Isomorphic Classification of Reflexive Müntz Spaces

The article is devoted to reflexive Müntz spaces MΛ,p of Lp functions with 1 < p < ∞. The Stieltjes transform and a potential transform are studied for these spaces. Isomorphisms of the reflexive Müntz spaces fulfilling the gap and Müntz conditions are investigated.

The reflexive Müntz spaces M Λ,p (F) are defined as completions of a F-linear span of the monomials t λ with λ ∈ Λ on the segment [0, 1] relative to the L p norm, where Λ ⊂ [0, ∞), t ∈ [0, 1], 1 < p < ∞, where F is either the real field F = R or the complex field F = C.It is worth mentioning that generally monomials t λ do not form a Schauder basis in the Müntz space M Λ,p .For a long time, whether or not they have Schauder bases remained a problem [1,2,8,16].
This article is devoted to the reflexive Müntz spaces M Λ,p fulfilling the gap and Müntz conditions.For this purpose, the Stieltjes transform and a potential transform are studied (see Propositions 1 and 2 and Corollary 2).This study is based on certain useful properties of the Fourier transform in the reflexive Müntz spaces with a change of the variable (Lemmas 4, 5, and 6 and Corollary 1).Their Banach space geometry is investigated in Propositions 3 and 4 and Theorem 1.A relation with the Banach space l p = l p (F) over the field F is elucidated.
It is proven in Theorem 1 that under the aforementioned conditions M Λ,p (F) is isomorphic with l p (F).
All main results of this paper are obtained for the first time.They can be used for further studies of Banach space geometry, measures and stochastic processes in Banach spaces, approximations of functions.

The Müntz M Λ,p Spaces
To avoid misunderstandings, we first give our notation and some useful Lemmas 1-3.

Notation 1.
As is usual, L p (Ω, F , ν, F) denotes the Banach space of all ν-measurable functions f : Ω → F having a finite norm where 1 < p < ∞ is a marked number, F is a σ-algebra of subsets in a set Ω, ν is a σ-finite nonnegative measure on F , either F is the real field R or F stands for the complex field C.Then, the closure of the F-linear span These spaces are also denoted by M Λ,E , where E = L p ([0, 1], F).
Henceforward, measures are considered on Borel σ-algebras, and for brevity short B([a, b]) will be omitted from the notation of the corresponding Banach spaces.
Henceforth, it is supposed that Λ is an increasing sequence contained in (0, ∞) and satisfying the gap condition inf and the Müntz condition Proof.The change of the variable Proof.The measures µ and ν are equivalent; consequently, these Banach spaces are linearly topologically isomorphic (see also [19,22]).
Next, the Fourier transform is studied in the reflexive Müntz spaces with a change of the variable.

Lemma 4.
A continuous linear operator exists, induced by the Fourier sine transform F s since ν(dx) = e −x µ(dx) so that the operator is the isometry from L p ([0, ∞), ν, R) onto L p ([0, ∞), µ, R).We consider the odd extension There is the natural embedding (see Section 2).Then, the sine transform is continuous from L p ([0, ∞), µ, R) into L q (R, µ, R) according to Theorem 33.5 [23], where i = √ −1.Indeed, the latter theorem states that if h ∈ L p (R, µ, C), then the sequence of functions and the Fourier transform is non-expanding: At the same time, In the case 1 ≤ p ≤ 2, the Fourier transform in L p spaces is defined as usual.For p > 2, the Fourier transform in L p is understood in the sense of the dual pair (L q , L p ) so that (F( f ), g) = ( f , F(g)) for each f ∈ L q and g ∈ L p , where 1/q + 1/p = 1 (see, for example, [23][24][25][26] and references therein).
For the dual pair (L p (R, µ, C); L q (R, µ, C)) of Banach spaces, a continuous adjoint operator F to F exists (see Proposition 1 and Corollary 6 in Section 8.6 [3]).On the other hand, we infer that where [φ, f ] denotes the value of a functional φ at f , where the notation is used while F stands for the adjoint operator, which exists due to Formulas ( 8), (11), and ( 13) in the proof of Lemma 4 above.

The Potential Transform for Reflexive Müntz Spaces
Lemma 5.For every f is satisfied.
Proof.Recall that, on [0, ∞), the measure ν has density e −x with respect to the Lebesgue measure µ.
The operator B p given by Formula ( 8) is the linear isometry from where 1/p + 1/q = 1.That is, for each continuous linear functional g on L p ([0, ∞), ν, R) there exists a function h ∈ L q ([0, ∞), µ, R) for which g has the form where for any v ∈ L q ([0, ∞), µ, R), where µ denotes the Lebesgue measure on the Borel σ-algebra B(R) as above.The space for every h, g ∈ L 2 (R, µ, C).Therefore, by continuity, for the dual pair (L p , L q ) this equality is also valid for any h ∈ L p (R, µ, C) and g ∈ L q (R, µ, C), where z denotes the conjugated number of a complex number z ∈ C. Using odd extensions of functions and Formulas (11) and (19), we deduce Equality (16).
Further, the Stieltjes transform and a potential transform are investigated in the reflexive Müntz spaces with a change of the variable.Proposition 1.There is the identity for each k ∈ N 0 and g ∈ L q ([0, ∞), µ, R).
Proof.This follows from Formula ( 16), since for each a > 0 (see Formulas 2-7-2 on page 49 and S7 on page 518 in [27]) and putting a = k + 1/p in the considered case.
for each k ∈ Λ.
Proof.From Embedding Formulas (10),it follows that for the space (see also Section 9.8 in [15] and Formula (17) above).Thus, Formula (21) follows from Identities ( 16)-( 20), the definition of the Müntz space, Lemma 1 and Condition (2) , since the real linear span is called the potential transform whenever this integral converges, where x ∈ R (see Section 7.2 in [28]).Generally, this integral is considered as the improper integral: It is related with the Stieltjes transform by the change of variables t = y 2 : where s = x 2 , h(t) = g( √ t).In the Stieltjes transform, S(h)(s) generally the complex variable s = σ + iτ ∈ C is considered with σ, τ ∈ R (see Chapter 7 in [29] or [30]).

Remark 3. Let α be a function of bounded variation on each segment
whenever the limit exists, where ρ > 0, If a function α is of bounded variation on each segment [ , R] = {t : ≤ t ≤ R}, where 0 < < R < ∞, and if the limits both exist for some s ∈ C, then one writes Next, we shall use Theorems 13, 14.2 from [28], 2a, 2b, and 7b from [29,30] about the potential and Stieltjes transforms.Lemma 6. Suppose that g is an odd function on R such that g ∈ L q (R, µ, R) and its support is contained in [−2π(m + 1), −2πm] ∪ [2πm, 2π(m + 1)] for some nonnegative integer m, and F s (g)(y) is continuous in the variable y ∈ R, where 1 < q < ∞.Then, its Fourier sine transform coefficients b n (g) are given by the formula: for each n ∈ N, where h(t) = F s (g)( √ t) while t ≥ 0.
Proof.Recall that S([0, ∞)) denotes a space of all infinite differentiable functions h : [0, ∞) → R such that lim x→∞ x n D m h(x) = 0 for all nonnegative integers n and m.As is traditional, the space S([0, ∞)) is supplied with the family of semi-norms ρ n,m (h where n and m are nonnegative integers. Since f provides a continuous linear functional on L p ([0, ∞), µ, R) with 1 < p < ∞, and since the topological dual space of L p is L q , where 1/p + 1/q = 1, then the function f belongs to L q ([0, ∞), µ, R).This means that the value < g, f > of f at g ∈ L p is given by Without loss of generality for each function g ∈ L r ([0, ∞), µ, R) with 1 ≤ r < ∞ we can consider the identically zero extension of g onto (−∞, 0).The Fourier transform is defined on L r (R, µ, R) for 1 ≤ r ≤ 2 as: The Fourier transform is a unitary operator on L 2 , and by the duality has the weakly continuous extension for the dual pair < L p , L q > such that the Parseval identity is satisfied for each g ∈ L p and f ∈ L q (see [24,25]).The sine Fourier transform Thus, Formula (38) follows from Formulas (30)- (32).
Proof.Since the Müntz space X = M Λ,p ([0, ∞), µ, R) is the linear subspace in the normed space Y, the Hahn-Banach Theorem (8.4.7) in [15] implies that each continuous linear functional f on X has a continuous linear extension to space in which the linear subspace of all functionals satisfying the conditions of Proposition 3 is dense.

Proof.
(I).In view of Theorem 2.1 and Lemma 2.2 in [8], up to the Banach spaces isomorphism it is sufficient to consider the case Λ ⊂ N. It is clear that if the theorem is proved over the real field R, then from it the case over the complex field C follows.
In view of Theorem 33.3 [23], if 1 < p ≤ 2 and 1/p + 1/q = 1, f ∈ L p ([2πn, 2π(n + 1)), µ, R) and if â( f ) := {a n ( f ) : n ∈ Z} is the sequence of Fourier coefficients of f , then â( f ) ∈ l q and â( f where Moreover, Theorem 33.4 [23] asserts that if 1 < p ≤ 2 and 1/p is the m-th partial sum of the trigonometric series ∑ ∞ k=−∞ b k e ikx , then the sequence of sums S m (x) tends to a function f ∈ L q ([0, 2π), µ, C) in the L q norm as m tends to infinity; furthermore, b = â( f ) and f L q ≤ b l p . (46) For odd real-valued functions, we consider the partial sums of the form where a k ∈ R. On the other hand, there exists an isomorphism of l q ⊕ l q with l q .Therefore, there exists a linear projection operator E from l q (R) onto the space where Λ ⊂ N. At the same time, there exists the continuous projection linear operator A from l q (l q (R), N) onto where for a sequence of Banach spaces J n over the same field either R or C, while l q (J, N 0 ) := l q (J n : Its adjoint operator A relative to the dual pair (l q , l p ) is the continuous embedding from l p (l p , N 0 ) into l p (Q , N 0 ) due to Corollary 8.6.4 in [26] and Formulas (47)-(49), since A 2 = A and hence (A ) 2 = (A 2 ) = A , where 1 < p < ∞ and 1/p + 1/q = 1.
By virtue of Theorem (9.11.1) in [15], if M is a closed subspace of a topological vector space X, then (X/M) is algebraically isomorphic to M ⊥ , where X denotes the topological dual space of X.In view of Theorem (9.11.3) in [15], if X and Y are paired topological vector spaces and M is a closed subspace in X, then X/M and M ⊥ are paired spaces and σ(X/M, M ⊥ ) is the quotient topology on X/M induced by σ(X, Y).
Let S denote the operator from l where 1 < q ≤ 2, 1/p + 1/q = 1.We consider its extension from V into L p ([0, ∞), µ, R): If B : H 0 → K is a continuous linear operator from the normed space H 0 into the Banach space K, while H 0 is a dense linear subspace in a normed space H such that the norm on H 0 is inherited from H, then B has a continuous extension from H into K [15].

Conclusions
The results of this paper can be used not only in Banach space geometry, function approximations, but also for periodic function analysis of perturbations to almost periodic functions with trend [32], also for distortions in high-frequency pulse acoustic signals [33].