Uniqueness of minimal Fourier-type extensions in L1-spaces

We give a characterization of uniqueness of finite rank Fourier-type minimal extensions in L1-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 Cn .


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 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 ∈ X is called a norming point for f ∈ X * if x ∈ S X and f (x) = x . For z ∈ C, sgnz = z/|z| for z = 0 and 0 for z = 0.
Let Y be a linear subspace of X and let L(X, Y ) (L(X ) if X = Y ) be the space of all linear, continuous operators from X into Y . Given A ∈ L(Y ), an operator P ∈ L(X, Y ) is called an extension of A (or a projection in the case of A = Id Y ) if P| Y = A. The set of all extensions of A will be denoted by P A (X, Y ). An extension P 0 ∈ P A (X, Y ) is minimal if We write briefly P(X, Y ) and λ(Y, X ) instead of P Id Y (X, Y ) and λ Id Y (Y, X ) respectively. Basic results on minimal projections and extensions can be found in [1,7,9,12,13,[15][16][17][18]20,22,24]. Set Let π n denote the space of all trigonometric polynomials of degree n and let C 0 (2π) be the space of all continuous, real valued, 2π -periodic functions. The classical Fourier projection from C 0 (2π) onto π n is defined by a formula where D n (t) = n j=−n e i jt . It is well-known that F n is the unique operator of minimal norm in the space P(C 0 (2π), π n ) [10,23]. Moreover, F n = λ(π n , L p [0, 2π ]) for 1 p ∞, 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, 2π ] onto π n for p = 1, 2, +∞. 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, -σ -algebra of subsets of M, ν a positive measure on such that (M, , ν) is a complete measure space. By L 1 (M, , ν) denote a space of complex-valued, ν-measurable functions on M satisfying a condition To the end of this paper we assume that (L 1 (M, , ν)) * = L ∞ (M, , ν), which is satisfied, for example, if ν is σ -finite. Definition 1 It is said that w ∈ V ⊂ L ∞ (M, , ν), w = 0 is determined by its roots in the space V if and only if for any g ∈ V the condition g/w ∈ L ∞ (M, , ν) implies that g = cw for some c ∈ C. Take V a smooth, finite-dimensional subspace of L 1 (M, , ν) with a basis {v 1 , . . . , v k } and fix an operator A ∈ L(V ). Observe that any P ∈ P A (L 1 (M, , ν), V ) has a form [a i, j ] k i, j=1 is a matrix of the operator A in the basis {v 1 , . . . , v k } and u ∈ (L 1 (M, , ν)) * denotes a functional associated with u ∈ L ∞ (M, , ν) by Let P ∈ P A (L 1 (M, , ν), V ) be given by (4) and z, w ∈ M. Define A map z → P z is called the Lebesgue function of the operator P. It is well known that P = ess sup z∈M P z (see [14,Lem. 2]).

Proof
Let Then For ν a.a. z ∈ M, so in the above inequalities we get equalities. In particular, for ν a.a. z, or equivalently and (P 1 ) z = (P 2 ) z = P 0 .

Main results
Now we introduce some notation which we will use in this section. We write briefly re it ∈ C n instead of (r 1 e it 1 , . . . , r n e it n ) ∈ C n , and put r = (r 1 , . . . , r n ) ∈ [0, ∞) n , t = (t 1 , . . . , t n ) ∈ I , I = [0, 2π ] n . The symbol T stands for the unit circle in a complex plane, i.e. T = {z ∈ C : |z| = 1}.
A unit ball with p-norm for p 1, T n or D 1 × · · · × D n , where D j ⊂ C for j = 1, . . . , 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 λ W be a nonnegative measure on W such that 0 < λ W (W ) < ∞. For example, for Z = {z ∈ C : |z| 1} and a Borel set A ⊂ [0, 1], λ W (A) = A rdr or for Z = p j=1 {z ∈ C : |z| = r j }, λ W is a counting measure on W = {r 1 , . . . , r p }. Let λ I denote the normalized Lebesgue measure on I . Set μ = λ W × λ I (the product measure of λ W and λ I on the set W × I ).
Define a measure ν on Z associated with μ by Throughout the remainder of this paper the symbol L 1 (Z ) stands for the space of all ν-measurable complex-valued functions on Z and such that For β ∈ N n , α ∈ Z n define a function e β,α ∈ L 1 (Z ) by e β,α (re it ) = e β (r )e α (e it ), where e γ (z) = n j=1 z γ j for γ ∈ Z n , z ∈ C n \{0}. (11) Fix for j = 1, . . . , k a j ∈ C, β j ∈ N n and α j ∈ Z n , α i = α j for i = j. Set and Define for t ∈ I an operator T t : 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

Lemma 6
The Lebesgue function of the operator F w is constant and F w = w 1 .
Proof By (13), (14) and (16), Hence Combining it with (6) and (17) we get that for ν a.a. ue is ∈ Z , and By (7), 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. (11)]. (20) Observe that (12) and (20) imply that V ⊂ X . Set and We say that It is easy to see that P

Theorem 10
The operator F w is the unique extension of R w belonging to the space P X (L 1 (Z ), V ) and commutative with a group {T t : t ∈ I }.
Proof By the Stone-Weierstrass Theorem and a density of the continuous functions in the space L 1 (Z ), By linear independence of the functions {e β j ,α j } k j=1 we get that w j (e β,α ) = 0 for j = 1, . . . , k. Hence P(e β,α ) = 0 = F w (e β,α ) for α / ∈ {α 1 , . . . , α k }, β ∈ N n , which by (24) shows that P = F w . Theorem 11 F w is a minimal extensionof R w in the set P X (L 1 (Z ), V ) and for any P ∈ P X (L 1 (Z ), V ), (23)]. Define By (15) we obtain that Q P ∈ 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 ∈ I }. By Theorem 10, Q P = F w . Since {T t : t ∈ I } are the isometries, F w = Q P P , which completes the proof of minimality of an operator F w in the space P X (L 1 (Z ), V ).
A convenient tool for studying minimality of Fourier-type extensions in the space P A (L 1 (M, , ν), V ) is the following: Theorem 13 Assume that #{a j = 0 : j = 1, . . . , k} 2 [see (13)]. Let Z be an n-circular setsuch that {e β j | W } k j=1 are linearly independent functions. Then F w is not a minimal extensionof an operator R w .
Proof Assume on the contrary that F w ∈ 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 ∈ I }. Taking ν 1 a Haar measure on G and I T s −1 BT s dν 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 , . . . , B k ). We calculate [see (6) and (19)], s). Now the condition (ii) of Theorem 12 [see (25)] is equivalent to (re it )(C 1 e α 1 (e it ), . . . , C k e α k (e it )) = (B 1 e β 1 ,α 1 (re it ), . . . , B k e β k ,α k (re it )). (26) By [5,Lem. 4], dim span{V 1 , . . . , V k } 2. Hence there exist j 1 , j 2 ∈ {1, . . . , k} such that C j 1 = 0 i C j 2 = 0 and which leads to a contradiction with a linear independence of the functions e β j 1 | W and e β j 2 | W . By Theorem 12, F w is not a minimal extensionof R w .
Remark 14 [19] In the case Z = T n , the operator F w is a minimal extensionof R w in the whole space P R w (L 1 (T n ), V ) [it is sufficient to take ≡ 1 and B j = C j for j = 1, . . . , k in the equality (26)].

Theorem 15 An operator F w is an extreme point of the set S
Proof Assume that there exists g ∈ S V , g = w, g/w ∈ L ∞ (Z ) such that g is a norming point for sgn(w) [see (18)], i.e. sgn(w) = sgn(g) ν a.e. Define h = g − w ∈ V . By Lemma 8, there exists a real function f ∈ L ∞ (Z ) which is orthogonal to e β,α for β ∈ N n , α ∈Ṽ −Ṽ , whereṼ = {α : e β,α ∈ V }. We can assume that For any k ∈ L 1 (Z ) a function Lk ∈ V because h = k j=1 B j e β j ,α j for some B j ∈ C, j = 1 . . . , k and Observe that L = 0. We calculate, By the properties of f , Hence Q 1 and Q 2 are the minimal extensions of R w and Q j = F w : j = 1, 2 (since L = 0). By (19) and (27)-(29) for ν a.a. ue is ∈ Z and j = 1, 2, . Now assume that F w is not an extreme point of the set P R w (L 1 (Z ), V )∩S(0, w 1 ). Hence there exist P 1 , P 2 ∈ S(0, w 1 ) ∩ P R w (L 1 (Z ), V ) such that P j = 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 F w + L = F w − L = w 1 and by Lemma 3, for ν a.a. ue is ∈ Z , Let us fix ue is ∈ Z satisfying (30) and such that x L ue is = 0 and x F w ue is (re i(t+s) ) = w(re it ) [see (19)].
The above inequality implies that a contradiction with (34).
Lemma 16 F w is the unique minimal extensionof R w in the space P X (L 1 (Z ), V ) if and only ifF 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 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 P X (L 1 (Z ), V ) if and only ifw/ w 1 is a unique norming point g ∈ V for a functional Proof Assume that there exists g ∈ S V , g = w, g/w ∈ L ∞ (Z ) such that g is a norming point for sgn(w) [see (18)], i.e. sgn(w) = sgn(g) ν a.e. Notice that the operator L constructed in Theorem 15 [see (28)] is actually an element of the space L X (L 1 (Z ), V ) and extensions Q 1 = F w + L and Q 2 = F w − L belong to the set 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/ w 1 is a unique norming point g ∈ V for a functional such that g/w ∈ L ∞ (Z ). By Theorem 15, F w is not an extreme point of the set and F w is not an extreme point of the set M X w . By Lemma 16 the proof is complete.
Directly from Theorem 17, reasoning as in the proof of [19, Cor. 1.9], we get the following result:

Corollary 18
Let w ∈ V, w = 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 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 Theorem 20 (Daugavet [11]) Let K be a compact set without isolated points. If L : is a compact operator, then We say that an operator P 0 * ∈ (P X (L 1 (Z ), V )) * is an element of best approximation to A : 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 I d : L ∞ (Z ) → L ∞ (Z ) has the unique element of best approximation in (P X (L 1 (Z ), V )) * if and only ifw/ w 1 is a unique norming point g ∈ V for a functional such that g/w ∈ L ∞ (Z ).

Applications
Now we show some applications of Theorem 17. Example 23 (Uniqueness) Let D be a bounded n-circular domain in C n , n 2 and let Z = ∂ D. Assume that V is the space of algebraic polynomials of n complex variables of degree k and fix w ∈ V . We prove that the operator F w is the unique minimal extensionof R w in the set P X (L 1 (Z ), V ). Indeed, by Theorem 17 it is sufficient to show that if g ∈ V satisfies the conditions sgn(g)(z) = sgn(w)(z) for z ∈ Z , g| Z 1 = 1 and g/w ∈ L ∞ (Z ), then g| Z = w| Z / w| Z 1 . Take g as we mentioned above. Define Observe that F 1 and F 2 are holomorphic functions on D and continuous inD. By assumptions, g(z)w(z) ∈ R] for z ∈ Z and Applying twice Theorem 22, we get that 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 ∈ C. A condition Reasoning as in Example 23 we get: Example 24 (Uniqueness) Let Z be an n-circular domain, n 2. Assume that V is a space of algebraic polynomials of n complex variables of degree k and fix w ∈ V . Then the operator F w is the unique minimal extensionof R w in the set P X (L 1 (Z ), V ). Now we give an example of n-circular setZ , a smooth space V and w ∈ V , for which the operator F w is not a unique minimal extensionof R w in the set P X (L 1 (Z ), V ).
Example 25 (Nonuniqueness) Let V be a space of algebraic polynomials of n complex variables of degree k. Set Z = T n . For s ∈ T define Consider any polynomial l of n variables of degree k − 2. Put w(s, t) = h(s)l(s, t), g(s, t) = k(s)l(s, t) for s ∈ T, t ∈ T n−1 .
Then g, w ∈ V , g/w ∈ L ∞ (Z ) and w(s, t)g(s, t) = h(s)k(s)|l(s, t)| 2 0 for (s, t) ∈ T n . Hence sgn(w) = 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, 2π ] n ×T n . More precisely, let L 1 ([0, 2π ] n ×T n ) denote the space of complex-valued functions, Lebesgue measurable on [0, 2π ] n ×T n and such that Let G α = {G α j : j = 1, . . . , 2 n }.

Theorem 26
The operator F w is the unique minimal extensionof R w in the set P R w (L 1 ([0, 2π ] n × T n ), V ) if and only ifw/ w 1 is a unique norming point g ∈ V for a functional (r, t) such that g/w ∈ L ∞ ([0, 2π ] n × T n ).
Here we have the uniqueness in the whole space P R w (L 1 ([0, 2π ] n × T n ), V ) because the groupG is so big that F w is the unique operator in P R w (L 1 ([0, 2π ] n × T n ), V ) commuting withG. As a consequence, F w has a minimal norm in the space P R w (L 1 (Z ), V ), which is not true in the case of an arbitrary n-circular set.
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