PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 155–163 DOI: 10.2298/PIM1613155C

APPROXIMATIONS OF PERIODIC FUNCTIONS BY ANALOGUE OF ZYGMUND SUMS IN THE SPACES L^{p(·)}

Stanislav Chaichenko

Abstract. We found order estimates for the upper bounds of the deviations of analogue of Zygmund’s sums on the classes of (ψ; β)-differentiable functions in the metrics of generalized Lebesgue spaces with variable exponent.

1. Definition and formulation of the problem

Let p = p(x) be a 2π-periodic measurable and essentially bounded function and let L^{p(·)} be space of measurable 2π-periodic functions f such that

∫_{−π}^{π} |f(x)|^{p(x)} dx < ∞.

If \underline{p} := essinf_{x} |p(x)| ⩾ 1 and p̄ := esssup_{x} |p(x)| < ∞, then L^{p(·)} are Banach spaces [16] (see also [8]) with the norm, which can be given by

∥f∥_{p(·)} := inf \{α > 0 : ∫_{−π}^{π} | \frac{f(x)}{α} |^{p(x)} dx ⩽ 1\}.

Here are some definitions which will be used in the statement and proof of the results of this article.

Definition 1.1. It is said that a function p = p(x) satisfies the Dini–Lipschitz condition of order γ, if ω(p; δ)( ln \frac{1}{δ})^{γ} ⩽ K = const, 0 < δ < 1, where

ω(p; δ) = sup_{x_{1} ,x_{2} ∈[−π;π]} {|p(x_{1}) − p(x_{2})| : |x_{1} − x_{2}| ⩽ δ}.

The set of 2π-periodic exponents p = p(x) > 1, satisfying the Dini–Lipschitz condition of order γ ⩾ 1 in the period, is denoted by P^{γ}. Obviously, if p ∈ P^{γ}, then \underline{p} > 1 and p̄ < ∞. In the work [16] shown that when 1 < \underline{p}, p̄ < ∞, space L^{q(·)}, where q(x) = \frac{p(x)}{p(x)−1} , is conjugate for L^{p(·)} and for arbitrary functions f ∈ L^{p(·)} and g ∈ L^{q(·)} an analogue of the classical Hölder inequality is true:

2010 Mathematics Subject Classification: 46E30, 42A10, 41A17, 41A20, 41A25.

Key words and phrases: Lebesgue spaces with variable exponent, analogue of Zygmund sums, (ψ; β)-derivative.

Partially supported by State Fund of Fundamental Research of Ukraine (state registration number: N 0112U000372).

Communicated by Gradimir Milovanovć.

∫_{−π}^{π} |f(x)g(x)| dx ⩽ K_{p,q}∥f∥_{p(·)}∥g∥_{q(·)}, (K_{p,q} ⩽ 1/\underline{p}+ 1/\underline{q}),

which, in particular, implies embedding: L^{p(·)} ⊂ L, where L is space of 2π-periodic Lebesgue integrable on the period functions.

The spaces L^{p(·)} are called generalized Lebesgue spaces with variable exponent. It is clear, that if p = p(x) ≡ const > 0, spaces L^{p(·)} coincide with the classical Lebesgue spaces L_{p}. In its turn, if p̄ < ∞, spaces L^{p(·)} are a special case of the so-called Orlicz–Musielak spaces [10].

For the first time, a Lebesgue space with variable exponent appeared in the literature in the article of W. Orlicz [12]. In the work [11] spaces L^{p(·)} considered as an example of the more general function spaces and, furthermore, have been studied by many authors in different directions. The basic results of the theory of these spaces are available, for example, in [1, 2, 4, 8, 10–12, 14, 16, 18, 19]. Note also that the generalized Lebesgue spaces with variable exponent used in the theory elastic mechanics, the theory of differential operators, variations calculus [3,13,15].

Next, we need the definitions of the (ψ; β)-derivative and the sets L_{β}^{ψ} , which belongs to A. I. Stepanetz [20, pp. 142–143].

Definition 1.2. Let f ∈ L and

S[f] = \frac{a_{0}(f)}{2} + ∑_{k=1}^{∞} (a_{k}(f) coskx + b_{k}(f) sinkx) ≡ ∑_{k=0}^{∞} A_{k}(f, x)

be its Fourier series. Let, further, ψ(k) be arbitrary function of natural argument and β ∈ ℝ. Assume that the series

∑_{k=1}^{∞} \frac{1}{ψ(k)} ( a_{k}(f) cos( kx + \frac{βπ}{2} ) + b_{k}(f) sin( kx + \frac{βπ}{2} ) )

is the Fourier series of some function from L. This function is denoted by f_{β}^{ψ} (·) (or (D_{β}^{ψ} f)(·)) and called (ψ; β)-derivative of a function f(·). The set of functions f(·), satisfying this condition is denoted by L_{β}^{ψ} .

Denote by Ẑ_{n} (f; x) (see [5]) the trigonometric polynomials of the form

(1.1)  Ẑ_{n} (f; x) := \frac{a_{0}(f)}{2} + ∑_{k=1}^{n−1} ( 1 − \frac{ψ(n)}{ψ(k)} ) A_{k}(f; x).

Note that in the case ψ(k) = 1/k^{r}, r > 0, the sums (1.1) are Zygmund’s well known sums

Z_{n}(f; x) := \frac{a_{0}(f)}{2} + ∑_{k=1}^{n−1} ( 1 − ( \frac{k}{n})^{r}) A_{k}(f; x).

In this paper, we study the value

E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} := sup_{f∈L_{β,p(·)}^{ψ}} ∥f − Ẑ_{n} (f)∥_{s(·)}

upper bounds of deviations analogues of Zygmund’s sums Ẑ_{n} (f; x) on the classes L_{β,p(·)}^{ψ} := {f ∈ L_{β}^{ψ} : f_{β}^{ψ} ∈ U_{p(·)}}, where U_{p(·)} := {φ ∈ L^{p(·)} : ∥φ∥_{p(·)} ⩽ 1} is the unit ball of L^{p(·)}.

2. Auxiliary results

In the proof of the main assertions of this work we use the following well-known results.

Theorem 2.1. [19] If p ∈ P^{γ}, then for an arbitrary function f ∈ L^{p(·)} the inequalities hold

(2.1)  ∥S_{n}(f)∥_{p(·)} ⩽ C_{p}∥f∥_{p(·)},
(2.2)  ∥f̃∥_{p(·)} ⩽ K_{p}∥f∥_{p(·)},

where

S_{n}(f; x) = \frac{a_{0}(f)}{2} + ∑_{k=1}^{n−1} (a_{k}(f) coskx + b_{k}(f) sinkx), n = 0, 1, …,

is the Fourier partial sums of the order n of function f, f̃(·) is trigonometric conjugate to f(·) functions, and C_{p}, K_{p} are positive constants which don’t depend on n and f.

From inequality (2.1), it follows that for an arbitrary function f ∈ L^{p(·)}, on condition p ∈ P^{γ}, its Fourier series converges to f in the metric of the spaces L^{p(·)}, that is ∥f − S_{n}(f)∥_{p(·)} → 0, as n → ∞, and the relation holds

E_{n}(f)_{p(·)} ⩽ ∥f − S_{n−1} (f)∥_{p(·)} ⩽ K_{p}E_{n}(f)_{p(·)},

where

E_{n}(φ)_{p(·)} := inf_{t_{n−1} ∈T_{2n−1}} ∥φ − t_{n−1} ∥_{p(·)}, φ ∈ L^{p(·)},

is the best approximation of function φ by subspace T_{2n−1} of trigonometric polynomials of order, not higher than n − 1, and K_{p} is the value which depends only on p = p(x).

Lemma 2.1. [7] Let the sequence μ(k), k = 0, 1, 2, …, satisfies the conditions

ν_{0} = ν_{0}(μ) = sup_{k}|μ(k)| ⩽ C, σ_{0} = σ_{0}(μ) = sup_{m∈ℕ} ∑_{k=2^{m}}^{2^{m+1}} |μ(k + 1) − μ(k)| ⩽ C,

where C is the value which does not depend on k and m. Then, if p ∈ P^{γ}, for a given function f ∈ L^{p(·)} there exists a function F ∈ L^{p(·)} such that the series

\frac{μ(0)a_{0}(f)}{2} + ∑_{k=1}^{∞} μ(k)(a_{k}(f) coskx + b_{k}(f) sinkx)

is the Fourier series of F and the inequality is true

(2.3)  ∥F ∥_{p(·)} ⩽ Kλ∥f∥_{p(·)}, λ = max{ν_{0}, σ_{0}},

where the value K does not depend on the function f.

In the case p = p(x) ≡ const this statement is a well-known lemma of Marcinkievicz for multipliers [9].

We will also use the following theorem of Hardy–Littlewood.

Theorem 2.2. [6] Let 1 < p < s < ∞, p, s = const, α = p^{−1} − s^{−1} and D_{α} (t) := ∑ _{k=1}^{∞} k^{−α} coskt. Then, for an arbitrary function φ ∈ L_{p} the convolution

Φ_{α} (x) = \frac{1}{π} ∫_{−π}^{π} φ(x + t)D_{α} (t)dt

belongs to L_{s}, and ∥Φ_{α} ∥_{s} ⩽ C_{s,p} ∥φ∥_{p}, where C_{s,p} depends on s and p only.

Note that if φ ∈ L_{p} and S[φ] = ∑ _{k=0}^{∞} A_{k}(φ; x), then

S[Φ_{α}] = ∑_{k=0}^{∞} k^{−α} A_{k}(φ; x),

that is Φ_{α} = M_{α} (φ), where M_{α} is an operator-multiplier, which is determined by the sequence of μ_{α} (k) = k^{−α} , k = 0, 1, 2, …, and it acts from L_{p} to L_{s}, where indicators 1 < p < s < ∞, p, s = const, are related by the equation p^{−1} − s^{−1} = α.

3. Approximation by analogue of Zygmund’s sums

We define the sequences μ(k) and μ̃(k), k ∈ ℕ, as follows:

μ(k) = μ_{n,α} (k) := \{ \begin{matrix}k^{α} ψ(n) cos \frac{βπ}{2} , & 1 ⩽ k ⩽ n − 1, \\ k^{α} ψ(k) cos \frac{βπ}{2} , & n ⩽ k, \end{matrix} (3.1)

μ̃(k) = μ̃_{n,α} (k) := \{ \begin{matrix}k^{α} ψ(n) sin \frac{βπ}{2} , & 1 ⩽ k ⩽ n − 1, \\ k^{α} ψ(k) sin \frac{βπ}{2} , & n ⩽ k, \end{matrix} (3.2)

where n ∈ ℕ, α ⩾ 0, β ∈ ℝ. For each fixed α ⩾ 0 we denote by Υ_{α,n} the set of pairs (ψ; β), such that for any positive number n the conditions

(3.3)  ν_{α} (ψ; β; n) := sup_{k}|μ_{n,α} (k)| ⩽ Cν(n)n^{α} < K,

(3.4)  σ_{α} (ψ; β; n) := sup_{m∈ℕ} ∑_{k=2^{m}}^{2^{m+1}} |μ_{n,α} (k + 1) − μ_{n,α} (k)| ⩽ Cν(n)n^{α} < K,

and similar conditions for the function μ̃_{n,α} (k) hold, where ν(n) = sup_{k⩾n} |ψ(k)|, C and K are positive constants uniformly bounded on n.

At first we consider the case when the functions p = p(x) and s = s(x) on the period satisfy the inequality s(x) ⩽ p(x). In our notation, the following assertion is true.

Theorem 3.1. Let (ψ; β) ∈ Υ_{0,n} p, s ∈ P^{γ}, s(x) ⩽ p(x), x ∈ [0; 2π]. Then, for

all n ∈ ℕ the inequality

C_{p,s} ν(n) ⩽ E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩽ K_{p,s} ν(n),

holds, where C_{p,s} and K_{p,s} are some constants depending on p = p(x) and s = s(x) only.

Proof. Suppose first that p(x) ≡ s(x), x ∈ [0; 2π]. For an arbitrary function f ∈ L_{β,p(·)}^{ψ}, the equality is true

f(x) − Ẑ_{n} (f; x) = ∑_{k=1}^{n−1} \frac{ψ(n)}{ψ(k)} A_{k}(f; x) + ∑_{k=n}^{∞} A_{k}(f; x)

= ∑_{k=1}^{∞} μ_{n,0} A_{k}(f_{β}^{ψ} ; x) + ∑_{k=1}^{∞} μ̃_{n,0} Ã_{k}(f_{β}^{ψ} ; x) := M_{0}(f_{β}^{ψ} ) + M̃_{0}(f̃_{β}^{ψ} ), (3.5)

where M_{0} and M̃_{0} are operators-multipliers, which are defined by the sequences (3.1) and (3.2) respectively, α = 0. According to the conditions of the theorem, the couples (ψ; β) belong to the set Υ_{0,n}, therefore, sequence (3.1) and (3.2) satisfy the conditions of Lemma 2.1. Applying this lemma, given inequalities (2.2), (3.3) and (3.4), for an arbitrary function f ∈ L_{β,p(·)}^{ψ} on the basis of the equality (3.5) we find

(3.6)  ∥f(·) − Ẑ_{n} (f; ·)∥_{p(·)} = ∥M_{0}(f_{β}^{ψ} ) + M̃_{0}(f̃_{β}^{ψ} )∥_{p(·)}
⩽ Kν(n)(∥f_{β}^{ψ} ∥_{p(·)} + ∥f̃_{β}^{ψ} ∥_{p(·)}) ⩽ C_{p}ν(n),

where C_{p} is a positive constant, which depends only on the function p = p(x). In the article [17] it was shown that if 1 ⩽ s(x) ⩽ p(x) ⩽ p̄ < ∞, then for an arbitrary function f ∈ L^{p(·)} the inequality hold

(3.7)  ∥f∥_{s(·)} ⩽ K_{s,p} ∥f∥_{p(·)}.

From relations (3.6) and (3.7) we obtain the estimate

E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩽ E(L_{β,s(·)}^{ψ} ; Ẑ_{n})_{s(·)} ⩽ C_{p,s} ν(n),

where C_{p,s} is a positive constant, which depends only on the functions p = p(x) and s = s(x). We now obtain the lower estimate for the value of E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)}. If for given sequence ψ(k) and the number n ∈ ℕ, there exists the natural number k_{n}, for which the equality

(3.8)  ν(n) = sup_{k⩾n}|ψ(k)| = ψ(k_{n}),

is true, then the corresponding lower estimate can be obtained with help of the function

(3.9)  f_{n}(x) = \frac{ψ(k_{n})}{∥ cosk_{n}x∥_{p(·)}} cos( k_{n}x − \frac{βπ}{2} ) .

Indeed, since

∥(f_{n}(x))_{β}^{ψ} ∥_{p(·)} = ∥ \frac{cosk_{n}x}{∥ cosk_{n}x∥_{p(·)}} ∥_{p(·)} = 1,

then f_{n} ∈ L_{β,p(·)}^{ψ} and

E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩾ ∥f_{n} − Ẑ(f_{n})∥_{s(·)} = \frac{ψ(k_{n})}{∥ cosk_{n}x∥_{p(·)}}∥ cosk_{n}x∥_{s(·)} = C_{p,s} ν(n).

But if for the sequence ψ(k) and the number n ∈ ℕ, there exists no natural number k_{n}, for which equality (3.8) holds, due to the limitation of the set {|ψ(k)|} of values of the function ψ(k) we will have

ν(n) = sup_{k⩾n}|ψ(k)| = sup_{k⩾n}{|ψ(k)|}.

In this case, there exists a sequence k_{j} , j ∈ ℕ such that k_{j} ⩾ n and the numbers ψ(k_{j}) don’t decrease and converge to ν(n). Let 𝔸 = ∪_{j} f_{j} (x), where the function f_{j} (x) is defined by equality (3.9). Since f_{j} ∈ L_{β,p(·)}^{ψ} for any j ∈ ℕ, then

E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} = sup_{f∈L_{β,p(·)}^{ψ}} ∥f − Ẑ_{n} (f)∥_{s(·)} ⩾ sup_{f∈𝔸}∥f − Ẑ_{n} (f)∥_{s(·)}

= sup_{j∈ℕ} \frac{ψ(k_{j})}{∥ cosk_{j} x∥_{p(·)}}∥ cosk_{j} x∥_{s(·)} = C_{p,s} ν(n).  □

We now obtain an estimate of the sequence E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} in the case when the function p = p(x) and s = s(x) on the period satisfies the inequality p(x) < s(x). The following result gives the upper estimate.

Theorem 3.2. Let p, s ∈ P^{γ}, p(x) < s(x), x ∈ [0; 2π] and (ψ; β) ∈ Υ_{α,n} ,

α = 1/\underline{p}− 1/\overline{s}. Then, for all n ∈ ℕ the following inequality

E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩽ C_{p,s} n^{α} ν(n),

holds, where C_{p,s} is a positive constant depending on p = p(x) and s = s(x) only.

Proof. For an arbitrary function f ∈ L_{β,p(·)}^{ψ}, the equality

(3.10)  f(x) − Ẑ_{n} (f; x) = ∑_{k=1}^{n−1} \frac{ψ(n)}{ψ(k)} A_{k}(f; x) + ∑_{k=n}^{∞} A_{k}(f; x)

= ∑_{k=1}^{∞} μ_{n,α} k^{−α} A_{k}(f_{β}^{ψ} ; x) + ∑_{k=1}^{∞} μ̃_{n,α} k^{−α} Ã_{k}(f_{β}^{ψ} ; x)

: = M_{α} (g_{α}) + M̃_{α} (g̃_{α}),

holds, where M_{α} and M̃_{α} are operators-multipliers, which are defined by the sequences (3.1) and (3.2) respectively, α = 1/\underline{p}− 1/\overline{s} and

g_{α} (x) := ∑_{k=1}^{∞} k^{−α} A_{k}(f_{β}^{ψ} ; x) = \frac{1}{π} ∫_{0}^{2π} f_{β}^{ψ} (x + t)D_{α} (t)dt,

g̃_{α}(x) := ∑_{k=1}^{∞} k^{−α} Ã_{k}(f_{β}^{ψ} ; x) = \frac{1}{π} ∫_{0}^{2π} f̃_{β}^{ψ} (x + t)D_{α} (t)dt,

D_{α} (t) is function defined in Theorem 2.2.

Since f ∈ L_{β,p(·)}^{ψ}, then f_{β}^{ψ} ∈ L^{p(·)}, and moreover f_{β}^{ψ} ∈ L\underline{p}. By Theorem 2.2 we conclude that the convolution g_{α} (x) belongs L\overline{s}, and moreover g_{α} ∈ L^{s(·)}. From the condition (ψ; β) ∈ Υ_{n,α} by Lemma 2.1 we conclude that the operator-multiplier M_{α} acts from L^{s(·)} to L^{s(·)} for any s ∈ P^{γ}. Using analogous arguments for the function g̃_{α}(x), taking into account inequalities (2.2), (2.3), (3.3) and (3.4), for an arbitrary function f ∈ L_{β,p(·)}^{ψ} on the basis of equality (3.10) we find

∥f − Ẑ_{n} (f)∥_{s(·)} ⩽ ∥M_{α} (g_{α})∥_{s(·)} + ∥M̃_{α} (g̃_{α})∥_{s(·)} ⩽ Kn^{α} ν(n)(∥g_{α} ∥_{s(·)} + ∥g̃_{α}∥_{s(·)})
⩽ C_{p,s} n^{α} ν(n)(∥f_{β}^{ψ} ∥_{p(·)} + ∥f̃_{β}^{ψ} ∥_{p(·)}) ⩽ C_{p,s} n^{α} ν(n).  □

To make formulate the following assertion, which gives a lower estimate for the quantity E(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} in the case, if the function p = p(x) and s = s(x) satisfies the inequality p(x) < s(x) on the period, we need the following definition.

Denoted by 𝔅 the set of pairs (ψ; β), such that for any n ∈ ℕ the relations are true:

sup_{n⩽k⩽2n} | \frac{ν(n)}{ψ(k)} | ⩽ C, sup_{m∈ℕ} ∑_{k=2^{m}}^{2^{m+1}} |τ(k + 1) − τ(k)| ⩽ C,

where C is a positive constant, which is independent of n, ν(n) = sup_{k⩾n} ψ(k) and

(3.11)  τ(k) := \{ \begin{matrix}0, & 1 ⩽ k ⩽ n − 1, \\ \frac{ν(n)}{ψ(k)} , & n ⩽ k ⩽ 2n. \end{matrix}
Theorem 3.3. Let p, s ∈ P^{γ}, p(x) < s(x), x ∈ [0; 2π] and (ψ; β) ∈ 𝔅. Then,

for all n ∈ ℕ we have E_{n}(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩾ C_{p,s} ν(n)n^{1/\overline{p}−1/\underline{s}} , where C_{p,s} is a positive constant depending on p = p(x) and s = s(x) only.

Proof. For obtaining a lower estimate, let us show that for any positive integer n in class L_{β,p(·)}^{ψ} there exists a function f_{n}^{∗}, for which the inequality is true

∥f_{n}^{∗} − Ẑ_{n} ∥_{s(·)} ⩾ C_{p,g}ν(n)n^{1/\overline{p}−1/\underline{s}} .

For this we fix n ∈ ℕ and consider the function f_{n}^{∗}(x) = ∑ _{k=n}^{2n} ψ(k) cos( kx − \frac{βπ}{2} ). Since

(f_{n}^{∗}(x))_{β}^{ψ} = ∑_{k=n}^{2n} coskx = \frac{sinnx/2 cos(3n + 1)x/2}{sinx/2} ,

then using relation (3.7) and also the well-known inequality

(3.12)  \frac{x}{π} ⩽ sinx/2, sinx ⩽ x, x ∈ [0; π],

162 CHAICHENKO

we obtain

∥(f_{n}^{∗})_{β}^{ψ} ∥_{p(·)} = ∥ ∑_{k=n}^{2n} coskx∥_{p(·)} ⩽ K_{p}∥ ∑_{k=n}^{2n} coskx∥\overline{p}

= ( 2 ∫_{0}^{π} | ∑_{k=n}^{2n} coskx|\overline{p}dx) ^{1/\overline{p}} ⩽ ( 2 ∫_{0}^{π} | \frac{sinnx/2}{sinx/2} |\overline{p}dx) ^{1/\overline{p}}

⩽ C_{p}^{∗}n^{1−1/\overline{p}} .

This implies that the function

g_{n}^{∗}(x) = \frac{n^{1/\overline{p}−1}}{C_{p}^{∗}} f_{n}^{∗}(x) = \frac{n^{1/\overline{p}−1}}{C_{p}^{∗}} ∑_{k=n}^{2n} ψ(k) cos( kx − \frac{βπ}{2} )

belongs to the class L_{β,p(·)}^{ψ}.

Again using the inequalities (3.7) and (3.12), we find

(3.13)  ∥ ∑_{k=n}^{2n} coskx∥_{s(·)} ⩾ ( 2 ∫_{0}^{π} | \frac{sinnx/2 cos(3n + 1)x/2}{sinx/2} |\underline{s}dx) ^{1/\underline{s}}

⩾ C_{s}n^{1−1/\underline{s}} ( ∫_{0}^{π/2} (cosx)\underline{s} dx) ^{1/\underline{s}} ⩾ K_{s}n^{1−1/\underline{s}} .

If now by T_{ψ} we denote the operator-multiplier that generates sequence (3.11), then by applaying Lemma 2.1 to the condition (ψ; β) ∈ 𝔅 we will have

∥ ∑_{k=n}^{2n} cos( kx − \frac{βπ}{2} ) ∥_{s(·)} = ∥T_{ψ} ( ∑_{k=n}^{2n} \frac{ψ(k)}{ν(n)} cos( kx − \frac{βπ}{2} ) ) ∥_{s(·)}

⩽ C∥ ∑_{k=n}^{2n} \frac{ψ(k)}{ν(n)} cos( kx − \frac{βπ}{2} ) ∥_{s(·)}.

Hence, considering inequality (3.13) we find

(3.14)  ∥ ∑_{k=n}^{2n} \frac{ψ(k)}{ν(n)} cos( kx − \frac{βπ}{2} ) ∥_{s(·)} ⩾ K∥ ∑_{k=n}^{2n} cos( kx − \frac{βπ}{2} ) ∥_{s(·)}

⩾ C_{s} ∥ ∑_{k=n}^{2n} coskx∥\underline{s} ⩾ K_{s}n^{1−1/\underline{s}} .

Using relation (3.14), we obtain

E_{n}(L_{β,p(·)}^{ψ}; Ẑ_{n})_{s(·)} ⩾ ∥g_{n}^{∗} − Ẑ_{n} (g_{n}^{∗})∥_{s(·)} = ∥ \frac{n^{1/\overline{p}−1}}{C_{p}^{∗}} ∑_{k=n}^{2n} ψ(k) cos( kx − \frac{βπ}{2} ) ∥_{s(·)}

⩾ \frac{n^{1/\overline{p}−1}}{C_{p}^{∗}} ν(n)∥ ∑_{k=n}^{2n} \frac{ψ(k)}{ν(n)} cos( kx − \frac{βπ}{2} ) ∥_{s(·)}

⩾ C_{p,s} n^{1/\overline{p}−1} ν(n)n^{1−1/\underline{s}} = C_{p,s} ν(n)n^{1/\overline{p}−1/\underline{s}} .  □

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Department of Mathematics (Received 15 01 2015) Donbas State Pedagogical University Slavyansk Ukraine

 s.chaichenko@gmail.com