aligned} \text{ sgn}(x^{P_1}_{z})=\text{ sgn}(x^{P_2}_{z})=\text{ sgn}(x^{P_0}_{z}) \end{aligned}$$

and

$$\begin{aligned} (P_1)_z=(P_2)_z=\Vert P_0\Vert . \end{aligned}$$

$\square$

2 Main results

Now we introduce some notation which we will use in this section. We write briefly $re^{it}\in \mathbb{C }^n$ instead of $(r_1e^{it_1},\ldots ,r_ne^{it_n})\in \mathbb{C }^n$, and put $r=(r_1,\ldots ,r_n)\in [0,\infty )^n$, $t=(t_1,\ldots ,t_n)\in I$, $I=[0,2\pi ]^n$. The symbol $\mathbb{T }$ stands for the unit circle in a complex plane, i.e. $\mathbb{T }=\{z\in \mathbb{C }: |z|=1\}$.

Definition 4

A subset $Z$ of $\mathbb{C }^{n}$ is called an $n$-circular set if for any $(z_1,\ldots ,z_n)\in Z$ and $(\delta _1,\ldots ,\delta _n)\in \mathbb{T }^n$, $(\delta _1z_1,\ldots ,\delta _nz_n)$ belongs to $Z$.

A unit ball with $p$-norm for $p\geqslant 1$, $\mathbb{T }^n$ or $D_1\times \dots \times D_n$, where $D_j\subset \mathbb{C }$ for $j=1,\ldots , n$ denotes a classical geometric ring, are the examples of the $n$-circular sets.

Observe that any $n$-circular set $Z$ can be written in a form

$$\begin{aligned} Z=\{re^{it}: r\in W\subset [0,\infty )^n, \ t\in I \}. \end{aligned}$$

(8)

Let $\lambda _W$ be a nonnegative measure on $W$ such that $0<\lambda _W(W)<\infty$. For example, for $Z=\{z\in \mathbb{C }: |z|\leqslant 1\}$ and a Borel set $A\subset [0,1]$, $\lambda _W(A)=\int _A rdr$ or for $Z=\bigcup _{j=1}^p\{z\in \mathbb{C }: |z|=r_j\}$, $\lambda _W$ is a counting measure on $W=\{r_1,\ldots ,r_p\}$. Let $\lambda _I$ denote the normalized Lebesgue measure on $I$. Set

$$\begin{aligned} \mu =\lambda _W\times \lambda _I \ (\text{ the} \text{ product} \text{ measure} \text{ of} \lambda _W \text{ and} \lambda _I \text{ on} \text{ the} \text{ set} W\times I). \end{aligned}$$

(9)

Define a measure $\nu$ on $Z$ associated with $\mu$ by

$$\begin{aligned} \nu (A)=\mu (\{(r,t)\in W\times I:re^{it}\in A\}). \end{aligned}$$

(10)

Throughout the remainder of this paper the symbol $L_1(Z)$ stands for the space of all $\nu$-measurable complex-valued functions on $Z$ and such that

$$\begin{aligned} \Vert f\Vert _1=\int _Z|f(z)|\,d\nu (z)=\int \int _{W\times I}|f(re^{it})|\,d\mu (r,t)<\infty . \end{aligned}$$

For $\beta \in \mathbb{N }^n, \alpha \in \mathbb{Z }^n$ define a function $e^{\beta ,\alpha }\in L_1(Z)$ by

$$\begin{aligned} e^{\beta ,\alpha }(re^{it})\!=\!e^{\beta }(r)e^{\alpha }(e^{it}),\ \ \text{ where}\ e^{\gamma }(z)\!=\!\prod _{j=1}^nz^{\gamma _j}\ \text{ for}\ \gamma \in \mathbb{Z }^n, z\in \mathbb{C }^n\!\setminus \!\{0\}.\qquad \end{aligned}$$

(11)

Fix for $j=1, \ldots , k$ $a_j\in \mathbb{C }$, $\beta ^j\in \mathbb{N }^n$ and $\alpha ^j\in \mathbb{Z }^n$, $\alpha ^{i}\ne \alpha ^{j}$ for $i\ne j$. Set

$$\begin{aligned} V=\text{ span}\{e^{\beta ^1,\alpha ^1},\ldots , e^{\beta ^k,\alpha ^k}\} \end{aligned}$$

(12)

and

$$\begin{aligned} w=\sum _{j=1}^{k}a_je^{\beta ^j,\alpha ^j}. \end{aligned}$$

(13)

Define for $t\in I$ an operator $T_t:L_1(Z)\rightarrow L_1(Z)$ by

$$\begin{aligned} T_t(f)(ue^{is})=f(ue^{i(s+t}), \ s\in I. \end{aligned}$$

(14)

Observe that $T_t$ is an isometry and $V$ is an invariant subspace of $T_t$. One can find that

$$\begin{aligned} T_t(e^{\beta ,\alpha })=e^{\beta ,\alpha }\cdot e^{\alpha }(e^{i t}). \end{aligned}$$

(15)

Now we will search for a minimal extension of an operator $R_w=F_w|_V$, where $F_w\in \mathcal L (L_1(Z),V)$ is given by

$$\begin{aligned} (F_wf)(re^{it})=(f*w)(re^{it})=\int \int \limits _{W\times I} f(ue^{is})w(re^{i(t-s)})\,d\mu (u,s). \end{aligned}$$

(16)

Remark 5

Let $n=1$, $Z=\mathbb{T }$, $V=\text{ span}\{e^{-k},\ldots , e^{k}\}$ and $w=\sum _{j=-k}^{k}e^{j}$. Then $F_w$ is a classical Fourier projection from $L_1(\mathbb{T })$ onto $V$.

Lemma 6

The Lebesgue function of the operator $F_w$ is constant and $\Vert F_w\Vert =\Vert w\Vert _1$.

Proof

By (13), (14) and (16),

$$\begin{aligned} (F_wf)(re^{it})&= \int \int \limits _{W\times I} f(ue^{is})\left(\sum _{j=1}^{k}a_je^{\beta ^j,\alpha ^j}\right)(re^{i(t-s)})\,d\mu (u,s)\\&= \sum _{j=1}^{k}a_j\int \int \limits _{W\times I} f(ue^{is})e^{0,-\alpha _j}(ue^{is})\,d\mu (u,s)e^{\beta ^j,\alpha ^j}(re^{it}). \end{aligned}$$

Hence

$$\begin{aligned} F_wf=\sum _{j=1}^{k} a_j\widehat{e^{0,-\alpha _j}}(f)e^{\beta ^j,\alpha ^j}, \end{aligned}$$

(17)

where for $v\in L_{\infty }(Z)$,

$$\begin{aligned} \hat{v}(f)=\int \int \limits _{W\times I} f(re^{it})v(re^{it})\,d\mu (r,t). \end{aligned}$$

(18)

Combining it with (6) and (17) we get that for $\nu$ a.a. $ue^{is}\in Z$,

$$\begin{aligned} x^{F_w}_{ue^{is}}(re^{it})&= \sum _{j=1}^{k} a_je^{0,-\alpha _j}(ue^{is})e^{\beta ^j,\alpha ^j}(re^{it})\nonumber \\&= \sum _{j=1}^{k} a_je^{\beta ^j,\alpha ^j}(re^{i(t-s)})=w(re^{i(t-s)}) \end{aligned}$$

(19)

and

$$\begin{aligned} (F_w)_{ue^{is}}&= \int \int \limits _{W\times I}|x^{F_w}_{ue^{is}}(re^{it})|\,d\mu (r,t)=\int \int \limits _{W\times I}|w(re^{i(t-s)})|\,d\mu (r,t)\\&= \int \int \limits _{W\times I}|w(re^{it})|\,d\mu (r,t)=\Vert w\Vert _1. \end{aligned}$$

By (7),

$$\begin{aligned} \Vert F_w\Vert = \text{ ess}\sup _{ue^{is}\in Z}(F_w)_{ue^{is}}=\Vert w\Vert _1. \end{aligned}$$

$\square$

Now we formulate three lemmas whose proofs go in the same manner as the proofs of Lemmas 1.3–1.5 in [19], so we omit them.

Lemma 7

A subspace $V\subset L_1(Z)$ defined by (12) is smooth $($see Definition 2$)$.

Lemma 8

For N a finite subset of $\mathbb{Z }^n$ there exists a real function $f\in L_{\infty }(Z)$, $f\ne 0$ such that

$$\begin{aligned} \int \int \limits _{W\times I} f(re^{it})e^{\beta ,\alpha }(re^{it})\,d\mu (r,t)=0\ \ \text{ for}\ \ \alpha \in N \ \ \text{ and}\ \ \beta \in \mathbb{N }^n. \end{aligned}$$

Lemma 9

Assume that $g,w\in S_V,g/w\in L_{\infty }(Z), \text{ sgn}(g)=\text{ sgn}(w)$ $\nu$ almost everywhere in $Z$. Then for any $\varepsilon \in \mathbb R$ such that $|\varepsilon |<(\Vert (g-w)/w\Vert _{\infty })^{-1}$ we have

$$\begin{aligned} \text{ sgn}(w+\varepsilon (g-w))=\text{ sgn}(w)\ \nu \text{ a.e.} \end{aligned}$$

Now define

$$\begin{aligned} X=\text{ span}\{e^{\beta ,\alpha ^{j}}: \beta \in \mathbb{N }^n,\quad j=1, \ldots , k\}\ \text{[see} \text{(11)]}. \end{aligned}$$

(20)

Observe that (12) and (20) imply that $V\subset X$. Set

$$\begin{aligned} \mathcal L _X(L_1(Z), V)=\{L\in \mathcal L (L_1(Z), V): L|_{X}=0\} \end{aligned}$$

(21)

and

$$\begin{aligned} \mathcal P _X(L_1(Z), V)=F_w+\mathcal L _X(L_1(Z), V) \ [\text{ see} \text{(2)}]. \end{aligned}$$

(22)

We say that $P_0\in \mathcal P _X(L_1(Z), V)$ is a minimal extension of the operator $R_w$ in the set $\mathcal P _X(L_1(Z), V)$, if

$$\begin{aligned} \Vert P_0\Vert =\inf \{\Vert P\Vert : P\in \mathcal P _X(L_1(Z), V)\}. \end{aligned}$$

It is easy to see that $\mathcal P _X(L_1(Z), V)\subset \mathcal P _{R_w}(L_1(Z), V)$. Denote

$$\begin{aligned} M^{X}_{w} \text{---the} \text{ set} \text{ of} \text{ minimal} \text{ extensions} \text{ of} R_w \text{ in} \text{ the} \text{ space} \mathcal P _X(L_1(Z), V). \end{aligned}$$

(23)

Theorem 10

The operator $F_w$ is the unique extension of $R_w$ belonging to the space $\mathcal P _X(L_1(Z),V)$ and commutative with a group $\{T_t: t\in I\}$.

Proof

By the Stone–Weierstrass Theorem and a density of the continuous functions in the space $L_1(Z)$,

$$\begin{aligned} L_1(Z)=\overline{\text{ span}\{e^{\beta ,\alpha }:\alpha \in \mathbb{Z }^n, \beta \in \mathbb{N }^n \}}. \end{aligned}$$

(24)

Let $P=\sum _{j=1}^{k}\widehat{w_j}(\cdot )e^{\beta ^j,\alpha ^j}$ be a minimal extensionof $R_w$ in the space $\mathcal P _X(L_1(Z), V)$ which commutes with the group of isometries $\{T_t: t\in I\}$. We show that $P(e^{\beta ,\alpha })=F_w(e^{\beta ,\alpha })$ for $\beta \in \mathbb{N }^n, \alpha \in \mathbb{Z }^n$. If $\beta \in \mathbb{N }^n, \alpha \in \{\alpha ^j: j=1,\ldots , k\}$ the above inequality follows from the fact that $P\in \mathcal P _X(L_1(Z), V)$. If $\beta \in \mathbb{N }^n, \alpha \notin \{\alpha ^j: j=1,\ldots , k\}$ the condition

$$\begin{aligned} T_s\circ P(e^{\beta ,\alpha })=P\circ T_s (e^{\beta ,\alpha })\quad \text{ for} s\in I \end{aligned}$$

is equivalent to

$$\begin{aligned} \sum _{j=1}^{k}\widehat{w_j}(e^{\beta ,\alpha })e^{\beta ^j,\alpha ^j}\cdot e^{\alpha ^j }(e^{is})=\sum _{j=1}^{k}\widehat{w_j}(e^{\beta ,\alpha })e^{\beta ^j,\alpha ^j}\cdot e^{\alpha }(e^{is}),\quad s\in I. \end{aligned}$$

Hence

$$\begin{aligned} \sum _{j=1}^{k}\widehat{w_j}(e^{\beta ,\alpha })e^{\beta ^j,\alpha ^j}\cdot ( e^{\alpha ^j }(e^{is})- e^{\alpha }(e^{is}))=0,\quad s\in I. \end{aligned}$$

By linear independence of the functions $\{e^{\beta ^j,\alpha ^j}\}_{j=1}^{k}$ we get that $\widehat{w_j}(e^{\beta ,\alpha })=0$ for $j=1, \ldots , k$. Hence $P(e^{\beta ,\alpha })=0=F_w(e^{\beta ,\alpha })$ for $\alpha \notin \{\alpha ^1,\ldots , \alpha ^k\}$, $\beta \in \mathbb{N }^n$, which by (24) shows that $P=F_w$. $\square$

Theorem 11

$F_w$ is a minimal extensionof $R_w$ in the set $\mathcal P _X(L_1(Z), V)$ and for any $P\in \mathcal P _X(L_1(Z), V)$,

$$\begin{aligned} F_w=\int \limits _IT_{s}^{-1}PT_s \, d \lambda _I(s). \end{aligned}$$

Proof

Let $P\in M^X_w$ [see (23)]. Define

$$\begin{aligned} Q_P=\int \limits _{I}T_{s}^{-1}PT_s\,d\lambda _I(s). \end{aligned}$$

By (15) we obtain that $Q_P\in \mathcal P _X(L_1(Z), V)$. Properties of the Lebesgue measure imply that $Q_P$ is an operator commutative with a group of isometries $\{T_t: t\in I\}$. By Theorem 10, $Q_P=F_w$. Since $\{T_t: t\in I\}$ are the isometries, $\Vert F_w\Vert =\Vert Q_P\Vert \leqslant \Vert P\Vert$, which completes the proof of minimality of an operator $F_w$ in the space $\mathcal P _X(L_1(Z), V)$. $\square$

A convenient tool for studying minimality of Fourier-type extensions in the space $\mathcal P _A(L_1(M,\Sigma , \nu ),V)$ is the following:

Theorem 12

([5, Cor. 1]) Let $V$ be a smooth, $k$-dimensional subspace of $L_1(M,\Sigma , \nu )$ with a basis $\{v_1,\ldots , v_k\}$. Then $P$ is a minimal extension of the operator $A$ if and only if two conditions are satisfied:

• (i) the Lebesgue function of the operator $P$ is constant on $M$;

• (ii) there exist a matrix $B=[B_{ij}]_{i,j=1}^k$ and a positive function $\Phi$ such that for $v=(v_1,\ldots ,v_k)$,

$$\begin{aligned} \Phi (z)(V_1(z),\ldots , V_k(z))=Bv(z)=\left[\sum _{j=1}^{k}B_{ij}v_j(z)\right]_{i=1}^{k}. \end{aligned}$$

(25)

Theorem 13

Assume that $\#\{a_j\ne 0: j=1,\ldots , k\}\geqslant 2$ [see (13)]. Let $Z$ be an $n$-circular setsuch that $\{e^{{\beta ^j}}|_W\}_{j=1}^{k}$ are linearly independent functions. Then $F_w$ is not a minimal extensionof an operator $R_w$.

Proof

Assume on the contrary that $F_w\in \mathcal P _{R_w}(L_1(Z),V)$ is a minimal extensionof an operator $R_w$. By Theorem 10, $F_w$ commutes with a group $G=\{T_t: t\in I\}$. Taking $\nu _1$ a Haar measure on $G$ and $\int _I {T_s}^{-1}BT_sd\nu _1(s)$ instead of a matrix $B$ we can assume that the matrix $B$ from Theorem 12 is commutative with $G$. It is easy to check that such a matrix is diagonal. Set $B=diag(B_1,\ldots , B_k)$. We calculate [see (6) and (19)],

$$\begin{aligned} V_j(re^{it})&= \int \int \limits _{W\times I} e^{\beta ^j,\alpha ^j}(ue^{is})\text{ sgn}(x^{F_w}_{re^{it}}(ue^{is}))\,d\mu (u,s)\\&= \int \int \limits _{W\times I} e^{\beta ^j,\alpha ^j}(ue^{is})\text{ sgn}(w(ue^{i(s-t)}))\,d\mu (u,s)\\&= \int \int \limits _{W\times I} e^{\beta ^j,\alpha ^j}(ue^{i(s+t)})\text{ sgn}(w(ue^{is}))\,d\mu (u,s)\\&= e^{\alpha ^j}(e^{it})\int \int \limits _{W\times I} e^{\beta ^j,\alpha ^j}(ue^{is})\text{ sgn}(w(ue^{is}))\,d\mu (u,s)=C_je^{\alpha ^j}(e^{it}), \end{aligned}$$

where $C_j=\int \int \limits _{W\times I} e^{\beta ^j,\alpha ^j}(ue^{is})\text{ sgn}(w(ue^{is}))d\mu (u,s)$. Now the condition (ii) of Theorem 12 [see (25)] is equivalent to

$$\begin{aligned} \Phi (re^{it})(C_1e^{\alpha ^1}(e^{it}), \ldots ,C_ke^{\alpha ^k}(e^{it}))\!=\!(B_1e^{\beta ^1,\alpha ^1}(re^{it}), \ldots , B_ke^{\beta ^k,\alpha ^k}(re^{it})).\qquad \quad \end{aligned}$$

(26)

By [5, Lem. 4], $\dim \text{ span}\{V_1,\ldots , V_k\}\geqslant 2$. Hence there exist $j_1, j_2\in \{1,\ldots ,k\}$ such that $C_{j_1}\ne 0$ i $C_{j_2}\ne 0$ and

$$\begin{aligned} \Phi (re^{it})=\frac{B_{j_1}}{C_{j_1}}e^{\beta ^{j_1}}(r)= \frac{B_{j_2}}{C_{j_2}}e^{\beta ^{j_2}}(r)\quad \text{ for} r\in W, \end{aligned}$$

which leads to a contradiction with a linear independence of the functions $e^{{\beta ^{j_1}}}|_W$ and $e^{{\beta ^{j_2}}}|_W$. By Theorem 12, $F_w$ is not a minimal extensionof $R_w$. $\square$

Remark 14

[19] In the case $Z=\mathbb{T }^n$, the operator $F_w$ is a minimal extensionof $R_w$ in the whole space $\mathcal P _{R_w}(L_1(\mathbb{T }^n), V)$ [it is sufficient to take $\Phi \equiv 1$ and $B_j=C_j$ for $j=1,\ldots ,k$ in the equality (26)].

Theorem 15

An operator $F_w$ is an extreme point of the set $S(0,\Vert w\Vert _1)\cap \mathcal P _{R_w}(L_1(Z), V)$ if and only if $w/\Vert w\Vert _1$ is a unique norming point $g\in S_V$ for a functional

$$\begin{aligned} L_1(Z)\ni h\mapsto \int \limits _{Z} h(z)\text{ sgn}(w)(z)\,d\nu (z)=\int \int \limits _{W\times I} h(re^{it})\text{ sgn}(w)(re^{it})\,d\mu (r,t) \end{aligned}$$

such that $g/w\in L_{\infty }(Z)$.

Proof

Assume that there exists $g\in S_V, g\ne w, g/w\in L_{\infty }(Z)$ such that $g$ is a norming point for $\widehat{\text{ sgn}(w)}$ [see (18)], i.e. $\text{ sgn}(w)=\text{ sgn}(g)\ \nu$ a.e. Define $h=g-w\in V$. By Lemma 8, there exists a real function $f\in L_{\infty }(Z)$ which is orthogonal to $e^{\beta ,\alpha }$ for $\beta \in \mathbb{N }^n$, $\alpha \in \tilde{V}-\tilde{V}$, where $\tilde{V}=\{\alpha : e^{\beta ,\alpha }\in V\}$. We can assume that $\Vert f\Vert _{\infty }<(\Vert h/w\Vert _{\infty })^{-1}$. By Lemma 9,

$$\begin{aligned} \text{ sgn}(w)(re^{it})=\text{ sgn} \left(w\pm f(ue^{is})\cdot h\right)(re^{it})\quad \text{ for}\ \nu \ \text{ a.a.}\ re^{it}, ue^{is}\in Z. \end{aligned}$$

(27)

Set $Q_1=F_w+L$ and $Q_2=F_w-L$, where

$$\begin{aligned} (Lk)(re^{it})=\int \int \limits _{W\times I} f(ue^{is})k(ue^{is})h(re^{i(t-s)})\,d\mu (u,s). \end{aligned}$$

(28)

For any $k\in L_1(Z)$ a function $Lk\in V$ because $h=\sum _{j=1}^{k}B_je^{\beta _j,\alpha ^{j}}$ for some $B_j\in \mathbb{C }, j=1\ldots , k$ and

$$\begin{aligned} Lk(re^{it})&= \int \int \limits _{W\times I}f(ue^{is})k(ue^{is})\left(\sum _{j=1}^{k}B_je^{\beta ^j,\alpha ^{j}}(re^{i(t-s)})\right)\,d\mu (u,s)\\&= \sum _{j=1}^{k}B_j \widehat{f e^{0,-\alpha ^{j}}}(k)e^{\beta ^j,\alpha ^{j}}(re^{it}). \end{aligned}$$

Observe that $L\ne 0$. We calculate,

$$\begin{aligned} x^{L}_{ue^{is}}(re^{it})&= \sum _{j=1}^{k}B_jf(ue^{is}) e^{0,-\alpha ^{j}}(ue^{is})e^{\beta ^j,\alpha ^{j}}(re^{it})\\&= \sum _{j=1}^{k}B_je^{\beta ^j, \alpha ^{j}}(re^{i(t-s)})f(ue^{is})=h(re^{i(t-s)})f(ue^{is}).\nonumber \end{aligned}$$

(29)

By the properties of $f$,

$$\begin{aligned} L(e^{\beta ,\alpha })&= \sum _{j=1}^{k}B_j\widehat{fe^{0,-\alpha ^{j}}}(e^{\beta , \alpha })\\&= \sum _{j=1}^{k}B_j\int \int \limits _{W\times I} f(ue^{is})e^{\beta ,\alpha -\alpha ^j}(ue^{is})\,d\mu (u,s)=0\quad \text{ for}\ \ \alpha \in \tilde{V}. \end{aligned}$$

Hence $Q_1$ and $Q_2$ are the minimal extensions of $R_w$ and $Q_j\ne F_w: j=1, 2$ (since $L\ne 0$). By (19) and (27)–(29) for $\nu$ a.a. $ue^{is}\in Z$ and $j=1,2$,

$$\begin{aligned} (Q_j)_{ue^{is}}&= \int \int \limits _{W\times I}|x^{F_w\pm L}_{ue^{is}}(re^{it})|\,d\mu (r,t)=\int \int \limits _{W\times I}|(x^{F_w}_{ue^{is}}\pm x^{L}_{ue^{is}})(re^{it})|\,d\mu (r,t)\\&= \int \int \limits _{W\times I}|w(re^{i(t-s)})\pm h(re^{i(t-s)})f(ue^{is})|\,d\mu (r,t)\\&= \int \int \limits _{W\times I} \text{ sgn}[(w\pm h)(re^{i(t-s)})f(ue^{is})][(w\pm h)(re^{i(t-s)})f(ue^{is})]\,d\mu (r,t)\\&= \int \int \limits _{W\times I} \text{ sgn}(w)(re^{i(t-s)})(w(re^{i(t-s)})\pm h(re^{i(t-s)})f(ue^{is}))\,d\mu (r,t)\\& = int \int \limits _{W\times I}|w(re^{i(t-s)})|\,d\mu (r,t)\!\pm \! f(ue^{is})\int \int \limits _{W\!\times \! I}(\text{ sgn}(w)\cdot (g\!-\!w))(re^{i(t-s)})\,d\mu (r,t)\\&= \Vert w\Vert _1\pm f(ue^{is})(\Vert g\Vert _1-\Vert w\Vert _1)=\Vert w\Vert _1=\Vert F_w\Vert . \end{aligned}$$

Applying (7) we obtain that $\Vert Q_1\Vert =\Vert Q_2\Vert =\Vert F_w\Vert$. Since $F_w=(Q_1+Q_2)/2$, $F_w$ is not an extreme point of the set $\mathcal P _{R_w}(L_1(Z),V)\cap S(0,\Vert w\Vert _1)$.

Now assume that $F_w$ is not an extreme point of the set $\mathcal P _{R_w}(L_1(Z),V)\cap S(0,\Vert w\Vert _1)$. Hence there exist $P_1, P_2\in S(0,\Vert w\Vert _1)\cap \mathcal P _{R_w}(L_1(Z),V)$ such that $P_j\ne F_w: j=1,2$ and $F_w=(P_1+P_2)/2$. Define $L=(P_1-P_2)/2$. Then $P_1=F_w+L$ and $P_2=F_w-L$. By Lemma 6, the Lebesgue function of the operator $F_w$ is constant. We have $\Vert F_w+L\Vert =\Vert F_w-L\Vert =\Vert w\Vert _1$ and by Lemma 3, for $\nu$ a.a. $ue^{is}\in Z$,

$$\begin{aligned} (F_w+L)_{ue^{is}}&= (F_w-L)_{ue^{is}} =\Vert w\Vert _1,\nonumber \\ \text{ sgn}(x^{F_w+L}_{ue^{is}})&= \text{ sgn}(x^{F_w-L}_{ue^{is}})= \text{ sgn}(x^{F_w}_{ue^{is}}). \end{aligned}$$

(30)

Let us fix $ue^{is}\in Z$ satisfying (30) and such that

$$\begin{aligned} x^{L}_{ue^{is}}\ne 0 \text{ and} x^{F_w}_{ue^{is}}(re^{i(t+s)})=w(re^{it})\ \text{[see} \text{(19)]}. \end{aligned}$$

(31)

Without loss of generality we can assume that $\Vert w\Vert _1=1$. Set

$$\begin{aligned} h=T_s(x^{L}_{ue^{is}})\ \text{[see} \text{(14)]},\ \ g=w+h. \end{aligned}$$

(32)

Since $h\ne 0$, $g\ne w$. Observe that by (30)–(32) for $\nu$ a.a. $re^{it}\in Z$,

$$\begin{aligned} g(re^{it})&= (w+h)(re^{it})=w(re^{it})+T_{s}(x^{L}_{ue^{is}})(re^{it})\nonumber \\&= x^{F_w}_{ue^{is}}(re^{i(t+s)})+x^{L}_{ue^{is}}(re^{i(t+s)})= x^{F_w+L}_{ue^{is}}(re^{i(t+s)}) \end{aligned}$$

(33)

and

$$\begin{aligned} \text{ sgn} (g)(re^{it})&= \text{ sgn} (x^{F_w+L}_{ue^{is}})(re^{i(t+s)})=\text{ sgn} (x^{F_w}_{ue^{is}})(re^{i(t+s)})=\text{ sgn}(w)(re^{it}),\\ \text{ sgn} (w-h)(re^{it})&= \text{ sgn} (x^{F_w-L}_{ue^{is}})(re^{i(t+s)})=\text{ sgn} (x^{F_w}_{ue^{is}})(re^{i(t+s)})=\text{ sgn}(w)(re^{it}). \end{aligned}$$

Hence

$$\begin{aligned} \text{ sgn}(g)=\text{ sgn}(w)=\text{ sgn}(w-h)\ \nu \ \text{ a.e.} \end{aligned}$$

(34)

By (30) and (33),

$$\begin{aligned} \Vert g\Vert _1&= \int \int \limits _{W\times I}|g(re^{it})|d\mu (r,t)=\int \int \limits _{W\times I}|x^{F_w+L}_{ue^{is}}(re^{i(t+s)})|\,d\mu (r,t)\\&= \int \int \limits _{W\times I}|x^{F_w+L}_{ue^{is}}(re^{it})|\,d\mu (r,t)=(F_w+L)_{ue^{is}}=\Vert w\Vert _1=1. \end{aligned}$$

Now we show that $\frac{g}{w}\in L_{\infty }(Z)$. Assume on the contrary that $\frac{g}{w}\notin L_{\infty }(Z)$. Then for any $k\in \mathbb{N }$ there exists $A_k\subset Z: \nu (A_k)>0$ and $|g(z)/w(z)|>k$ for $z\in A_k$. Let $z\in A_k$ and $a_z=\text{ sgn}(w)(z)$. By (34),

$$\begin{aligned} a_z(w(z)+h(z))>ka_zw(z) \end{aligned}$$

and

$$\begin{aligned} a_z(w(z)-h(z))<(2-k)|w(z)|\leqslant 0 \quad \text{ for} \ k> 2. \end{aligned}$$

The above inequality implies that

$$\begin{aligned} \text{ sgn}(w-h)(z)\ne \text{ sgn}(w)(z)\quad \text{ for}\ z\in A_k, k>2; \end{aligned}$$

a contradiction with (34). $\square$

Lemma 16

$F_w$ is the unique minimal extensionof $R_w$ in the space $\mathcal P _X(L_1(Z), V)$ if and only if$F_w$ is an extreme point of the set $M^{X}_{w}$ [see (23)].

The proof of Lemma 16 goes in the same manner as the proof of [19, Lem. 1.2], so we omit it.

Now we can formulate a main result of this paper, a theorem, which characterize the uniqueness of minimal Fourier-type extensions in the set $\mathcal P _{X}(L_1(Z),V)$.

Theorem 17

The operator $F_w$ is the unique minimal extensionof $R_w=F_w|_V$ in the set $\mathcal P _X(L_1(Z), V)$ if and only if$w/\Vert w\Vert _1$ is a unique norming point $g\in V$ for a functional

$$\begin{aligned} L_1(Z)\ni h\mapsto \int \int \limits _{W\times I}h(re^{it})\text{ sgn}(w)(re^{it})\,d\mu (r,t) \end{aligned}$$

such that $g/w\in L_{\infty }(Z)$.

Proof

Assume that there exists $g\in S_V, g\ne w, g/w\in L_{\infty }(Z)$ such that $g$ is a norming point for $\widehat{\text{ sgn}(w)}$ [see (18)], i.e. $\text{ sgn}(w)=\text{ sgn}(g)\ \nu$ a.e. Notice that the operator $L$ constructed in Theorem 15 [see (28)] is actually an element of the space $\mathcal L _X(L_1(Z), V)$ and extensions $Q_1=F_w+L$ and $Q_2=F_w-L$ belong to the set $\mathcal P _X(L_1(Z), V)$. By the first part of the proof of Theorem 15, $F_w$ is not an extreme point of the set $M^{X}_{w}$ [see (23)].

Now assume that $w/\Vert w\Vert _1$ is a unique norming point $g\in V$ for a functional

$$\begin{aligned} L_1(Z)\ni h\mapsto \int \int \limits _{W\times I}h(re^{it})\text{ sgn}(w)(re^{it})\,d\mu (r,t) \end{aligned}$$

such that $g/w\in L_{\infty }(Z)$. By Theorem 15, $F_w$ is not an extreme point of the set $S(0,\Vert w\Vert _1)\cap \mathcal P _{R_w}(L_1(Z), V)$. Since $M^{X}_{w}\subset S(0,\Vert w\Vert _1)\cap \mathcal P _{R_w}(L_1(Z), V)$, we get that

$$\begin{aligned} \text{ ext} \left[S(0,\Vert w\Vert _1)\cap \mathcal P _{R_w}(L_1(Z), V)\right]\subset \text{ ext}M^{X}_{w} \end{aligned}$$

and $F_w$ is not an extreme point of the set $M^{X}_{w}$. By Lemma 16 the proof is complete. $\square$

Directly from Theorem 17, reasoning as in the proof of [19, Cor. 1.9], we get the following result:

Corollary 18

Let $w\in V, w\ne 0$ is determined by its roots in $V$ (Definition 1). Then the operator $F_w$ is the unique minimal extensionof $R_w$ in the set $\mathcal P _X(L_1(Z), V)$

The next theorem shows how large the set of minimal extensions can be. We present it without proof. The reader interested in the method of proof is referred to [19, Th. 1.7].

Theorem 19

Let

$$\begin{aligned} S_w=\text{ span}\{P-F_w: P\in M^{X}_{w}\}. \end{aligned}$$

If the operator $F_w$ is not a unique minimal extensionof the operator $R_w$ in the set $\mathcal P _X(L_1(Z), V)$, then $\dim (S_w)=\infty$.

Theorem 20

(Daugavet [11]) Let $K$ be a compact set without isolated points. If $L: \mathcal C _\mathbb{K }(K)\rightarrow \mathcal C _\mathbb{K }(K)$ $(\mathbb{K }=\mathbb{R }$ or $\mathbb{K }=\mathbb{C })$ is a compact operator, then

$$\begin{aligned} \Vert Id+L\Vert =1+\Vert L\Vert . \end{aligned}$$

Denote

$$\begin{aligned} \left(\mathcal{P }_X(L_1(Z), V)\right)^*=\{P^*: P\in \mathcal{P }_X(L_1(Z), V)\}. \end{aligned}$$

We say that an operator ${P_0}^*\in \left(\mathcal{P }_X(L_1(Z), V)\right)^*$ is an element of best approximation to $A: L_{\infty }(Z)\rightarrow L_{\infty }(Z)$ in the set $\left(\mathcal{P }_X(L_1(Z), V)\right)^*$ if

$$\begin{aligned} \Vert A-{P_0}^*\Vert =\inf \{ \Vert A-P^*\Vert : {P}^*\in \left(\mathcal{P }_X(L_1(Z), V)\right)^*\}. \end{aligned}$$

In the same manner as in [20, Th.1.9] we can prove:

Theorem 21

Let $Z$ be a compact $n$-circular set. An identity operator $Id: L_{\infty }(Z)\rightarrow L_{\infty }(Z)$ has the unique element of best approximation in $\left(\mathcal{P }_X(L_1(Z), V)\right)^*$ if and only if$w/\Vert w\Vert _1$ is a unique norming point $g\in V$ for a functional

$$\begin{aligned} L_1(Z)\ni h\mapsto \int \int \limits _{W\times I} h(re^{it})\text{ sgn}(w)(re^{it})\,d\mu (r,t) \end{aligned}$$

such that $g/w\in L_{\infty }(Z)$.

3 Applications

Now we show some applications of Theorem 17.

Theorem 22

[25, Th.14.3.3] Let $\Omega$ be a bounded domain in $\mathbb{C }^n$, $n>1$. If $f$ and $g$ are holomorphic functions on $\Omega$, continuous in $\bar{\Omega }$ and such that

$$\begin{aligned} |f(z)|\leqslant |g(z)| \quad \text{ for} \quad z\in \partial \Omega , \end{aligned}$$

then

$$\begin{aligned} |f(z)|\leqslant |g(z)| \quad \text{ for} \quad z\in \Omega . \end{aligned}$$

Directly from Theorem 17 and Theorem 22 we get the following two examples:

Example 23

(Uniqueness) Let $D$ be a bounded $n$-circular domain in $\mathbb{C }^n$, $n\geqslant 2$ and let $Z=\partial D$. Assume that $V$ is the space of algebraic polynomials of $n$ complex variables of degree $\leqslant k$ and fix $w\in V$. We prove that the operator $F_w$ is the unique minimal extensionof $R_w$ in the set $\mathcal P _X(L_1(Z), V)$. Indeed, by Theorem 17 it is sufficient to show that if $g\in V$ satisfies the conditions $\text{ sgn}(g)(z)=\text{ sgn}(w)(z)$ for $z\in Z$, $\Vert g|_{Z}\Vert _1=1$ and $g/w\in L_{\infty }(Z)$, then $g|_Z=w|_Z/\Vert w|_Z\Vert _1$. Take $g$ as we mentioned above. Define $F_1=g+iw$, $F_2=g-iw$. Observe that $F_1$ and $F_2$ are holomorphic functions on $D$ and continuous in $\bar{D}$. By assumptions, $g(z)\overline{w(z)}\in \mathbb{R }]$ for $z\in Z$ and

$$\begin{aligned} |F_1(z)|=|F_2(z)|=\sqrt{|g(z)|^2+|w(z)|^2}, z\in Z. \end{aligned}$$

Applying twice Theorem 22, we get that $|F_1(z)|=|F_2(z)|$ for $z\in D$. Put $G=D\setminus \{F_2=0\}$ and $h(z)=F_1(z)/F_2(z)$ for $z\in G$. Note that $h$ is holomorphic in the domain $G$ and $|h|=1$ on $G$. Since nonconstant holomorphic functions are open mappings, $h|_{G}=c$ for some $c\in \mathbb{C }$. A condition

$$\begin{aligned} \Vert g|_{Z}\Vert _1=\left\Vert\frac{w|_{Z}}{\Vert w|_{Z}\Vert _1}\right\Vert_1=1 \end{aligned}$$

implies that $g|_{Z}=w|_{Z}/\Vert w|_{Z}\Vert _1$.

Reasoning as in Example 23 we get:

Example 24

(Uniqueness) Let $Z$ be an $n$-circular domain, $n\geqslant 2$. Assume that $V$ is a space of algebraic polynomials of $n$ complex variables of degree $\leqslant k$ and fix $w\in V$. Then the operator $F_w$ is the unique minimal extensionof $R_w$ in the set $\mathcal P _X(L_1(Z), V)$.

Now we give an example of $n$-circular set$Z$, a smooth space $V$ and $w\in V$, for which the operator $F_w$ is not a unique minimal extensionof $R_w$ in the set $\mathcal P _X(L_1(Z), V)$.

Example 25

(Nonuniqueness) Let $V$ be a space of algebraic polynomials of $n$ complex variables of degree $\leqslant k$. Set $Z=\mathbb{T }^n$. For $s\in \mathbb{T }$ define

$$\begin{aligned} h(s)&= s^2+l(1+b)s+b,\\ k(s)&= s^2+m(1+b)s+b, \end{aligned}$$

where $l,m\in (0,\infty )\setminus \{1\}, m\ne l, |b|=1, b\ne -1$ are such that polynomials $h$ and $k$ have all roots outside of $\mathbb{T }$ and

$$\begin{aligned} ml(1+Reb)+(m+l)Re((1+\overline{b})s)\geqslant 0 \text{ for} s\in \mathbb{T }. \end{aligned}$$

Observe that by our assumptions,

$$\begin{aligned} h(s)\overline{k(s)}&= (s^2+b+l(1+b)s)(\overline{s}^2+\overline{b}+m(1+\overline{b})\overline{s})\\&= |s^2+b|^2\!+\!m(1+\overline{b})s\!+\!m(1\!+\!b)\overline{s}\!+\!l(1+b)\overline{s}\!+\!l(1+\overline{b})s\!+\!ml|1+b|^2 \\&= |s^2+b|^2+2ml(1+Re b)+2(m+l)Re[(1+\overline{b})s]\geqslant 0. \end{aligned}$$

Consider any polynomial $l$ of $n$ variables of degree $k-2$. Put

$$\begin{aligned} w(s,t)=h(s)l(s,t),\ \ g(s,t)=k(s)l(s,t)\quad \text{ for} s\in \mathbb{T }, t\in \mathbb{T }^{n-1}. \end{aligned}$$

Then $g,w\in V$, $g/w\in L_{\infty }(Z)$ and $w(s,t)\overline{g(s,t)}=h(s)\overline{k(s)}|l(s,t)|^2\geqslant 0$ for $(s,t)\in \mathbb{T }^n$. Hence $\text{ sgn}(w)=\text{ sgn}(g)$ and by Theorem 17, $F_w$ is not a minimal extensionof $R_w$ in the set $M^{X}_{w}$. In this case the assertion of Theorem 19 is fulfilled.

Other examples in which there is more than one minimal extensionof $R_w$ can be found in [19].

Notice that methods of proofs used in this paper can be applied not only in the case of $n$-circular sets, but also to $Z=[0,2\pi ]^n\times \mathbb{T }^n$. More precisely, let $L_1([0,2\pi ]^n\times \mathbb{T }^n)$ denote the space of complex-valued functions, Lebesgue measurable on $[0,2\pi ]^n\times \mathbb{T }^n$ and such that

$$\begin{aligned} \Vert f\Vert _1=(1/2\pi )^{2n}\int \int \limits _{I\times I}|f(u,e^{is})|\,duds=\int \int \limits _{I\times I}|f(u,e^{is})|\,d\mu (u,s)<\infty , \end{aligned}$$

where

$$\begin{aligned} \mu =\lambda _I\times \lambda _I, \ \lambda _I\text{---a} \text{ normalized} \text{ Lebesgue} \text{ measure} \text{ on} I=[0,2\pi ]^n. \end{aligned}$$

(35)

For $\alpha =(\alpha _1,\ldots , \alpha _n)\in \mathbb{Z }^n$ and $r=(r_1,\ldots , r_n)\in [0,2\pi ]^n$ put

$$\begin{aligned} G^{\alpha }&= \{f: f(r)=f_1(r_1)\cdot \ldots \cdot f_n(r_n),\quad f_j=\cos (\alpha _j\cdot )\ \text{ or}\nonumber \\ f_j&= \sin (\alpha _j\cdot ):j=1,\ldots ,n\} \end{aligned}$$

(36)

Let

$$\begin{aligned} G^{\alpha }= \{G^{\alpha }_{j}: j=1,\ldots , 2^n\}. \end{aligned}$$

Let $\alpha \in \mathbb{Z }^n$, $\beta \in \mathbb{Z }^n$ and $j\in \{1,\ldots , 2^n\}$. Define a function

$$\begin{aligned} G^{\alpha ,\beta }_{j}: L_1([0,2\pi ]^n\times \mathbb{T }^n)\ni (r,e^{it})\mapsto G^{\alpha }_{j}(r)e^{\beta }(e^{it})\in \mathbb{C } , \end{aligned}$$

(37)

where $e^{\beta }$ is given by the formula (11). Let

$$\begin{aligned} V&= \text{ span}\{G^{\alpha ^p,\beta ^p}_{j}: p=1,\ldots ,k, j=1,\ldots , 2^n\ \},\nonumber \\ w&= \sum \limits _{\begin{array}{c} p=1\ldots ,k\\ j=1,\ldots , 2^n \end{array}}b_{p,j}G^{\alpha ^p,\beta ^p}_{j}, \end{aligned}$$

(38)

for some $\alpha ^p\in \mathbb{Z }^n$, $\beta ^p\in \mathbb{Z }^n$ such that $(\alpha ^p,\beta ^p)\ne (\alpha ^m,\beta ^m)$ for $p\ne m$ and $b_{p,j}\in \mathbb{C }$. Set

$$\begin{aligned} (F_wf)(r,e^{it})=(f*w)(r,e^{it})=\int \int \limits _{I\times I} f(u,e^{is})w(r-u,e^{i(t-s)})\,d\mu (u,s).\qquad \end{aligned}$$

(39)

Applying a group of isometries $\tilde{G}=\{T_{u,s}(f)(r,e^{it})=f(r+u,e^{i(s+t)})\}_{u,s\in [0,2\pi ]^n}$ instead of $G=\{T_t\}_{t\in [0,2\pi ]^n}$ [see (14)] and reasoning in the same manner as in the case of $n$-circular sets, we can obtain the following:

Theorem 26

The operator $F_w$ is the unique minimal extensionof $R_w$ in the set $\mathcal P _{R_w}(L_1([0,2\pi ]^n\times \mathbb{T }^n),V)$ if and only if$w/\Vert w\Vert _1$ is a unique norming point $g\in V$ for a functional

$$\begin{aligned} L_1([0,2\pi ]^n\times \mathbb{T }^n)\ni h\mapsto \int \int \limits _{I\times I} h(r,e^{it})\text{ sgn}(w)(r,e^{it})\,d\mu (r,t) \end{aligned}$$

such that $g/w\in L_{\infty }([0,2\pi ]^n\times \mathbb{T }^n)$.

Here we have the uniqueness in the whole space $\mathcal P _{R_w}(L_1([0,2\pi ]^n\times \mathbb{T }^n),V)$ because the group $\tilde{G}$ is so big that $F_w$ is the unique operator in $\mathcal P _{R_w}(L_1([0,2\pi ]^n\times \mathbb{T }^n),V)$ commuting with $\tilde{G}$. As a consequence, $F_w$ has a minimal norm in the space $\mathcal P _{R_w}(L_1(Z), V)$, which is not true in the case of an arbitrary $n$-circular set.