Approximation and Schauder bases in M\"untz spaces $M_{\Lambda ,C}$ of continuous functions

In this article M\"untz spaces $M_{\Lambda ,C}$ of continuous functions supplied with the absolute maximum norm are considered. An approximation of functions in M\"untz spaces $M_{\Lambda ,C}$ of continuous functions by Fourier series is studied. An existence of Schauder bases in M\"untz spaces $M_{\Lambda ,C}$ is investigated.

defined as completions of the linear span over the real field R or the complex field C of monomials t λ with λ ∈ Λ on the segment [0, 1] relative to the absolute maximum norm, where Λ ⊂ [0, ∞), t ∈ [0, 1]. It was K. Weierstrass who in 1885 had proved his theorem about polynomial approximations of continuous functions on the segment. But the space of continuous functions possesses the algebra structure. Later on in 1914 C. Müntz had considered generalizations so that his spaces generally had not such algebraic structure.
A problem was whether they have bases [10,27]. Then a progress was for Müntz spaces satisfying the lacunary condition lim n→∞ λ n+1 /λ n > 1 with countable Λ, but in general this problem was unsolved [9,28]. It is worth to mention that the monomials t λ with λ ∈ Λ generally do not form a Schauder All main results of this paper are obtained for the first time. They can be used for further studies of function approximations and geometry of Banach spaces. This is important not only for progress of mathematical analysis and functional analysis, but also in different applications including measure theory and stochastic processes in Banach spaces.
To avoid misunderstanding we first present our notation and definitions.
where −∞ < a < b < ∞, while F = R is the real field or F = C the complex field.
Then L p ((a, b), F) denotes the Banach space of all Lebesgue measurable functions f : (a, b) → F possessing a norm as defined by the Lebesgue integral: Let Q = (q n,k ) be a lower triangular infinite matrix with matrix elements q n,k having values in the field F = R or F = C so that q n,k = 0 for each k > n, where k, n are nonnegative integers. To each 1-periodic function f : where a k = a k (f ) and b = b k (f ) are the Fourier coefficients of a function f (x), whilst on R the Lebesgue measure is considered.
For measurable 1-periodic functions h and g their convolution is defined whenever it exists: Putting the kernel of the operator U n to be: (3) U n (x, Q) := q n,0 2 + n k=1 q n,k cos(2πkx) one gets The norms of these operators are: where * C and * L 1 denote norms on Banach spaces C([a, a + 1], F) and L 1 ([a, a + 1], F) respectively, while a ∈ R is a marked real number. These numbers L n (Q) are called Lebesgue constants of a summation method.
uniformly in x ∈ [a, a + 1] will be considered.
For each f ∈ M Υ k ,C we consider the power series f 1 (t) = ∞ l=1 a n t µ k+1,n , where the power series decomposition f (t) = ∞ l=1 a n t µ k,n converges for each 0 ≤ t < 1, since f is analytic on [0, 1) (see [5,9]). Then we infer that so that u n (t) is a monotone decreasing sequence by n and hence according to Dirichlet's criterium for each 0 ≤ t < 1. Therefore, the function f 1 (t) has the continuous extension onto [0, 1] and since the mapping t → t 2 is the order preserving diffeomorphism of [0, 1] onto itself. Thus the series ∞ l=1 a n t µ k+1,n converges on [0, 1). Similarly to each This implies that there exists the linear isomorphism T k of M Υ k ,C with Next we take the sequence of operators The space M Λ∪Υ,C is complete and the sequence {S n : n} converges in the operator norm uniformity to an operator S : M Λ,C → M Λ∪Υ,C so that where I denotes the unit operator. Therefore, the operator S is invertible.
6. Remark. In view of Lemmas 3, 4 and Theorem 5 it suffices to consider a set Λ satisfying the gap and Müntz conditions such that Λ ⊂ N up to an isomorphism of Müntz spaces.
Next we recall necessary definitions and notations of the Fourier approximation. Then auxiliary Proposition 10 is given which is used for proving Theorem 11 about the property that for Müntz spaces satisfying the Müntz and gap conditions their functions belong to Weil-Nagy's class.
7. Notation. Suppose that (ψ(k) : k ∈ N) is a sequence of non-zero numbers tending to zero, β is a marked real number. By F is denoted the set of all pairs (ψ, β), for which is the Fourier series of some function belonging to L 1 [0, 1]. Then F 1 denotes the family of all positive sequences (ψ(k) : k ∈ N) tending to zero with converges.
An approximation of a function f by the Fourier series S(f, x) is estimated by the function is the partial Fourier sum. That is a trigonometric polynomial approximating a summable (i.e. Lebesgue integrable) 1-periodic function f ∈ L 1 [0, 1].

8.
Definitions. Let f ∈ L 1 [a, a + 1] and S(f ) be its Fourier series with coefficients a k and b k , let also ψ(k) be an arbitrary sequence real or complex.
If the function Let for a Banach space N of some functions on [a, a + 1]: where (ψ(k) : k) is a sequence with non-zero elements for each k and β is a real parameter.
In particular, let C ψ β M[a, a + 1] (or C ψ β [a, a + 1] for short) be the space of all continuous 1-periodic functions f having a continuous Weil derivative 1] and considered relative to the absolute maximum norm and such that Particularly, for ψ(k) = k −r there is the Weil-Nagy class denotes the family of all trigonometric polynomials T n−1 of degree not greater than n − 1. Proof. Let f ∈ M Λ,C [0, 1]. In view of Theorem 6.2.3 and Corollary 6.2.4 [9] a function f is analytic onḂ 1 (0) and the series (1) f (z) = ∞ n=1 a n z λn converges onḂ 1 (0), whereḂ r (x) := {y : y ∈ C, |y − x| < r} denotes the open disk in C of radius r > 0 with center at x ∈ C, where a n ∈ F is an expansion coefficient for each n ∈ N. That is, the function f has a holomorphic univalent extension from [0, 1) onḂ 1 (0), since Λ ⊂ N (see Theorem 20.5 in [30]).
Remind that for Lebesgue measurable functions f : for each x > 0 whenever this integral exists, where χ A denotes the characteristic function of a subset A in R such that χ A (y) = 1 for each y ∈ A, also χ A (y) = 0 for each y outside A, y ∈ R \ A. In particular, T ] (t)dt (see also [8,13]). This can be applied to formula 1(2) putting α = 0 there and with the help of the equality for each 0 ≤ x ≤ 1 and 1-periodic functions f and g and using also that 1] for the considered here types of norms for each Mention that according to the weak Young inequality (4) ξ * η r ≤ K p,q ξ p η q,w for each ξ ∈ L p and η ∈ L q,w , where 1 ≤ p, q, r ≤ ∞ and p −1 + q −1 = 1 + r −1 , K p,q > 0 is a constant independent of ξ and η (see theorem 9.5.1 in [6], §IX.4 in [26]).

Proof. Due to Theorem 11 the inclusion is valid
Then estimate (1) follows from Theorems 3.12.3 and 3.12.3' in [32].
14. Proposition. Let X be a Banach space over R and let Y be its Banach subspace so that they fulfill conditions (1 − 4) below: (1) there is a sequence (e i : i ∈ N) in X such that e 1 , ..., e n are linearly independent vectors and e n X = 1 for each n and (2) there exists a Schauder basis (z n : n ∈ N) in X such that z n = n k=1 b k,n e k for each n ∈ N, where b k,n are real coefficients; (3) for every x ∈ Y and n ∈ N there exist x 1 , ..., x n ∈ R so that x i e i X ≤ s(n) x , where s(n) is a strictly monotone decreasing positive function with lim n→∞ s(n) = 0 and (4) u n = k(n) l=m(n) u n,l e l , where u n,l ∈ R for each natural numbers k and l, where a sequence (u n : n ∈ N) of normalized vectors in Y is such that its real linear span is everywhere dense in Y and 1 ≤ m(n) ≤ k(n) < ∞ and m(n) < m(n+ 1) for each n ∈ N.
Then Y has a Schauder basis.
Proof. Without loss of generality one can select and enumerate (5) vectors u 1 ,...,u n so that they are linearly independent in Y for each natural number n. By virtue of Theorem (8.4.8) in [23] their real linear span span R (u 1 , ...., u n ) is complemented in Y for each n ∈ N. Put L n,∞ := cl X span R (u k : k ≥ n) and L n,m := cl X span R (u k : n ≤ k ≤ m), where cl X A denotes the closure of a subset A in X, where span R A denotes the real linear span of A. Since Y is a Banach space and u k ∈ Y for each k, then L n,∞ ⊂ Y and L n,m ⊂ Y for each natural numbers n and m. Then we infer that L n,j ⊂ span R (e l : m(n) ≤ l ≤ k n,j ), where k n,j := max(k(l) : n ≤ l ≤ j).
Take arbitrary vectors f ∈ L 1,j and g ∈ L j+1,q , where 1 ≤ j < q. Therefore, there are real coefficients f i and g i such that i=a f i e i := 0, when a > b. When 0 < δ < 1/4 and s(m(j) + 1) < δ we infer using the triangle X /(1−δ) ≤ δs(m(j+1)−1) f X /(1− δ) for the best approximation h of f [j+1] in L j+1,∞ , since m(j) < m(j + 1) for each j. Therefore, the inequality − g X and s(n) ↓ 0 imply that there exists n 0 such that the inclination of L 1,j to L j+1,∞ is not less than 1/2 for each j ≥ n 0 . Condition (4) implies that L 1,n 0 is complemented in Y . By virtue of Theorem 1.
There exists a countable subset {f n : n ∈ N} in X such that f n = D ψ,β * g n with g n ∈ L(0, 1) for each n ∈ N and so that span R {f n : n ∈ N} is dense in X, since X is separable. From Formulas (1, 2) and Theorem 11 and Lemma 12 we infer that a countable set K and a sufficiently large natural number n 0 exist so that the Banach space Y K,n 0 is isomorphic with (X ⊖ X n 0 ) and Y K,n 0 | (0,1) ⊂ W γ β,C (0, 1), where 0 < γ < 1 and β = 1 − γ. Thus the Banach space Y K,n 0 is the C[0, 1] completion of the real linear span of a countable family (s l : l ∈ N) of trigonometric polynomials s l .
Without loss of generality this family can be refined by induction such that s l is linearly independent of s 1 , ..., s l−1 over F for each l ∈ N. With the help of transpositions in the sequence {s l : l ∈ N}, the normalization and the Gaussian exclusion algorithm we construct a sequence {r l : l ∈ N} of trigonometric polynomials which are finite real linear combinations of the initial trigonometric polynomials {s l : l ∈ N} and satisfying the conditions (3) r l C(0,1) = 1 for each l; (4) the infinite matrix having l-th row of the form ..., a l,k , b l,k , a l,k+1 , b l,k+1 , ... [a l,k cos(2πkx) + b l,k sin(2πkx)] with a 2 l,m(l) + b 2 l,m(l) > 0 and a 2 l,n(l) + b 2 l,n(l) > 0, where 1 ≤ m(l) ≤ n(l), deg(r l ) = n(l), or r 1 (x) = a 1,0 2 when deg(r 1 ) = 0; a l,k , b l,k ∈ R for each l ∈ N and 0 ≤ k ∈ Z.