On approximations by trigonometric polynomials of classes of functions defined by moduli of smoothness

TL;DR

This paper characterizes Nikol'skii-Besov function classes using moduli of smoothness and Fourier coefficients, providing necessary and sufficient conditions for membership based on recent inequalities.

Contribution

It offers a new characterization of function classes via series over moduli of smoothness and Fourier coefficients, advancing understanding of function approximation.

Findings

Characterization of Nikol'skii-Besov classes through moduli of smoothness.

Necessary and sufficient conditions using Fourier coefficients.

Application of reverse Copson- and Leindler-type inequalities.

Abstract

In this paper, we give a characterization of Nikol'ski\u{\i}-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.

Equations

$$f(x)\sim \sum _{n=0}^{\infty }{a}_n\cos nx,a_n\downarrow 0.$$

$$f(x)\sim \sum _{\nu =1}^{\infty }{\lambda }_{\nu }\cos \nu x,$$

$${\lambda }_{\nu }=\{\begin{array}{ll}{a}_{\mu }\ge 0& \text{for ν=2μ},\\ 0& \text{for ν=2μ},\end{array}$$

$$f(x)\sim \sum _{\mu =0}^{\infty }{a}_{\mu }\cos 2^{\mu }x,a_{\mu }\ge 0.$$

$${\omega }_k(f,t{)}_p=\underset{|h|\le t}{\sup }\Vert {\Delta }_h^{k}f{\Vert }_p,$$

$${\Delta }_h^{k}f(x)=\sum _{\nu =0}^k(-1{)}^{k-\nu }(\genfrac{}{}{0ex}{}{k}{\nu })f(x+\nu h)$$

$$E_n(f{)}_p=\underset{T_{n-1}}{\inf }\Vert f-T_{n-1}{\Vert }_p,$$

$$\biggl{(}\int_{0}^{\delta}t^{-r\theta-1}\omega_{k}(f,t)_{p}^{\theta}\,dt+\delta^{\lambda\theta}\int_{\delta}^{1}t^{-(r+\lambda)\theta-1}\omega_{k}(f,t)_{p}^{\theta}\,dt\biggr{)}^{1/\theta}\leq C\varphi(\delta),$$

$$\biggl{(}\sum_{\nu=n+1}^{\infty}\omega_{k}\left(f,\frac{1}{\nu}\right)_{p}^{\theta}\nu^{r\theta-1}+n^{-\lambda\theta}\sum_{\nu=1}^{n}\omega_{k}\left(f,\frac{1}{\nu}\right)_{p}^{\theta}\nu^{(r+\lambda)\theta-1}\biggr{)}^{1/\theta}\\ \leq C\varphi\left(\frac{1}{n}\right),$$

$$f(x)\sim \sum _{\nu =1}^{\infty }{a}_{\nu }\cos \nu x,a_{\nu }\downarrow 0,$$

$$\biggl{(}\sum_{\nu=n+1}^{\infty}a_{\nu}^{\theta}\nu^{r\theta+\theta-\theta/p-1}+n^{-\lambda\theta}\sum_{\nu=1}^{n}a_{\nu}^{\theta}\nu^{r\theta+\lambda\theta+\theta-\theta/p-1}\biggr{)}^{1/\theta}\leq C\varphi\left(\frac{1}{n}\right),$$

$$a_{\nu }\le \frac{C}{{\nu }^{r+\alpha +1-\frac{1}{p}}},$$

$$\sum _{\nu =1}^{\infty }{a}_{\nu }^{\theta }{\nu }^{r\theta +\theta -\theta /p-1}<\infty$$

$$f(x)\sim \sum _{\nu =1}^{\infty }{\lambda }_{\nu }\cos \nu x,$$

$${\lambda }_{\nu }=\{\begin{array}{ll}{a}_{\mu }\ge 0& \text{for ν=2μ},\\ 0& \text{for ν=2μ},\end{array}$$

$$\biggl{(}\sum_{\nu=m+1}^{\infty}\lambda_{\nu}^{\theta}\nu^{r\theta}+m^{-\lambda\theta}\sum_{\nu=1}^{m}\lambda_{\nu}^{\theta}\nu^{(r+\lambda)\theta}\biggr{)}^{1/\theta}\leq C\varphi\left(\frac{1}{m}\right),$$

$$a_{\mu }\le C{2}^{-\mu (r+\alpha )}$$

$$\sum _{\mu =1}^{\infty }{a}_{\mu }^{\theta }{2}^{\mu r\theta }<\infty$$

$$f(x)\sim \sum _{\mu =0}^{\infty }{a}_{\mu }\cos 2^{\mu }x,$$

$$a_{\mu }=2^{-\mu r}(\mu +1{)}^{-(\alpha +1/\theta )},\alpha >0.$$

$$C_{1}n^{-\alpha}\leq\biggl{(}\sum_{\mu=n+1}^{\infty}a_{\mu}^{\theta}2^{\mu r\theta}\biggr{)}^{1/\theta}\leq C_{2}n^{-\alpha}$$

$$C_{3}n^{-(\alpha+1/\theta)}\leq\biggl{(}2^{-n\lambda\theta}\sum_{\mu=0}^{n}a_{\mu}^{\theta}2^{\mu(r+\lambda)\theta}\biggr{)}^{1/\theta}\leq C_{4}n^{-(\alpha+1/\theta)},$$

$$\biggl{(}\sum_{\nu=1}^{n}a_{\nu}^{\beta}\biggr{)}^{1/\beta}\leq\biggl{(}\sum_{\nu=1}^{n}a_{\nu}^{\alpha}\biggr{)}^{1/\alpha}.$$

$$\sum_{\mu=m}^{n}\mu^{\alpha-1}\biggl{(}\sum_{\nu=\mu}^{n}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\leq C_{1}\sum_{\mu=m}^{n}\mu^{\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p},$$

$$\sum_{\mu=m}^{n}\mu^{-\alpha-1}\biggl{(}\sum_{\nu=m}^{\mu}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\leq C_{2}\sum_{\mu=m}^{n}\mu^{-\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p};$$

$$\sum_{\mu=m}^{n}\mu^{\alpha-1}\biggl{(}\sum_{\nu=\mu}^{n}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\geq C_{3}\sum_{\mu=m}^{n}\mu^{\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p},$$

$$\sum_{\mu=m}^{n}\mu^{-\alpha-1}\biggl{(}\sum_{\nu=m}^{\mu}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\geq C_{4}\sum_{\mu=m}^{n}\mu^{-\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p},$$

$$\sum_{\mu=m}^{n}\mu^{\alpha-1}\biggl{(}\sum_{\nu=\mu}^{n}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\geq C_{1}\sum_{\mu=8m}^{n}\mu^{\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p},$$

$$\sum_{\mu=m}^{n}\mu^{-\alpha-1}\biggl{(}\sum_{\nu=m}^{\mu}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\geq C_{2}\sum_{\mu=4m}^{n}\mu^{-\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p};$$

$$\sum_{\mu=4m}^{n}\mu^{\alpha-1}\biggl{(}\sum_{\nu=\mu}^{n}a_{\nu}\nu^{\lambda}\biggr{)}^{p}\leq C_{3}\sum_{\mu=m}^{n}\mu^{\alpha-1}(a_{\mu}\mu^{\lambda+1})^{p},$$

Full text

1. Introduction

Let 111The authors declare that there is no conflict of interest regarding the publication of this paper. $f\in L_{p}[0,2\pi]$, $1<p<\infty$, be a $2\pi$-periodic function. We say that the function $f$ has monotone Fourier coefficients if it has a cosine Fourier series with

[TABLE]

We say that the function $f$ has lacunary Fourier coefficients if

[TABLE]

where

[TABLE]

that is

[TABLE]

By $\omega_{k}(f,t)_{p}$ we denote the modulus of smoothness of order $k$ in $L_{p}$ metrics of a function $f\in L_{p}$, $1<p<\infty$:

[TABLE]

where

[TABLE]

is the $k$-th order shift operator.

By $E_{n}(f)_{p}$ we denote the best approximation in $L_{p}$ metrics of a function $f\in L_{p}$, $1<p<\infty$, by means of trigonometric polynomials whose degree is not greater than $n-1$, i.e.

[TABLE]

where $T_{n-1}=\sum_{\nu=0}^{n-1}(\alpha_{\nu}\cos\nu x+\beta_{\nu}\sin\nu x)$, $\alpha_{\nu}$ and $\beta_{\nu}$ are arbitrary real numbers.

We say that a $2\pi$-periodic function $f$ belongs to the Nikol’skiĭ-Besov class $N(p,\theta,r,\lambda,\varphi)$, $1<p<\infty$, if the following conditions are satisfied

1. (1)

$f\in L_{p}[0,2\pi]$; 2. (2)

Numbers $\theta$, $r$, $\lambda$ belong to the interval $(0,\infty)$, and $k$ is an integer satisfying $k>r+\lambda$; 3. (3)

The following inequality holds true

[TABLE]

while the function $\varphi$ satisfies the conditions

1. (4)

$\varphi$ is a non-negative continuous function on $(0,1)$ and $\varphi\neq 0$; 2. (5)

For every $\delta_{1}$, $\delta_{2}$ such that $0\leq\delta_{1}\leq\delta_{2}\leq 1$ holds $\varphi(\delta_{1})\leq C_{1}\varphi(\delta_{2})$; 3. (6)

For every $\delta$ such that $0\leq\delta\leq\frac{1}{2}$ holds $\varphi(2\delta)\leq C_{2}\varphi(\delta)$,

where constants222Without mentioning it explicitly, we will consider all the constants positive. $C$, ${C_{1}}$ and $C_{2}$ do not depend on $\delta_{1}$, $\delta_{2}$ and $\delta$.

A more detailed approach to the classes $N(p,\theta,r,\lambda,\varphi)$ is given in [6] and [12] (see also [2]). In the paper, we give a characterization of $N(p,\theta,r,\lambda,\varphi)$ classes of functions in terms of series over their moduli of smoothness. Then we give the necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function $f\in L_{p}[0,2\pi]$ to belong to a class $N(p,\theta,r,\lambda,\varphi)$. In the process of proving the results, we make use of certain recent reverse $l_{p}$-type inequalities [10], closely related to Copson’s and Leindler’s inequalities.

Finally, by making use of our results, we construct an example of a function having a lacunary Fourier series, which shows that $N(p,\theta,r,\lambda,\varphi)$ classes are properly embedded between the appropriate Nikol’skiĭ classes and Besov classes.

2. Statement of Results

Now we formulate our results.

Theorem 2.1**.**

A function $f$ belongs to the class $N(p,\theta,r,\lambda,\varphi)$ if and only if333Here and below we assume that the parameters $\theta$, $r$, $\lambda$ and $k$ satisfy the condition 2, and the function $\varphi$ satisfies the conditions 4–6 of the definition of the class $N(p,\theta,r,\lambda,\varphi)$.

[TABLE]

where constant $C$ does not depend on $n$.

Theorem 2.2**.**

For a function $f\in L_{p}[0,2\pi]$, $1<p<\infty$, such that

[TABLE]

to belong to the class $N(p,\theta,r,\lambda,\varphi)$ it is necessary and sufficient that its Fourier coefficients satisfy the condition

[TABLE]

where constant $C$ does not depend on $n$.

Regarding Theorem 2.1, a very interesting open question remains its analogue for functions with general monotone Fourier coefficients, generalized in the sense of [13, 9].

Corollary 2.1**.**

Put $\varphi(\delta)=\delta^{\alpha}$, $0<\alpha<\lambda$, in the definition of the class $N(p,\theta,r,\lambda,\varphi)$, we obtain [6] the Nikol’skiĭ class $H_{p}^{r+\alpha}$. Thus Theorems 2.1 and 2.2 give the single coefficient condition

[TABLE]

for $f\in H_{p}^{r+\alpha}$, given in [5], where the function $f$ is given by (2).

Corollary 2.2**.**

If $\varphi(\delta)\geq C$, then we obtain [6] the Besov class $B_{p}^{\theta r}$. Thus Theorems 2.1 and 2.2 give the necessary and sufficient condition

[TABLE]

for $f\in B_{p}^{\theta r}$, given in [11], where the function $f$ is given by (2).

Theorem 2.3**.**

For a function $f\in L_{p}$, $1<p<\infty$, such that

[TABLE]

and

[TABLE]

to belong to the class $N(p,\theta,r,\lambda,\varphi)$ it is necessary and sufficient that its Fourier coefficients satisfy the condition444Here and below we assume that the parameters $\theta$, $r$, $\lambda$ and $k$ satisfy the condition 2, and the function $\varphi$ satisfies the conditions 4–6.

[TABLE]

where constant $C$ does not depend on $m$.

Corollary 2.3**.**

Putting $\varphi(\delta)=\delta^{\alpha}$, $0<\alpha<\lambda$, in the definition of the class $N(p,\theta,r,\lambda,\varphi)$, we obtain [6] the Nikol’skiĭ class $H_{p}^{r+\alpha}$. Thus Theorem 2.3 gives the single coefficient condition

[TABLE]

for $f\in H_{p}^{r+\alpha}$, where the function $f$ is given by (3).

Corollary 2.4**.**

If $\varphi(\delta)=C$, then we obtain [6] the Besov class $B_{p}^{\theta r}$. Thus Theorem 2.3 gives the necessary and sufficient condition

[TABLE]

for $f\in B_{p}^{\theta r}$, given in [11], where the function $f$ is given by (3).

Example 2.1*.*

Let

[TABLE]

where are

[TABLE]

Then, we have

[TABLE]

and

[TABLE]

thus implying (see the proof of Theorem 2.3) $f\in N(p,\theta,r,\lambda,\varphi)$ for $\varphi(\delta)=\left(\ln\frac{1}{\delta}\right)^{-\alpha}$. This means that classes $N$ are classes of embedding between classes $H$ and $B$.

3. Auxiliary statements

In order to establish our results, we use the following lemmas.

Lemma 3.1**.**

Let $0<\alpha<\beta<\infty$ and $a_{\nu}\geq 0$. The following inequality holds true

[TABLE]

Proof of the lemma is due to Jensen [4, p. 43].

Lemma 3.2**.**

Let $\{a_{\nu}\}_{\nu=1}^{\infty}$ be a sequence of non-negative numbers, $\alpha>0$, $\lambda$ a real number, $m$ and $n$ positive integers such that $m<n$. Then

1. (1)

for $1\leq p<\infty$ the following equalities hold

[TABLE]

[TABLE] 2. (2)

for $0<p\leq 1$ the following equalities hold

[TABLE]

[TABLE]

where constants ${C_{1}}$, ${C_{2}}$, ${C_{3}}$ and $C_{4}$ depend only on numbers $\alpha$, $\lambda$ and $p$, and do not depend on $m$, $n$ as well as on the sequence $\{a_{\nu}\}_{\nu=1}^{\infty}$.

Proof of the lemma is given in [4, p. 308].

Lemmas 3.3 and 3.4 that follow state certain $l_{p}$-type inequalities which are reversed to the ones given in Lemma 3.2 and closely related to Copson’s and Leindler’s inequalities (see, e.g., [3, 7, 8, 14]).

We write $a_{\nu}\downarrow$ if $\{a_{\nu}\}_{\nu=1}^{\infty}$ is a monotone-decreasing sequence of non-negative numbers, i.e. if $a_{\nu}\geq a_{\nu+1}\geq 0$ $(\nu=1,2,\dotsc)$.

Lemma 3.3**.**

Let $a_{\nu}\downarrow$, $\alpha>0$, $\lambda$ a real number, $m$ and $n$ positive integers. Then

1. (1)

for $1\leq p<\infty$, $n\geq 16m$ the following equalities hold

[TABLE]

[TABLE] 2. (2)

for $0<p\leq 1$, $n\geq 4m$ the following equalities hold

[TABLE]

[TABLE]

where constants ${C_{1}}$, ${C_{2}}$, ${C_{3}}$ and $C_{4}$ depend only on numbers $\alpha$, $\lambda$ and $p$, and do not depend on $m$, $n$ as well as on the sequence $\{a_{\nu}\}_{\nu=1}^{\infty}$.

Proof of the lemma is given in [10].

Lemma 3.4**.**

Let $a_{\nu}\downarrow$, $\alpha>0$, $\lambda$ a real number, $m$ and $n$ positive integers. For $0<p<\infty$ the following inequalities hold

[TABLE]

[TABLE]

where constants ${C_{1}}$, ${C_{2}}$, ${C_{3}}$ and $C_{4}$ depend only on numbers $\alpha$, $\lambda$ and $p$, and do not depend on $m$, $n$ as well as on the sequence $\{a_{\nu}\}_{\nu=1}^{\infty}$.

The lemma is also proved in [10].

Lemma 3.5**.**

Let $f\in L_{p}[0,2\pi]$ for a fixed $p$ from the interval $1<p<\infty$ and let

[TABLE]

The following inequalities hold

[TABLE]

where constants ${C_{1}}$ and $C_{2}$ do not depend on $n$ and $f$.

The lemma is proved in [11].

Lemma 3.6**.**

A function $f$ belongs to the class $N(p,\theta,r,\lambda,\varphi)$ if and only if

[TABLE]

where constant $C$ does not depend on $n$.

Proof of the lemma is given in [6].

Lemma 3.7**.**

Let $f\in L_{p}$, $1<p<\infty$, and

[TABLE]

The following inequalities hold

[TABLE]

where constants $C_{2}$ and ${C_{1}}$ do not depend on $f$.

Proof of the lemma is due to Zygmund [16, vol. I, p. 326].

Corollary 3.1**.**

Lemma 3.7 yields the following estimate

[TABLE]

where constants $C_{2}$ and ${C_{1}}$ do not depend on $n$ and $f$.

4. Proofs

Now we prove our results.

Proof of Theorem 2.1.

Put

[TABLE]

We have [4, p. 55]

[TABLE]

and, taking into account properties of modulus of smoothness [15],

[TABLE]

In an analogous way we estimate

[TABLE]

and

[TABLE]

Let $f\in N(p,\theta,r,\lambda,\varphi)$. For a positive integer $n$ we put $\delta=\frac{1}{n+1}$. Then we have

[TABLE]

Hence we obtain

[TABLE]

which proves inequality (1).

Now we suppose that inequality (1) holds. For $\delta\in(0,1)$ we choose the positive integer $n$ satisfying $\frac{1}{n+1}<\delta\leq\frac{1}{n}$. Then, taking into consideration the estimates from above for $I_{1}$ and $I_{2}$ we have

[TABLE]

Hence

[TABLE]

implying $f\in N(p,\theta,r,\lambda,\varphi)$.

Proof of Theorem 2.1 is completed. ∎

Proof of Theorem 2.2.

Theorem 2.1 implies that the condition $f\in N(p,\theta,r,\lambda,\varphi)$ is equivalent to the condition

[TABLE]

where constant $C_{1}$ does not depend on $n$. Lemma 3.5 yields that the last estimate is equivalent to the estimate

[TABLE]

where constant $C_{2}$ does not depend on $n$. Hence, if we denote the terms on the left-hand side of the inequality by $J_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$ respectively, then condition $f\in N(p,\theta,r,\lambda,\varphi)$ is equivalent to the condition

[TABLE]

Now we estimate the terms $J_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$ from below and above by means of expression taking part in the condition of the theorem.

First we estimate $J_{1}$ and $J_{2}$ from below. We have

[TABLE]

For $k-r>0$, making use of Lemmas 3.2 and 3.3 we obtain

[TABLE]

In an analogous way, for $r\theta>0$ we get

[TABLE]

We estimate the term $J_{2}$ from above:

[TABLE]

For $J_{1}$ we have

[TABLE]

and applying once more Lemmas 3.2 and 3.3 we obtain

[TABLE]

Put

[TABLE]

Then for

[TABLE]

taking into account that $(k+1)p-2\geq 0$ and $a_{\nu}\downarrow 0$ we get

[TABLE]

Since $k-r-\lambda>0$, we have

[TABLE]

Applying Lemma 3.4 we obtain

[TABLE]

From (8) it follows that

[TABLE]

This way, inequalities (5), (6), (7) and (9) yield

[TABLE]

Now we estimate $J_{3}$ and $J_{4}$. Put

[TABLE]

and

[TABLE]

applying Lemma 3.4 for $r+\lambda-k<0$ we get

[TABLE]

We estimate $A_{2}$ in an analogous way:

[TABLE]

We estimate the series

[TABLE]

First let $\frac{\theta}{p}>1$. Applying Hölder inequality we have

[TABLE]

Since \bigl{(}rp-\frac{p}{\theta}+1\bigr{)}\frac{\theta}{\theta-p}=rp\frac{\theta}{\theta-p}+1>1, we get

[TABLE]

So, for $\frac{\theta}{p}>1$ we have proved that

[TABLE]

Let $\frac{\theta}{p}\leq 1$. For given $n$ we choose the positive integer $N$ such that $2^{N}\leq n+1<2^{N+1}$. Then we have

[TABLE]

Making use of Lemma 3.1 we obtain

[TABLE]

Since for $n\geq 3$ holds $\left[\frac{n+1}{4}\right]\geq\frac{n}{12}$, we get

[TABLE]

This way, for $0<\frac{\theta}{p}<\infty$ we proved that

[TABLE]

Hence (12) yields

[TABLE]

Now, from (11) it follows that

[TABLE]

Further, we estimate the series

[TABLE]

where is

[TABLE]

Hence

[TABLE]

Making use of (14) and (13) we have

[TABLE]

Hence, applying (14) in (10) we obtain

[TABLE]

Now we estimate $A_{1}$ and $A_{2}$ from below. Making use of Lemma 3.4 we get

[TABLE]

and in an analogous way

[TABLE]

Hence

[TABLE]

This way the following inequality holds

[TABLE]

From (10) it follows that

[TABLE]

Since

[TABLE]

holds, we have

[TABLE]

Now, estimates (16) and (15) imply

[TABLE]

This way we proved that condition (1) is equivalent to the condition of the theorem. Since condition (1) is equivalent to the condition $f\in N(p,\theta,r,\lambda,\varphi)$, proof of Theorem 2.2 is completed. ∎

Proof of Theorem 2.3.

Considering Lemma 3.6, condition $f\in N(p,\theta,r,\lambda,\varphi)$ is equivalent to the condition

[TABLE]

where constant $C$ does not depend on $n$. Corollary 3.1 yields that the last estimate is equivalent to the estimate

[TABLE]

where constant $C_{43}$ does not depend on $n$.

Put

[TABLE]

we estimate $J_{1}$ and $J_{2}$ from below and above.

Let $0<\frac{\theta}{2}\leq 1$. Using Lemma 3.1, changing the order os summation we get

[TABLE]

Therefrom, taking into consideration that $r\theta>0$ while computing the second sum we obtain

[TABLE]

Let $1\leq\frac{\theta}{2}<\infty$ and $0<\varepsilon<r$. Applying Hölder inequality we have

[TABLE]

where is $\frac{2}{\theta}+\frac{1}{\theta^{\prime}}=1$. Computing the second sum we obtain

[TABLE]

Now we have

[TABLE]

This way, for $0<\theta<\infty$ we have

[TABLE]

where constant $C_{48}$ does not depend on $n$.

Now we estimate $J_{1}$ from below.

Let $1\leq\frac{\theta}{2}<\infty$. Making use of Lemma 3.1 we get

[TABLE]

Computing the second sum we get

[TABLE]

Let $0<\frac{\theta}{2}\leq 1$ and $\varepsilon>0$. Applying Hölder inequality we have

[TABLE]

where is $\frac{\theta}{2}+\frac{1}{\theta^{\prime}}=1$. The last estimate implies

[TABLE]

Changing the order of summation and then computing the second sum we obtain

[TABLE]

where constant $C_{52}$ does not depend on $n$.

Consequently, for every $0<\theta<\infty$ the following estimate holds

[TABLE]

where constants ${C_{53}}$ and $C_{54}$ do not depend on $n$.

Now we estimate $J_{2}$. Obviously

[TABLE]

Let $1\leq\frac{\theta}{2}<\infty$. Applying Lemma 3.1, changing the order of summation, and then computing the second sum we obtain

[TABLE]

Let $0<\frac{\theta}{2}\leq 1$ and $\varepsilon>0$. Applying Hölder inequality we get

[TABLE]

where is $\frac{\theta}{2}+\frac{1}{\theta^{\prime}}=1$. The last estimate implies

[TABLE]

Changing the order of summation and computing the second sum we have

[TABLE]

Thus, for every $0<\theta<\infty$ holds

[TABLE]

Now we estimate $J_{2}$ from above. Taking into consideration that $(r+\lambda)\theta>0$, we have

[TABLE]

Since

[TABLE]

holds and an upper bound for $J_{1}$ is already found, we estimate from above the expression

[TABLE]

Let $0<\frac{\theta}{2}\leq 1$. Applying Lemma 3.1 we obtain

[TABLE]

Let $1\leq\frac{\theta}{2}<\infty$ and $0<\varepsilon<r+\lambda$. Then applying Hölder inequality we have

[TABLE]

where is $\frac{2}{\theta}+\frac{1}{\theta^{\prime}}=1$. Using the last estimate we get

[TABLE]

Changing the order of summation and computing the second sum we obtain

[TABLE]

Therefore, for every $0<\theta<\infty$ the following estimate holds

[TABLE]

Now making use of inequalities (20) and (18) we have

[TABLE]

This way, inequalities (18), (19) and the last inequality imply the estimate

[TABLE]

where constants ${C_{67}}$ and $C_{68}$ do not depend on $n$. Hence, considering the condition (17) we conclude that condition $f\in N(p,\theta,r,\lambda,\varphi)$ is equivalent to the condition

[TABLE]

where constant $C_{69}$ does not depend on $n$.

We put

[TABLE]

For given $m$ we choose the positive integer $n$ such that $2^{n}\leq m+1<2^{n+1}$.

First we consider the case $2^{n}<m+1<2^{n+1}$. We have

[TABLE]

Since $\lambda_{\nu}=0$ for $\nu\neq 2^{\mu}$, we get

[TABLE]

Further, since $\lambda_{2^{\mu}}=a_{\mu}$, we get

[TABLE]

Hence, for $2^{n}<m+1<2^{n+1}$ we obtain

[TABLE]

where constants ${C_{70}}$ and $C_{71}$ do not depend on $m$ and $n$.

Let us assume now that $m+1=2^{n}$. In an analogous way we have

[TABLE]

Thus, for $2^{n}\leq m+1<2^{n+1}$ the following estimate holds

[TABLE]

where constants ${C_{72}}$ and $C_{73}$ do not depend on $m$ and $n$. Hence, considering the condition (21) we conclude that condition $f\in N(p,\theta,r,\lambda,\varphi)$ is equivalent to the condition

[TABLE]

where constant $C_{74}$ does not depend on $m$ and $n$.

Since $\frac{1}{2^{n}}<\frac{2}{m+1}<\frac{2}{m}$, we get

[TABLE]

where constant $C_{76}$ does not depend on $m$ and $n$; and since $\frac{1}{2^{n}}\geq\frac{1}{m+1}\geq\frac{1}{2m}$ we get

[TABLE]

where constant $C_{78}$ does not depend on $m$ and $n$. This way, condition (22) is equivalent to the condition

[TABLE]

where constant $C_{79}$ does not depend on $m$.

This completes the proof of Theorem 2.3. ∎

Remark 4.1*.*

Notice that another way of proving Theorems 2.2 and 2.3 is presented in [12]. Our approach here is similar to that used in [1].