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 \(\mathbb{C }^n\).
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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
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
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
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
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
\([a_{i,j}]_{i,j=1}^{k}\) 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
Let \(P\in \mathcal P _A(L_1(M,\Sigma , \nu ),V)\) be given by (4) and \(z, w\in M\). Define
A map \(z\rightarrow P_z\) is called the Lebesgue function of the operator \(P\). It is well known that
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. \(z\in M\),
Proof
Let
for some \(u_{ji}\in L_{\infty }(M,\Sigma , \nu ): j=1,2, i=1,\ldots ,k\) [see (5)]. Then
For \(\nu \) a.a. \(z\in M\),
so in the above inequalities we get equalities. In particular, for \(\nu \) a.a. \(z,w\in M\),
or equivalently
and
\(\square \)
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
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
Define a measure \(\nu \) on \(Z\) associated with \(\mu \) by
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
For \(\beta \in \mathbb{N }^n, \alpha \in \mathbb{Z }^n\) define a function \(e^{\beta ,\alpha }\in L_1(Z)\) by
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
and
Define for \(t\in I\) an operator \(T_t:L_1(Z)\rightarrow L_1(Z)\) by
Observe that \(T_t\) is an isometry and \(V\) is an invariant subspace of \(T_t\). One can find that
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
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
Hence
where for \(v\in L_{\infty }(Z)\),
Combining it with (6) and (17) we get that for \(\nu \) a.a. \(ue^{is}\in Z\),
and
By (7),
\(\square \)
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
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
Now define
Observe that (12) and (20) imply that \(V\subset X\). Set
and
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
It is easy to see that \(\mathcal P _X(L_1(Z), V)\subset \mathcal P _{R_w}(L_1(Z), V)\). Denote
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 \(L_1(Z)\),
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
is equivalent to
Hence
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 \(P\in \mathcal P _X(L_1(Z), V)\),
Proof
Let \(P\in M^X_w\) [see (23)]. Define
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 \(v=(v_1,\ldots ,v_k)\),
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)],
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
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
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
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,
Set \(Q_1=F_w+L\) and \(Q_2=F_w-L\), where
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
Observe that \(L\ne 0\). We calculate,
By the properties of \(f\),
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 \(j=1,2\),
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. \(ue^{is}\in Z\),
Let us fix \(ue^{is}\in Z\) satisfying (30) and such that
Without loss of generality we can assume that \(\Vert w\Vert _1=1\). Set
Since \(h\ne 0\), \(g\ne w\). Observe that by (30)–(32) for \(\nu \) a.a. \(re^{it}\in Z\),
and
Hence
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),
and
The above inequality implies that
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
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
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
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
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
Denote
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
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
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
then
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
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
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
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
Observe that by our assumptions,
Consider any polynomial \(l\) of \(n\) variables of degree \(k-2\). Put
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
where
For \(\alpha =(\alpha _1,\ldots , \alpha _n)\in \mathbb{Z }^n\) and \(r=(r_1,\ldots , r_n)\in [0,2\pi ]^n\) put
Let
Let \(\alpha \in \mathbb{Z }^n\), \(\beta \in \mathbb{Z }^n\) and \(j\in \{1,\ldots , 2^n\}\). Define a function
where \(e^{\beta }\) is given by the formula (11). Let
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
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
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.
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Communicated by K. Gröchenig.
G. Lewicki was supported by the State Committee for Scientific Research, Poland (Grant No. N N201 541 438). A. Micek was supported by the State Commitee for Scientific Research, Poland (Grant No. N N201 609 740).
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Lewicki, G., Micek, A. Uniqueness of minimal Fourier-type extensions in \(L_1\)-spaces. Monatsh Math 170, 161–178 (2013). https://doi.org/10.1007/s00605-012-0468-8
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DOI: https://doi.org/10.1007/s00605-012-0468-8