Approximation in M\"untz spaces $M_{\Lambda ,p}$ of $L_p$ functions for $1

M\"untz spaces satisfying the M\"untz and gap conditions are considered. A Fourier approximation of functions in the M\"untz spaces $M_{\Lambda ,p}$ of $L_p$ functions is studied, where $1<p<\infty $. It is proved that up to an isomorphism and a change of variables these spaces are contained in Weil-Nagy's class. Moreover, existence of Schauder bases in the M\"untz spaces $M_{\Lambda ,p}$ is investigated.

example, [12,14,18]- [22,32] and references therein). It is not surprising that for concrete classes of Banach spaces many open problems remain, particularly for the Müntz spaces M Λ,p , where 1 < p < ∞ (see [1]- [6], [10,30] and references therein). These spaces are defined as completions of the linear span over R or C of monomials t λ with λ ∈ Λ on the segment [0, 1] relative to the L p norm, where Λ ⊂ [0, ∞), t ∈ [0, 1]. In his classical work K. Weierstrass had proved in 1885 the theorem about polynomial approximations of continuous functions on the segment. But the space of continuous functions also forms the algebra. Generalizations of such spaces were considered by C. Müntz in 1914 such that his spaces had not the algebra structure. The problem was whether they have bases. Then a progress was for lacunary Müntz spaces satisfying the condition lim n→∞ λ n+1 /λ n > 1 with a countable set Λ, but in its generality this problem was not solved [10]. It is worth to mention that the system {t λ : λ ∈ Λ} itself does not contain a Schauder basis for a nonlacunary set Λ satisfying the Müntz and gap conditions.  Lemma 14). There is proved that under the Müntz condition and the gap condition Schauder bases exist in the Müntz spaces M Λ,p , where 1 < p < ∞.
All main results of this paper are obtained for the first time. They can be used for further investigations of function approximations and geometry of Banach spaces. It is important not only for development of mathematical analysis and functional analysis, but also in their many-sided applications.
2 Approximation in Müntz L p spaces.
To avoid misunderstandings we first remind necessary definitions and notations.
1. Notation. Let C([α, β], F) denote the Banach space of all continuous functions f : [α, β] → F supplied with the absolute maximum norm where −∞ < α < β < ∞, F is either the real field R or the complex field C.
Suppose that Q = (q n,k ) is a lower triangular infinite matrix with real matrix elements q n,k satisfying the restrictions: 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).
For measurable 1-periodic functions h and g their convolution is defined whenever it exists by the formula: Putting the kernel of the operator U n to be: we get The norms of these operators are: which are constants of a summation method, where * E denotes a norm on As usually span F (v k : k) will stand for the linear span of vectors v k over a field F.
Henceforward the Fourier summation methods prescribed by sequences of operators {U m : m} which converge on E in the E norm will be considered.
Henceforth it is supposed that the set Λ satisfies the gap condition The completion of the linear space containing all monomials at λ with a ∈ F and λ ∈ Λ and t ∈ [α, β] relative to the L p norm is denoted by it is completed relative to the C norm. Shortly they will also be written as M Λ,p or M Λ,C respectively for α = 0 and β = 1, when F is specified.
Before subsections about the Fourier approximation in Müntz spaces aux-  Proof. We have that a sequence {λ k : k ∈ N} is strictly increasing and satisfies the gap condition and hence lim n λ n = ∞. We order a set Ξ ∪ (αΛ + β) into a strictly increasing sequence also.
In virtue of Theorem 9.1.6 [10] the Müntz space M Λ,p contains a complemented isomorphic copy of l p , consequently, M Λ,p and M Ξ∪Λ,p are linearly topologically isomorphic as normed spaces.
Then from Lemma 3 taking α > 0 we deduce that for each f ∈ M Λ,p and hence M αΛ,p is isomorphic with M Λ,p . Considering the set Λ 1 = Λ ∪ { β α } and then the set αΛ 1 we get that M Λ,p and M αΛ+β,p are linearly topologically isomorphic as normed spaces as well.
Therefore, for each f ∈ M Υ k ,p we consider the power series f 1 (t) = ∞ n=1 a n t υ k+1,n , where the power series decomposition f (t) = ∞ n=1 a n t υ k,n converges for each 0 ≤ t < 1, since f is analytic on [0, 1). Then we infer that so that u l (t) is a monotone decreasing sequence by l and hence according to Dirichlet's criterium (see, for example, [8]) for each 0 ≤ t < 1, where θ = θ k+1 . Therefore, the function f 1 (t) is analytic on [0, 1) and [10] and Thus the series ∞ n=1 a n t υ k+1,n converges on [0, 1). Inequality (5) implies that the linear isomorphism The space M Λ∪Υ,p is complete and the sequence {S n : n} operator norm converges to an operator S : and p ≥ 1, where I denotes the unit operator. Therefore, the operator S is invertible. On the other hand, from Conditions (1 − 4) it follows that Now we recall necessary definitions and notations of the Fourier approximation theory and then present useful lemmas.
6. Notation. Henceforth F denotes the set of all pairs (ψ, β) satisfying the conditions: (ψ(k) : k ∈ N) is a sequence of non-zero numbers for which lim k→∞ ψ(k) = 0 the limit is zero, β is a real number, also is the Fourier series of some function from L 1 [0, 1]. By F 1 is denoted the family of all positive sequences (ψ(k) : k ∈ N) tending to zero with is the approximation precision of f by the Fourier series S(f, x), where is the partial Fourier sum approximating a Lebesgue integrable 1-periodic function f on (0, 1).

7.
Definition. Suppose that f ∈ L 1 (α, α + 1) and S[f ] is its Fourier series with coefficients a k = a k (f ) and b k = b k (f ), while ψ(k) is an arbitrary sequence real or complex. If the function belongs to the space L(α, α + 1) of all Lebesgue integrable (summable) func- ) and the notation W r β,p can be used instead of L ψ β,p in this case. Put partic- where X is a subset in L p (α, α + 1) = L p ((α, α + 1), R), denotes the family of all trigonometric polynomials T n−1 of degree not greater than n − 1.
Proof. There is the natural embedding of L p (a, b) into L p (c, d) when notates the characteristic function of a set A. Since Q α < 1, then the operator I − Q α is invertible (see [13]).

Thus from Formulas
11. Note. We remind the following definition: the family of all Lebesgue measurable functions f : (a, b) → R satisfying the condition is called the weak L s space and denoted by L s,w (a, b), where µ notates the Lebesgue measure on the real field R, 0 < s < ∞, (a, b) ⊂ R (see, for example, §9.5 [7], §IX.4 [26], [29]).
The following proposition 12 is used below in theorem 13 to prove that functions of Müntz spaces M Λ,p for Λ satisfying the Müntz condition and the gap condition belong to Weil-Nagy's class, where 1 < p < ∞.
Evidently, 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 T ] (t)dt (see also [8,13]). This is applicable 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 p ≤ K r,s ξ r η s,w for each ξ ∈ L r and η ∈ L s,w , where 1 ≤ p, r ≤ ∞, 0 < s < ∞ and r −1 + s −1 = 1 + p −1 , K r,s > 0 is a constant independent of ξ and η (see theorem 9.5.1 in [7], §IX.4 in [26]).
In virtue of formula (3), the weak Young inequality (4) and Proposition 12 there exists a function s in L p (0, 1) so that where β = 1 − γ. Therefore φ v 0 ,γ,β = s and D ψ β v 0 = s according to (1) and (3). Thus v 0 ∈ W γ β L p (0, 1). Below Lemma 14 and Proposition 15 are given. They are used in subsection 16 for proving existence of a Schauder basis. On the other hand, Theorem 13 is utilized that to prove Lemma 14.
15. 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 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 On the other hand, 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 − 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 . In virtue of Theorem 1. For example, Cesaro's summation method of order 1 can be taken to which Fejér kernels F n correspond so that the limit lim n→∞ F n * f = f converges in L p (0, 1) (see Theorem 19.1 and Corollary 19.2 in [33]). That is, there exists a Schauder basis z n in L p (0, 1) such that z 2n (t) = a 0,2n + [ n−1 k=1 (a k,2n cos(2πkt) + b k,2n sin(2πkt)] + a n,2n cos(2πnt) and z 2n+1 (t) = a 0,2n+1 + n k=1 (a k,2n+1 cos(2πkt) + b k,2n+1 sin(2πkt)) for every t ∈ (0, 1) and n ∈ N, where a k,j and b k,j are real expansion coefficients.
There exists a countable subset {f n : n ∈ N} in Z Λ,p,2,δ such that f n • σ = 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 Z Λ,p,α,δ , since Z Λ,p,2,δ is separable. Using Formulas (1, 2), Proposition 12 and Lemma 14 we deduce 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 (Z Λ,p,2,δ ⊖ (I − Q 2 )X n 0 ) and Y K,n 0 | (0,1) ⊂ W γ β L p (0, 1), where 0 < γ < 1 and β = 1 − γ. Therefore, by the construction above the Banach space Y K,n 0 is the L p 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 Lp(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.