1. Introduction
The domain of mathematics concerning topology and geometry of topological vector spaces is one of the important branches of functional analysis (see, for example, [
1,
2,
3,
4]). Particularly, a great part of it consists in investigations of bases in Banach spaces (see, for example, [
1,
5,
6,
7,
8,
9,
10,
11] and references therein). Many open problems remain for concrete classes of Banach spaces.
Among them Müntz spaces
play very important role and there also remain unsolved problems (see [
12,
13,
14,
15,
16] and references therein). They are provided as completions of the linear span over the real field
or the complex field
of monomials
with
on the unit segment
by the absolute maximum norm, where
,
. It was K. Weierstrass [
17,
18] who in 1885 had proven his theorem about polynomial approximations of continuous functions on the segment. But the space of continuous functions also possesses the algebraic structure. Later on in 1914 C. Müntz [
19] considered generalizations to spaces which did not have such algebraic structure anymore.
There was a problem about an existence in them bases [
8,
20]. Further a result was for lacunary Müntz spaces which satisfy the restriction
with the countable set
, but in general this problem remained unsolved [
15,
16]. For Müntz spaces of
functions with
this problem was investigated in [
21]. It is worth to mention that the monomials
with
generally do not form a Schauder basis of the Müntz space
.
In this article results of investigations of the author on this problem are presented.
In
Section 2 a Fourier analysis in Müntz spaces
of continuous functions on the unit segment supplied with the absolute maximum norm is studied. For this purpose auxiliary Lemma 2 and Theorem 3 are proved. They are utilized for reducing consideration to a subclass of Müntz spaces
up to isomorphisms of Banach spaces such that a domain
is contained in the set of positive integers
. It is proved that for Müntz spaces subjected to the Müntz and gap conditions their functions belong to Weil-Nagy’s class (about this class of functions see, for example, [
22]). Then the theorem about existence of Schauder bases in Müntz spaces
under the Müntz condition and the gap condition is proven.
All main results of this paper are obtained for the first time.
2. Müntz Spaces
Henceforth the notations and definitions from [
15,
21] are used.
Definition 1. Let Λ be an increasing sequence in the set .
The completion of the linear space containing all monomials
with
and
and
relative to the absolute maximum norm:
is denoted by
, where
, where the symbol
stands for
or
. Particularly, for
it is also shortly written
. We consider also its subspace:
of 1-periodic functions.
Henceforth it is supposed that the set
satisfies the gap condition:
and the Müntz condition:
Lemma 1 and Theorem 1, which are proved below, deal with isomorphisms of Müntz spaces . Utilizing these results reduces our consideration to a subclass of Müntz spaces where a set is contained in the set of natural numbers .
Lemma 1. The Müntz spaces and are isomorphic for every and and a finite subset Ξ in .
Proof. The set
is infinite with
. By virtue of Theorem 9.1.6 [
15] Müntz space
contains a complemented isomorphic copy of
. Therefore,
and
are isomorphic.
The isomorphism of
with
follows from the equality:
for each continuous function
, since the mapping
is a diffeomorphism of the segment
onto itself. Taking
and then
we infer that
and
are isomorphic. ☐
Theorem 1. Suppose that increasing sequences and of positive numbers satisfy the Müntz and gap conditions and for each n. If , where , then and are isomorphic as Banach spaces.
Proof. There are isometric linear embeddings of
and
into
. Consider a sequence of sets
. Properties of the sequences
include:
for each
and
, where
;
for each
and
;
is an enumeration of the non-zero numbers of the form
by elimination of zero terms, also
is a monotone increasing sequence with:
For more details see (1–4) in the proof of Theorem 1 in [
21].
For each
we consider the power series
, where the power series expansion
converges for each
, since
f is analytic on
(see [
14,
15]). Then we infer that:
so that
is a monotone decreasing sequence in
n and hence:
according to Dirichlet’s criterium for each
. Therefore, the function
has a continuous extension onto
and:
since the mapping
is an order preserving diffeomorphism of
onto itself. Thus the series
converges on
. Analogously to each
there corresponds
which is continuous on
.
This implies that there exists a linear isomorphism
of
with
so that
,
. Take the sequence of operators
. The space
is complete and the sequence
operator norm converges to an operator
so that
, since
satisfies the conditions of this theorem and:
where
I denotes the identity operator. Therefore, the operator
S is invertible. From the conditions on
it follows that
. ☐
Remark 1. Next we recall necessary definitions and notations of the Fourier approximation. Then the auxiliary Proposition 1 about the weak -space is given. This proposition is used to prove Theorem 2 about the property that functions in a Müntz space satisfying the Müntz and gap conditions belong to Weil-Nagy’s class. For this purpose in the space of continuous functions is considered its subspace:of 1-periodic functions. Let
be a lower triangular infinite matrix with matrix elements
having values in the field
or
so that
for each
, where
are nonnegative integers. To each 1-periodic function
in the space
is counterposed a trigonometric polynomial:
where
and
are the Fourier coefficients of a function
, whilst on
the Lebesgue measure is considered.
For measurable 1-periodic functions
h and
g their convolution is defined whenever it exists:
The approximation methods by trigonometric polynomials use integral operators provided with the help of the convolution. We recall it briefly (for more details see [
22,
23,
24,
25]). We consider summation methods in the space of continuous periodic functions. Putting the kernel of the operator
to be:
one gets:
The norms of these operators are well-known:
where
and
denote norms on Banach spaces
and
respectively, while
is a marked real number. These numbers
are called Lebesgue constants of a summation method (see also [
22,
23]).
Henceforth, we consider spaces of real-valued functions if something other will not be specified, since an existence of a Schauder basis in the Müntz space over the real field implies its existence in the corresponding Müntz space over the complex field .
Definition 2. For a function by or is denoted its Fourier series with coefficients and :is the approximation precision of f by the Fourier series , where:is the partial Fourier sum approximating a Lebesgue integrable 1-periodic function f on . If the following function:
belongs to the space
of all Lebesgue integrable (summable) functions on
, then
is called the Weil
derivative of
f, where
is a sequence of non-zero numbers in
and
is a real parameter.
Let for a Banach space
of some functions on
:
(see in more details Notation 2 and Definition 2 in [
21]).
In particular, let
(or
for short) be the space of all continuous 1-periodic functions
f having a continuous Weil derivative
,
and considered relative to the absolute maximum norm and such that:
Particularly, for
there is the Weil-Nagy class:
Then let:
where
is described at the beginning of these Definitions 2:
where
is given just above, while
and
are described just below, where a set
X is contained in
:
denotes the family of all trigonometric polynomials
of degree not greater than
(see the definitions in more details in [
22]).
The family of all Lebesgue measurable functions
satisfying the condition:
is called the weak
space and denoted by
, where
notates the Lebesgue measure on the real field
,
,
(see, for example, §9.5 in [
26], §IX.4 in [
27]).
By
is denoted the set of all pairs
, for which:
is the Fourier series of some function belonging to
. Then
denotes the family of all positive sequences
tending to zero with
for each
k so that the series:
converges.
Proposition 1. Suppose that an increasing sequence of natural numbers satisfies the Müntz condition. If , then .
Proof. The
proof is similar to that of Proposition 1 in [
21] with the following modifications. Consider any
. From [
14] (or see Theorem 6.2.3 and Corollary 6.2.4 in [
15]) it follows that
f is analytic on the unit open disk
in
with center at zero and the series:
converges on
, where
is an expansion coefficient for each
.
Using the Riemann integral we have that:
due to Newton-Leibnitz’ formula (see, for example, §II.2.6 in [
28]), since
is continuous on
.
By virtue of the uniqueness theorem for holomorphic functions (see, for example, II.2.22 in [
29]), applied to the considered case, if a nonconstant holomorphic function
g on
has a set
of zeros in
, then either
is finite or infinite with the unique limit point 1. Then we take a linear function
with real coefficients
and
, put
and choose
and
so that
.
On the other hand,
and hence
f is nonconstant. The case
is trivial. So there remains the variant when
is nonconstant. Denote by
zeros in
of
of odd order so that
for each
. Therefore:
for each
according to Theorem II.2.6.10 in [
28]. If
is a finite set, then from Formulas
and
it follows that
and hence
.
Consider now the case when the set
is infinite. We take a convex connected domain
V such that
V is canonically closed,
,
,
for each
,
for each
and
,
, where
and
denote the closure and the interior of a set
A in the complex field
. According to Cauchy’s formula 21
in [
29]:
for each
, where
is a rectifiable path encompassing once a point
z in the positive direction so that
, for example, a circle with center at
z. A set
V can be taken as the disc
. For each
a circle can be chosen with center at
x and of radius
with
while
. Using the homotopy theorem and the continuity of
f on the compact disc
V one can take simply the circle
. Since
due to the Weierstrass theorem (see Vol. 1, Part III, Ch. 1, §12 in [
28]), then from the estimate of the Cauchy integral (see Ch. II, §7, subsection 24 in [
29]) it follows that:
for each
, hence
and consequently
. Therefore, from
we infer that:
where
denotes the Lebesgue measure on
. The latter means that
. ☐
Theorem 2. Let an increasing sequence of natural numbers satisfy the Müntz condition and let . Then for each there exists so that , where for each and is 1-periodic on .
Proof. We have that , since for each n. Therefore, we consider on and take its 1-periodic extension v on .
According to Proposition 1.7.2 [
22] (or see [
23]) a function
h belongs to
if and only if there exists a function
which is 1-periodic on
and Lebesgue integrable on
such that:
where
.
We take a sequence
given by (3) in Remark 1 or see Formula
in [
21] so that:
and write for short
instead of
. Under these conditions the limit exists:
in
norm for each
according to Chapters 2 and 3 in [
22] (see also [
23,
30]).
Put
for all
. Then for
we get that
(see the proof of Theorem 2 in [
21]).
With the help of Proposition 1 and Formula
we define the function
such that:
By virtue of the weak Young inequality (see Theorem 9.5.1 in [
26], §IX.4 in [
27]) and Proposition 1 this function
s is in
.
In view of Formula I
in [
22] if
, then
, where
. Therefore
and
according to Formula
, where
for each
. Thus
. For
such that
similarly
. On the other hand,
v is analytic on
, 1-periodic and continuous on
, consequently,
s is analytic on
and 1-periodic. Therefore, from the latter and Formulas (1)–(3) it follows that
and hence
. ☐
Lemma 2. Let Λ be an increasing sequence of natural numbers satisfying the Müntz condition. Define the subset Y of the unit ball of :
Then for each 0 <
γ < 1
a positive constant ω =
ω(
γ)
exists so that:
for each natural number Proof. Let and put for all . Suppose that the 1-periodic extension v of belongs to Y and let . By Theorem 2 it follows that .
Then estimate
follows from Theorems 3.12.3 and 3.12.3’ in [
22]. ☐
Lemma 3. If and (see at the end of Definitions 2), then is the Banach space relative to the norm given by the formula: Proof. Using the notation of Definitions 2 (see the notation 2 in [
21]) we have that
is the
-linear space and hence
is such also as the kernel of the linear functional
, since each
is integrable. Therefore, the assertion of this lemma follows from Propositions I.8.1 and I.8.3 [
22], since each
has the convolution representation:
for each
, but
for each
, while the convolution
is continuous for each
and
so that
where
is given in Definition 2. ☐
Theorem 3. If an increasing sequence Λ of positive numbers satisfies the Müntz condition and the gap condition, then the Müntz space has a Schauder basis.
Proof. By virtue of Lemma 1 and Theorem 1 it is sufficient to prove an existence of a Schauder basis in the Müntz space for . According to Definition 1 and the proof of Lemma 1 the Banach spaces and are isomorphic.
The functional:
is continuous on
, where
and
satisfy conditions of Lemma 3. Then
. Therefore,
.
In view of Theorem 6.2.3 and Corollary 6.2.4 [
15] each function
has an analytic extension on
and hence:
are the convergent series on the unit open disk
in
with center at zero, where
and
,
,
,
for each
.
Take the finite dimensional subspace in , where . Due to Lemma 1 the Banach space exists and is isomorphic with .
Consider the trigonometric polynomials
for
, where
(see Formula
in [
21] and Remark 1 above). Put
to be the completion in
of the linear span
, where
,
,
.
There exists a countable subset in X such that with for each and so that is dense in X, since X is separable. From Formulas and and Theorem 2 and Lemmas 2 and 3 we infer that a countable set K and a sufficiently large natural number exist so that the Banach space is isomorphic with and , where and . Thus the Banach space is the completion of the real linear span of a countable family of trigonometric polynomials .
Without loss of generality this family can be refined by induction such that is linearly independent of over for each . With the help of transpositions in the sequence , the normalization and the Gaussian exclusion algorithm we construct a sequence of trigonometric polynomials which are finite real linear combinations of the initial trigonometric polynomials and which satisfy the conditions:
for each l;
the infinite matrix having
l-th row of the form
for each
is upper trapezoidal (step), where:
with
and
, where
,
, or
when
;
for each
and
.
Then as
X and
Y in Proposition 2 of [
21] we take
and
. In view of the aforementioned proposition and Lemma 1 a Schauder basis exists in
and hence also in
. ☐