Riesz–Zygmund means and trigonometric approximation in Morrey spaces

Abstract

We estimate best approximations and moduli of smoothness of functions in Morrey spaces variable exponent spaces by norms of derivatives of Riesz–Zygmund means and partial Fourier sums in these spaces. As a consequence, we obtain a description of Hölder spaces based on the Morrey spaces. The direct results on approximation in these Hölder spaces are also obtained.

1 Introduction

Let $L^p_{2\pi }$, $1\le p<\infty$, be the Lebesgue space of measurable $2\pi$-periodic functions f on ${\mathbb {R}}$ such that $\Vert f\Vert ^p_p:=\int ^{2\pi }_0|f(x)|^p\,dx<\infty$. If $1\le p<\infty$, $0<\lambda \le 1$ and f is a measurable $2\pi$-periodic function for which

$$ \left\| f \right\|_{{p,\lambda }} = \mathop {\sup }\limits_{I} \left( {|I|^{{\lambda - 1}} \int_{I} | f(x)|^{p} {\mkern 1mu} dx} \right)^{{1/p}} <\infty,$$

where the supremum is taken over all $I=[a,b]$ with $0\le b-a\le 2\pi$ and |I| is the Lebesgue measure of I, then f belongs to the Morrey space $L^{p,\lambda }_{2\pi }$. We note that the norm $\Vert \cdot \Vert _{p,\lambda }$ is invariant with respect to usual translation and that $L^{p,\lambda }_{2\pi }\subset L^p_{2\pi }\subset L^1_{2\pi }$ for all $0<\lambda \le 1$ and $1\le p<\infty$ (for $\lambda =1$ the space $L^{p,\lambda }_{2\pi }$ coincides with $L^p_{2\pi }$). More about these spaces see in [1, ch. 1].

It is known that in general case $L^{p,\lambda }_{2\pi }$ is not a separable space (see [27, Prop. 2.16]). Therefore we consider a proper subspace $L^{p,\lambda }_{2\pi ,0}$ of $L^{p,\lambda }_{2\pi }$, which is the closure of the space of trigonometric polynomials in $L^{p,\lambda }_{2\pi }$ with the same norm $\Vert \cdot \Vert _{p,\lambda }$. Then $\lim _{h\rightarrow 0}\Vert f(\cdot +h)-f(\cdot )\Vert _{p,\lambda }=0$ for $f\in L^{p,\lambda }_{2\pi ,0}$ (see Lemma 1). Let

$$\begin{aligned} a_0(f)/2+\sum ^\infty _{k=1}(a_k(f)\cos kx+b_k(f)\sin kx)=\sum ^\infty _{k=0}A_k(f)(x) \end{aligned}$$

(1)

be the trigonometric Fourier series of $f\in L^1_{2\pi }$ and $S_n(f)(x)= \sum ^n_{k=0}A_k(f)(x)$ be its n-th partial sum. If $T_n$ is the space of trigonometrical polynomials of order at most $n\in {\mathbb {Z}}_+=\{0,1,\dots \}$ and $f\in L^{p,\lambda }_{2\pi ,0}$, then $E_n(f)_{p,\lambda }=\inf \{\Vert f-t_n\Vert _{p,\lambda }: t_n\in T_n\}$.

For a function $f\in L^{p,\lambda }_{2\pi ,0}$ and $m\in \mathbb N=\{1,2,\dots \}$ we consider the difference of order $m\in {\mathbb {N}}$ with step h

$$\begin{aligned} \Delta ^m_hf(x)=\sum ^m_{k=0}(-1)^{m-k}\left( {\begin{array}{c}m\\ k\end{array}}\right) f(x+kh) \end{aligned}$$

and the modulus of smoothness

$$\begin{aligned} \omega _m(f,\delta )_{p,\lambda }=\sup \limits _{|h|\le \delta }\Vert \Delta ^m_hf(x)\Vert _{p,\lambda }. \end{aligned}$$

Let $1<p<\infty$, $1/p+1/q=1$. A weight function w (a $2\pi$-periodic, measurable and positive a.e. on ${\mathbb {R}}$ function) belongs to the Muckenhoupt class $A_p({\mathbb {T}})$, if the inequality

$$\begin{aligned} \sup \limits _I\left( |I|^{-1}\int _I w^p(x)\,dx\right) ^{1/p}\left( |I|^{-1}\int _I w^{-q}(x)\,dx\right) ^{1/q}=[v]_{A_p[0,1]}<\infty , \end{aligned}$$

holds, where I are intervals of length at most $2\pi$ (see [19]). A weight function w belongs to the class $A_1({\mathbb {T}})$ if $Mw(x)\le Cw(x)$ a.e. on ${\mathbb {R}}$. Here $Mw(x)=\sup _{I\ni x}|I|^{-1}\int _Iw(x)\,dx$ is the maximal function of w. From the Hölder inequality it follows that $A_1(\mathbb T)\subset A_{p_1}({\mathbb {T}})\subset A_{p_2}({\mathbb {T}})$ for $1\le p_1\le p_2<\infty$. A measurable $2\pi$-periodic function f belongs to weighted space $L^p_{w,2\pi }$, $1\le p<\infty$, if $fw\in L^p_{2\pi }$.

Further we use the famous Riesz-Zygmund or typical means of order $r\in {\mathbb {N}}$ for $f\in L^1_{2\pi }$

$$\begin{aligned} Z^{r}_n(f)(x)=\sum ^n_{k=0}\left( 1-\frac{k^r}{(n+1)^r}\right) A_k(x)=\sum ^n_{k=0}\frac{(k+1)^r-k^r}{n^r}S_k(f)(x), \end{aligned}$$

(2)

where $n\in {\mathbb {Z}}_+$. The famous Fejér means $\sigma _n(f)=(n+1)^{-1}\sum ^n_{k=0}S_k(f)$ coincide with $Z^1_n(f)$.

According to [4, Ch. VIII, §§ 7,14] for a function $f(x)\in L^1_{2\pi }$ there exists a.e. the conjugate function

$$\begin{aligned} {\widetilde{f}}(x)=\lim _{\varepsilon \rightarrow 0+0}(2\pi )^{-1}\int ^\pi _\varepsilon (f(x-t)-f(x+t))/\tan (t/2)\,dt, \end{aligned}$$

the conjugation operator is bounded in $L^p_{2\pi }$, $1<p<\infty$ and the Fourier series of the function ${\widetilde{f}}$ (if ${\widetilde{f}}\in L^1_{2\pi }$) has the form

$$\begin{aligned} \sum ^\infty _{k=1}(a_k(f)\sin kx-b_k(f)\cos kx)=: \sum ^\infty _{k=1} B_k(f)(x). \end{aligned}$$

Let $\Phi$ be the space of strictly increasing and continuous on $[0,2\pi ]$ functions $\omega (t)$, with property $\omega (0)=0$.

We will write $\omega \in B$, if $\omega \in \Phi$ and $\sum ^\infty _{k=n}k^{-1}\omega (k^{-1})=O(\omega (n^{-1}))$, $n\in {\mathbb {N}}=\{1,2,\dots \}$.

This class and its equivalent definitions was studied by Bary and Stechkin [3]. If $\omega \in \Phi$ and $\omega (2t)\le \omega (t)$, $t\in [0,\pi ]$, then $\omega \in \Delta _2$ (or $\omega$ satisfies $\Delta _2$-condition).

For $\omega \in \Phi$ and $m\in {\mathbb {N}}$ let us consider a Hölder type space

$$\begin{aligned} H^{m,\omega }_{p,\lambda }=\{f\in L^{p,\lambda }_{2\pi ,0}: \omega _m(f,\delta )_{p,\lambda }\le C\omega (\delta ), 2\pi \ge \delta \ge 0\}, \end{aligned}$$

(3)

where C depends on f and no depends on $\delta$. The last space with the norm

$$\begin{aligned} \Vert f\Vert _{p,\lambda ,m,\omega }=\Vert f\Vert _{p,\lambda }+\sup \limits _{0<t\le 2\pi }\frac{\omega _m(f,t)_{p,\lambda }}{\omega (t)} \end{aligned}$$

(4)

is a Banach one.

Testici and Israfilov [23] proved

Proposition 1

Let $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi }$. Then

• (i) $\Vert S_n(f)\Vert _{p,\lambda }\le C_1\Vert f\Vert _{p,\lambda }$, $n\in {\mathbb {N}}$;

• (ii) $\Vert {\widetilde{f}}\Vert _{p,\lambda }\le C_2\Vert f\Vert _{p,\lambda }$;

• (iii) $\Vert f-S_n(f)\Vert _{p,\lambda }\le (C_1+1)E_n(f)_{p,\lambda }$, $n\in {\mathbb {N}}$;

where $C_1$, $C_2$ does not depend on n and f.

Let $f\in L^p_{2\pi }$, $1\le p<\infty$, and $t^*_n(f)\in T_n$ be such that $\Vert f-t^*_n(f)\Vert _p=\inf \limits _{t_n\in T_n}\Vert f-t_n\Vert _p$.

Sunouchi [21] established the following result and its analogue for continuous periodic functions.

Proposition 2

Let $f\in L^p_{2\pi }$, $1\le p<\infty$, $r\in {\mathbb {N}}$, $0<\alpha <r$. Then the conditions

$$\begin{aligned} \Vert f-t^*_n(f)\Vert _p=O(n^{-\alpha }), \quad n\in {\mathbb {N}}, \end{aligned}$$

and

$$\begin{aligned} \Vert (t^*_n(f))^{(r)}\Vert _p=O(n^{r-\alpha }), \quad n\in {\mathbb {N}}, \end{aligned}$$

are equivalent.

Zhuk and Natanson [30] obtained the estimate of modulus of smoothness in terms of norms of derivatives of polynomials of best approximation.

Proposition 3

Let $1\le p\le \infty$, $m\in {\mathbb {N}}$, $r\in {\mathbb {Z}}_+$, $f\in L^p_{2\pi }$. If the series

$$\begin{aligned} \sum ^\infty _{k=1}k^{-m-1}\Vert (t^*_k(f))^{(m+r)}\Vert _{p} \end{aligned}$$

converges, then for $r\ge 1$ a function f is equivalent to $f_0$ such that $f'_0,\dots , f^{(r-1)}_0$ are absolutely continuous on each period and $f^{(r)}_0\in L^p_{2\pi }$ (for $r=0$ we set $f_0=f$) and the inequality

$$\begin{aligned} \omega _m(f^{(r)}_0,1/n)_{p}\le C\sum ^\infty _{k=n+1}k^{-m-1}\Vert (t^*_k(f))^{(m+r)}\Vert _{p}, \quad n\in {\mathbb {N}}, \end{aligned}$$

holds.

The aim of the present paper is to obtain Sunouchi and Zhuk–Natanson type results in the Morrey space using Riesz–Zygmund means (or Fourier partial sums) instead of polynomials of best approximation. Also we establish a two-sided estimate for the degree of approximation by Riesz–Zygmund means in Morrey space and some direct approximation results for Riesz–Zygmund and Bernstein–Rogosinski means. The approximation in Hölder type spaces based on the Morrey spaces is studied.

2 Auxiliary lemmas

Lemma 1

Let $1\le p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} \lim \limits _{h\rightarrow 0}\Vert f(\cdot +h)-f(\cdot )\Vert _{p,\lambda }=0 \end{aligned}$$

Proof

By definition for $\varepsilon >0$ we can find $t_n\in T_n$ such that

$$\begin{aligned} \Vert f-t_n\Vert _{p,\lambda }=\Vert f(\cdot +h)-t_n(\cdot +h)\Vert _{p,\lambda }<\varepsilon /3. \end{aligned}$$

Since $t_n$ is uniformly continuous. there exists $\delta >0$ such that $|t_n(x+h)-t_n(x)|<\varepsilon /(6\pi )$ for all $x\in {\mathbb {R}}$ and $|h|<\delta$. Then

$$\begin{aligned}{} & {} \Vert t_n(\cdot +h)-t_n(\cdot )\Vert _{p,\lambda }=\sup \limits _I\left( |I|^{\lambda -1} \int _I|t_n(x+h)-t_n(x)|^p\,dx\right) ^{1/p}\\{} & {} \le \sup \limits _I\left( |I|^{\lambda -1}|I|(\varepsilon /(6\pi ))^p\right) ^{1/p}\le (2\pi )^{\lambda /p}\varepsilon /(6\pi )\le \frac{2\pi \varepsilon }{6\pi }=\frac{\varepsilon }{3} \end{aligned}$$

and

$$\begin{aligned} \Vert f(\cdot +h)-f(\cdot )\Vert _{p,\lambda }\le & {} \Vert f(\cdot +h)-t_n(\cdot +h)\Vert _{p,\lambda }+ \Vert t_n(\cdot +h)-t_n(\cdot )\Vert _{p,\lambda }\\{} & {} +\Vert f(\cdot )-t_n(\cdot )\Vert _{p,\lambda }<\varepsilon , \quad |h|<\delta . \end{aligned}$$

□

Lemma 2 is known, e.g., it is used in [8] without references. But we can not find it in monographs where Morrey spaces are treated (see [1, 17]) and give a proof here.

Lemma 2

Let $1\le p<\infty$, $0<\lambda \le 1$, f(x, y) is measurable on ${\mathbb {R}}^2$ and $2\pi$-periodic in each variable. Then

$$\begin{aligned} \left\| \int ^{2\pi }_0|f(\cdot ,y)|\,dy\right\| _{p,\lambda }\le \int ^{2\pi }_0\Vert f(\cdot ,y)\Vert _{p,\lambda }\,dy. \end{aligned}$$

Proof

Let $I=[a,b]\subset {\mathbb {R}}$ and $b-a\le 2\pi$, By the generalized Minkowski inequality we have

$$\begin{aligned} \left( |I|^{\lambda -1}\int _I\left( \int ^{2\pi }_0|f(x,y)|\,dy\right) ^p\,dx\right) ^{1/p} \end{aligned}$$

$$\begin{aligned} \le |I|^{(\lambda -1)/p}\int ^{2\pi }_0\left( \int _I|f(x,y)|^p\,dx\right) ^{1/p}\,dy\le \int ^{2\pi }_0\Vert f(\cdot ,y)\Vert _{p,\lambda }\,dy. \end{aligned}$$

(5)

Taking the supremum in the left-hand side of (5) over I we obtain the inequality of Lemma. □

Corollary 1

Let $1\le p<\infty$, $h\in L^{p,\lambda }_{2\pi }$, $g\in L^1_{2\pi }$. Then the convolution $h*g(x)=\int ^{2\pi }_0 h(x-y)g(y)\,dy$ belongs to $L^{p,\lambda }_{2\pi }$ and $\Vert h*g\Vert _{p,\lambda }\le \Vert h\Vert _{p,\lambda }\Vert g\Vert _1.$

Proof

We take $f(x,y)=h(x-y)g(y)$ in Lemma 2. Then

$$\begin{aligned} \Vert h*g\Vert _{p,\lambda }\le & {} \left\| \int ^{2\pi }_0|h(x-y)||g(y)|\,dy\right\| _{p,\lambda } \\\le & {} \int ^{2\pi }_0 \Vert h(\cdot -y)g(y)\Vert _{p,\lambda }\,dy =\Vert h\Vert _{p,\lambda }\int ^{2\pi }_0|g(y)|\,dy=\Vert h\Vert _{p,\lambda }\Vert g\Vert _1. \end{aligned}$$

□

Lemma 3 is a variant of Marcinkiewicz multiplier theorem and it is stated in [15] without proof (the author claims that the proof is similar to one of corresponding result in weighted Lebesgue space [18]. But in [18] the multipliers in ${\mathbb {R}}^n$ were studied). We prove Lemma 3 by the method of Israfilov and Testici [23].

Lemma 3

Let $\{\mu _k\}^\infty _{k=0}$ satisfy the conditions

$$\begin{aligned} |\mu _k|\le C_1, \quad k\in {\mathbb {Z}}_+; \quad \sum ^{2^m-1}_{k=2^{m-1}}|\mu _k-\mu _{k+1}|\le C_2, \quad m\in {\mathbb {N}}. \end{aligned}$$

If $1<p<\infty$, $0<\lambda \le 1$ and $f\in L^{p,\lambda }_{2\pi }$ has the Fourier series (1), then there exists a function $F(f)\in L^{p,\lambda }_{2\pi }$ with the Fourier series $\sum ^\infty _{k=0}\mu _kA_k(f)(x)$ and $\Vert F(f)\Vert _{p,\lambda }\le C_3\Vert f\Vert |_{p,\lambda }$, where $C_3$ does not depend on f, p and $\lambda$.

Proof

Since $L^{p,\lambda }_{2\pi }\subset L^p_{2\pi }$ and Lemma 3 is well-known for $\lambda =1$, i.e. in $L^p_{2\pi }$, $1<p<\infty$ (see [31, Ch. XV, Theorem 4.14]), the function F(f) is correctly defined for $f\in L^{p,\lambda }_{2\pi }$. Coifman and Rochberg [9] proved that for any interval I and its indicator $X_I$ the inequality $M(M(X_I))(x)\le C_1M(X_I)(x)$ holds a.e. on ${\mathbb {R}}$. In other words, $M(X_I)$ belongs to the Muckenhoupt class $A_1({\mathbb {T}})$. Since $A_1({\mathbb {T}})\subset A_p({\mathbb {T}})$ and F(f) is bounded in $L^p_{w,2\pi }$ for $1<p<\infty$, $w\in A_p({\mathbb {T}})$ (see [5, Theorem 4.4]), one has for $I\subset [0,2\pi ]$

$$\begin{aligned} \int _I|F(f)(x)|^p\,dx= & {} \int ^{2\pi }_0|F(f)(x)|^p X_I(x)\,dx\le \\\le & {} \int ^{2\pi }_0|F(f)(x)|^p M(X_I)(x)\,dx\le C_2\int ^{2\pi }_0|f(x)|^p M(X_I)(x)\,dx. \end{aligned}$$

It is known that for $x\in [0,2\pi ]$

$$\begin{aligned} M(X_I)(x)\asymp X_I(x)+\sum ^\infty _{k=0}2^{-k}X_{J_k}(x), \quad J_k=(2^{k+1}I\setminus 2^kI)\cap [0,2\pi ], \end{aligned}$$

(6)

where $A(x)\asymp B(x)$, $x\in Y$, means that $C_1A(x)\le B(x)\le C_2A(x)$ for some $C_2>C_1>0$ and $x\in Y$, and mI is the interval of length m|I| such that the centers of I and mI are the same (see [13]). Using (6) we obtain

$$\begin{aligned}{} & {} \sup \limits _I|I|^{\lambda -1}\int _I|F(f)(x)|^p\,dx\\{} & {} \quad \le C_3\sup \limits _I|I|^{\lambda -1}\int _I|f(x)|^p\,dx\\{} & {} \quad \quad +C_3\sum ^\infty _{k=0}2^{-k}\sup \limits _I|I|^{\lambda -1}\int _{J_k}|f(x)|^p\,dx \\{} & {} \quad \le C_3\left( \Vert f\Vert ^p_{p,\lambda }+\sum ^\infty _{k=0}2^{-k}(2^{k+1})^{1-\lambda } \sup \limits _I|2^{k+1}I|^{\lambda -1}\int _{2^{k+1}I}|f(x)|^p\,dx\right) \\{} & {} \quad \le C_4\Vert f\Vert ^p_{p,\lambda }\left( 1+\sum ^\infty _{k=0}2^{-\lambda }\right) = C_5\Vert f\Vert ^p_{p,\lambda }. \end{aligned}$$

□

For a technical purpose we define the following iterated means

$$\begin{aligned} Z^r_{n,*}(f)=\sum ^n_{k=0}\left( 1-\frac{k^r}{(n+1)^r}\right) \left( 1-\frac{k^r}{(n+2)^r}\right) A_k(f)= Z^r_n(Z^r_{n+1}(f)). \end{aligned}$$

The result of Lemma 4 for even r may be found in [7] and for odd r in [29].

Lemma 4

For $f\in L^1_{2\pi }$ the following equalities

$$\begin{aligned} |Z^r_{n,*}(f)-Z^r_{n-1,*}(f)|=\frac{(n+2)^r-n^r}{(n+2)^rn^r}|(U^r_{n}(f))^{(r)}|, \end{aligned}$$

where $U^r_n(f)=Z^r_n(f)$ for even r and $U^r_n(f)=\widetilde{Z^r_n(f)}$ for odd r

Lemma 5

Let $r\in {\mathbb {N}}$, $1\le p<\infty$, $0<\lambda \le 1$. Then the operators $Z^r_n$ are uniformly bounded in $L^{p,\lambda }_{2\pi ,0}$ and

$$\begin{aligned} \Vert f-Z^r_n(f)\Vert _{p,\lambda }\le C(n+1)^{-r}\sum ^n_{k=0}(k+1)^{r-1}E_k(f)_{p,\lambda }. \end{aligned}$$

Proof

It is easy to see that $Z^r_n(f)=f*F^r_n$, where $F^r_n(x)=\pi ^{-1}(1/2+\sum ^n_{k=1}(1-k^r/(n+1)^r))\cos kx$. Timan [26] proved that $\{\Vert F^r_n\Vert _1\}^\infty _{n=1}$ is bounded. From this result and Corollary 1 the first statement of Lemma 2.5 follows. We note that for $1<p<\infty$ this assertion may be proved using the equality

$$\begin{aligned} Z^r_n(f)(x)=\sum ^n_{k=0} \frac{(k+1)^r-k^r}{(n+1)^r}S_k(f)(x) \end{aligned}$$

and Proposition 1.

For the second statement we use Timan’s method of proof of inequality (1.15) in [26]. Since the last paper is in Russian, we recommend the proof of Lemma 3.8 in [29] for representation of Timan’s method. Here also for $1<p<\infty$ the proof is more brief and uses Proposition 1. Namely,

$$\begin{aligned} \Vert f-Z^r_n(f)\Vert _{p,\lambda }\le & {} \sum ^n_{k=0}\frac{(k+1)^r-k^r}{(n+1)^r}\Vert f-S_k(f)\Vert _{p,\lambda }\le \\\le & {} \frac{C_2}{(n+1)^r}\sum ^n_{k=0}k^{r-1}E_k(f)_{p,\lambda }. \end{aligned}$$

□

The result of Lemma 6 (i) is established by Israfilov and Tozman [12] while the result of (ii) may be found in their paper [13]. These inequalities are known as direct and inverse theorems of approximation by trigonometric polynomials and in general form for the first time were obtained by Stechkin [20] for continuous functions.

Lemma 6

Let $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

• (i) $E_n(f)_{p,\lambda }\le C\omega _m(f,(n+1)^{-1})_{p,\lambda }$, $n\in {\mathbb {Z}}_+$.

• (ii) $\omega _m(f,n^{-1})_{p,\lambda }\le Cn^{-m}\sum ^n_{k=0}(k+1)^{m-1}E_k(f)_{p,\lambda }$, $n\in \mathbb Z_+$.

Lemma 7 (i) is due to Nikolskii and Stechkin while part (ii) is established by Stechkin in [20] in the case of continuous functions. We give proof of this part for the utility of a reader.

Lemma 7

 

• (i) Let $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$. If $t_n\in T_n$, $n\in {\mathbb {N}}$, then

  $$\begin{aligned} \Vert t^{(m)}_n\Vert _{p,\lambda }\le \left( \frac{n}{2\sin nh}\right) ^m\Vert \Delta ^m_h t_n\Vert _{p,\lambda }, \quad 0<h\le \pi /(2n). \end{aligned}$$

• (ii) If $m\in {\mathbb {N}}$, $f\in L^{p,\lambda }_{2\pi ,0}$ and $\tau _n(f)\in T_n$, $n\in {\mathbb {N}}$, satisfy the inequality

  $$\begin{aligned} \Vert f-\tau _n(f)\Vert _{p,\lambda }\le K\omega _m(f,n^{-1})_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

  Then $\omega _m(\tau _n(f),\delta )_{p,\lambda }\le C(K)\omega _m(f,\delta )_{p,\lambda }$ for some $C(K)>0$ and all $\delta \in [0,2\pi ]$.

Proof

(i) The result of Lemma 7 (i) can be proved by the method of Civin (see [25, Ch. 4, sect. 4.8.61]) or by the method of Zamansky (see [10, Ch. VII, Lemma 2.6]).

(ii) For $\tau _n(f)$ satisfying conditions of (ii) we have

$$\begin{aligned} \omega _m(\tau _n(f),\delta )_{p,\lambda }\le & {} \omega _m(f,\delta )_{p,\lambda }+ \omega _m(f-\tau _n(f),\delta )_{p,\lambda }\nonumber \\\le & {} \omega _m(f,\delta )_{p,\lambda }+2^m\Vert f-\tau _n(f)\Vert _{p,\lambda }\nonumber \\\le & {} \omega _m(f,\delta )_{p,\lambda }+C_1\omega _m(f,n^{-1})_{p,\lambda }\le (C_1+1)\omega _m(f,\delta )_{p,\lambda } \end{aligned}$$

(7)

for all $\delta \ge 1/n$. From (i) and (7) we also deduce that

$$\begin{aligned} \Vert \tau ^{(m)}_n(f)\Vert _{p,\lambda }\le (n/(2\sin 1))^m\omega _m(\tau _n(f), n^{-1})_{p,\lambda } \le C_2n^m\omega _m(f,n^{-1}). \end{aligned}$$

(8)

It is known that usual modulus of smoothness in translation-invariant spaces has the properties $\omega _m(f,\eta )_{p,\lambda }\le C_3(\eta /\delta )^m\omega _m(f,\delta )_{p,\lambda }$, $0<\delta \le \eta \le 2\pi$, and $\omega (\tau _n(f),\delta )_{p,\lambda } \le \Vert \tau ^{(m)}_n(f)\Vert _{p,\lambda }\delta ^m$ (see (7.8) and the proof of (7.12) in [10, Ch. II]). Using these facts and (8) we obtain

$$\begin{aligned} \omega (\tau _n(f),\delta )\le & {} \delta ^m\Vert \tau ^{(m)}_n(f)\Vert _{p,\lambda }\le C_2(n\delta )^m\omega _m(f,n^{-1})_{p,\lambda }\le \\\le & {} C_4(n\delta )^m(n^{-1}/\delta )^m\omega (f,\delta )_{p,\lambda }=C_4\omega (f,\delta )_{p,\lambda } \end{aligned}$$

for $0<\delta \le n^{-1}$ and (ii) is proved. □

Lemma 8 may be found in [8, Lemma 2.3].

Lemma 8

Let $r\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$ and $f\in L^{p,\lambda }_{2\pi ,0}$ be such that f, $f',\dots ,f^{(r-1)}$ are absolutely continuous on each period and $f^{(r)}\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} E_n(f)_{p,\lambda }\le Cn^{-r}\Vert f^{(r)}\Vert _{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Lemma 9 is proved for $r=1$ by Alexits [2] while its general variant for $r>0$ is established by Joó [16].

Lemma 9

Let $r>0$, $(X,\Vert \cdot \Vert _X)$ be a Banach space and $a_k\in X$, $k\in {\mathbb {Z}}_+$. Let $R^{(r)}_n=\sum ^{n-1}_{k=0}(1-k^r/n^r)a_k$, $T^{(r)}_n=\sum ^{n-1}_{k=0}(1-k^r/n^r)k^ra_k$, $n\in {\mathbb {N}}$. Then the condition $\Vert T^{(r)}_n\Vert _X\le C_1$, $n\in {\mathbb {N}}$, holds if and only if there exists $R\in X$ such that $\Vert R-R^{(r)}_n\Vert _X\le C_2n^{-r}$, $n\in {\mathbb {N}}$. Note that $C_2=C(r)C_1$ and vice versa.

Lemma 10 is proved by Timan [24]. By $D_n(t)$ we denote the trigonometric Dirichlet kernel $\sin (n+1/2)t/(2\sin (t/2))$, $n\in {\mathbb {N}}$.

Lemma 10

Let $\alpha _n=\pi k(n)/(2n+1)+O((n\ln (n+1)^{-1}))$, $n\in {\mathbb {N}}$, k(n) be an even natural number, $|\alpha _n|\le \pi$. Then the norms $\Vert D_n(\cdot +\alpha _n)+D_n(\cdot -\alpha _n)\Vert _1$ are bounded.

3 Approximation by Riesz–Zygmund and Bernstein–Rogosinski means is Morrey spaces

In view of Lemma 6 (ii) Theorem 1 sharpens the estimate of Lemma 5. We note that a similar result in $L^p_{2\pi }$, $1<p<\infty$, was proved by Trigub (see [28, sect. 8.2.6]).

Theorem 1

Let $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} C^{-1}\omega _m(f,1/n)_{p,\lambda }\le \Vert f-Z^m_n(f)\Vert _{p,\lambda }\le C\omega _{m}(f,1/n)_{p,\lambda }, \quad n\in {\mathbb {N}}, \end{aligned}$$

(9)

for some $C>0$.

Proof

Suppose that $\tau _n(f)\in T_n$ satisfies the equality

$$\begin{aligned} \Vert f-\tau _n(f)\Vert _{p,\lambda } =E_n(f)_{p,\lambda }. \end{aligned}$$

Then by Lemma 5 we have

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{p,\lambda }\le & {} \Vert f-\tau _n(f)\Vert _{p,\lambda }+\Vert Z^m_n(\tau _n(f))-\tau _n(f))\Vert _{p,\lambda }+\\{} & {} +\Vert Z^m_n(f-\tau _n(f))\Vert _{p,\lambda }\le C_1E_n(f)_{p,\lambda }+\Vert Z^m_n(\tau _n(f))-\tau _n(f))\Vert _{p,\lambda }. \end{aligned}$$

It is easy to see that for $t_n\in T_n$ and even m the equality $|t_n-Z^m_n(t_n)|=(n+1)^{-m}|t^{(m)}_n|$ holds, while for odd m we have $|t_n-Z^m_n(t_n)|=(n+1)^{-m}|\widetilde{t^{(m)}_n}|$. Thus, by Lemmas 6 (i) and 7 (i)

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{p,\lambda }\le & {} C_2\omega _m(f,n^{-1})_{p,\lambda }+\\{} & {} +C_2(n+1)^{-m}n^m\omega _m(\tau _n(f),n^{-1})_{p,\lambda }\le C_3\omega _m(\tau _n(f),n^{-1})_{p,\lambda }. \end{aligned}$$

The right-hand side inequality (9) is proved.

Using again the property $\omega _m(t_n,\delta )_{p,\lambda }\le \Vert t^{(m)}_n\Vert _{p,\lambda }\delta ^m$ for $t_n\in T_n$, we obtain

$$\begin{aligned} \omega _m(f,n^{-1})_{p,\lambda }\le & {} \omega _m(f-Z^m_n(f),n^{-1})_{p,\lambda }+\omega _m(Z^m_n(f),n^{-1})_{p,\lambda }\le \\\le & {} C_4(\Vert f-Z^m_n(f)\Vert _{p,\lambda }+n^{-m}\Vert (Z^m_n(f))^{(m)}\Vert _{p,\lambda }. \end{aligned}$$

Using above equalities for $|t_n-Z^m_n(t_n)|$ we find that

$$\begin{aligned} n^{-m}\Vert (Z^m_n(f))^{(m)}\Vert _{p,\lambda }=\left( \frac{n+1}{n}\right) ^m \Vert U^m_n(f)-Z^m_n(U^m_n(f))\Vert _{p,\lambda }, \end{aligned}$$

(10)

where $U^m_n(f)=Z^m_n(f)$ for even m and $U^m_n(f)=\widetilde{Z^m_n}(f)$ for odd m. By Proposition 1 and Lemma 5

$$\begin{aligned} \omega _m(f,n^{-1})_{p,\lambda }\le & {} C_4\Vert f-Z^m_n(f)\Vert _{p,\lambda }+2^mC_4\Vert U^m_n(f-Z^m_n(f))\Vert _{p,\lambda }\le \\\le & {} C_6\Vert f-Z^m_n(f)\Vert _{p,\lambda } \end{aligned}$$

and the left-hand side inequality (9) is proved. □

We will write $g\in W^rL^{p,\lambda }_{2\pi ,0}$, where $r\in \mathbb N$, $1\le p<\infty$, $0<\lambda \le 1$, if $g,g',\dots ,g^{(r-1)}$ are absolutely continuous on any period and $g^{(r)}\in L^{p,\lambda }_{2\pi ,0}$. Theorem 2 is an analogue and extension of the result of Bustamante [6, Theorem 1] obtained in the case $r=1$ and $f\in L^p_{2\pi }$, $1<p<\infty$.

Theorem 2

Let $r\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f,{\widetilde{f}}\in W^rL^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} \Vert f-Z^r_n(f)\Vert _{p,\lambda }\le Cn^{-r}\Vert f^{(r)}\Vert _{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

(11)

For $r\in {\mathbb {N}}$, $p=1$, $0<\lambda \le 1$ and $f,{\widetilde{f}}\in W^rL^{p,\lambda }_{2\pi ,0}$ we have the inequality

$$\begin{aligned} \Vert f-Z^r_n(f)\Vert _{1,\lambda }\le Cn^{-r}\Vert (\varphi (f))^{(r)}\Vert _{1,\lambda }, \quad n\in {\mathbb {N}}, \end{aligned}$$

where $\varphi (f)=f$ for even r ($r\in 2{\mathbb {N}}$) and $\varphi (f)={\widetilde{f}}$ for odd r ($r\in 2{\mathbb {N}}-1$).

Proof

Let $r\in {\mathbb {N}}$ and $f,{\widetilde{f}}\in W^rL^{p,\lambda }_{2\pi ,0}$. Then the Fourier series of $f^{(r)}$ is

$$\begin{aligned}{} & {} \sum ^\infty _{k=1}k^r(a_k(f)\cos (kx+r\pi /2)+b_k(f)\cos (kx+r\pi /2))=\\{} & {} \quad ={\left\{ \begin{array}{ll} \pm \sum ^\infty _{k=1}k^rA_k(f)(x), \quad r\in 2{\mathbb {N}}; \\ \pm \sum ^\infty _{k=1}k^rB_k(f)(x), \quad r\in 2{\mathbb {N}}-1. \end{array}\right. } \end{aligned}$$

In a similar manner, the Fourier series of ${\widetilde{f}}^{(r)}$ for odd r has the form $\pm \sum ^\infty _{k=1}k^rA_k(f)(x)$. If $\varphi (f)\in W^rL^{p,\lambda }_{2\pi ,0}$, then by Lemma 5 the norms of $T^{(r)}_{n+1}=\sum ^n_{k=0}(1-k^r/(n+1)^r)k^rA_k(f)$ in $L^{p,\lambda }_{2\pi }$ are bounded by $C_1\Vert (\varphi (f))^{(r)}\Vert _{p,\lambda }$. Thus, by Lemma 9 Riesz-Zygmund means $Z^r_n(f)$ converges to Z(f) in $L^{p,\lambda }_{2\pi }$ and

$$\begin{aligned} \Vert Z^r_n(f)-Z(f)\Vert _{p,\lambda }\le C_2\frac{\Vert (\varphi (f))^{(r)}\Vert _{p,\lambda }}{(n+1)^r}, \quad n\in {\mathbb {N}} \end{aligned}$$

(12)

for all $1\le p<\infty$ and $0<\lambda \le 1$. It is easy to see that Fourier coefficients of Z(f) and f are the same and $Z(f)=f$. Finally, for $1<p<\infty$ by Proposition 1 (ii) we obtain $\Vert (\varphi (f))^{(r)}\Vert _{p,\lambda }\le C_3\Vert f^{(r)}\Vert _{p,\lambda }$ and (11) is valid. □

Now we obtain an analogue of Theorem 1 in the case $p=1$. For $f\in L^{1,\lambda }_{2\pi ,0}$, $m\in {\mathbb {N}}$ and $\varphi (g)$ defined as in Theorem 2 we introduce a K-functional

$$\begin{aligned} \widetilde{K_m}(f,t)_{1,\lambda }=\inf \{\Vert f-g\Vert _{1,\lambda }+t^m\Vert (\varphi (g))^{(r)}\Vert _{1,\lambda }: g,{\widetilde{g}}\in W^mL^{1,\lambda }_{2\pi }\} \end{aligned}$$

Theorem 3

Let $p=1$, $0<\lambda \le 1$, $m\in {\mathbb {N}}$, $f\in L^{1,\lambda }_{2\pi ,0}$. Then there exist $K_2>K_1>0$ such that

$$\begin{aligned} K_1\widetilde{K_m}(f,n^{-1})_{1,\lambda }\le \Vert f-Z^m_n(f)\Vert _{1,\lambda }\le K_2\widetilde{K_m}(f,n^{-1})_{1,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

It is clear that

$$\begin{aligned} \widetilde{K_m}(f,n^{-1})_{1,\lambda }\le \Vert f-Z^m_n(f)\Vert _{1,\lambda }+n^{-m}\Vert (\varphi (Z^m_n(f)))^{(m)}\Vert _{1,\lambda }. \end{aligned}$$

By the proof of Theorem 1 and Lemma 5 we have $(\varphi (Z^m_n(f)))^{(m)}=\pm (n+1)^m(Z^m_n(f)-Z^m_n(Z^m_n(f)))$ and

$$\begin{aligned} \widetilde{K_m}(f,n^{-1})_{1,\lambda }\le & {} \Vert f-Z^m_n(f)\Vert _{1,\lambda } +\\{} & {} +((n+1)/n)^m\Vert Z^m_n(f)-Z^m_n(Z^m_n(f))\Vert _{1,\lambda }\le C_1\Vert f-Z^m_n(f)\Vert _{1,\lambda }. \end{aligned}$$

On the other hand, for every g such that $g,{\widetilde{g}}\in W^mL^{1,\lambda }_{2\pi }$ we obtain by Lemma 5 and (12) from the proof of Theorem 2

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{1,\lambda }\le & {} \Vert f-g-Z^m_n(f-g)\Vert _{1,\lambda }+\Vert g-Z^m_n(g)\Vert _{1,\lambda }\le \\\le & {} C_2\Vert f-g\Vert _{1,\lambda }+C_3\frac{\Vert (\varphi (g))^{(m)}\Vert _{1,\lambda }}{n^m}. \end{aligned}$$

Taking the infimum over all such g we find that

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{1,\lambda }\le \max (C_2,C_3)\widetilde{K_m}(f,n^{-1})_{1,\lambda }. \end{aligned}$$

□

From Theorems 1 and 3 we deduce for Fejér means

Corollary 2

Let $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then for some $C_2>C_1>0$ we have

$$\begin{aligned} C_1\omega _1(f,n^{-1})_{p,\lambda }\le \Vert f-\sigma _n(f)\Vert _{p,\lambda }\le C_2\omega _1(f,n^{-1})_{p,\lambda }, n\in {\mathbb {N}}. \end{aligned}$$

If $p=1$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$, then for some $C_4>C_3>0$

$$\begin{aligned} C_3\widetilde{K_1}(f,n^{-1})_{1,\lambda }\le \Vert f-\sigma _n(f)\Vert _{1,\lambda }\le K_2\widetilde{K_1}(f,n^{-1})_{1,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Now we consider Bernstein–Rogosinski means

$$\begin{aligned} R_{n,\alpha }(f)=2^{-1}(S_n(f)(\cdot -\alpha _n)+S_n(f)(\cdot +\alpha _n)), \end{aligned}$$

where $\{\alpha _n\}^\infty _{n=1}$ satisfies the conditions of Lemma 10. Similar to Riesz–Zygmund means approximation by this means under some restrictions can be estimated by the modulus of smoothness of higher order, namely, of order 2. Such estimate was obtained for continuous functions by Stechkin. In the case $\alpha _n=\pi /(2n)$ see Stechkin’s result in [11, Ch. 5, Theorem 2.4].

Theorem 4

Let $1\le p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. If

$$\begin{aligned} \alpha _n=\frac{\pi k(n)}{2n+1}+O\left( \frac{1}{n\ln (n+1)}\right) , \quad n\in {\mathbb {N}}, \end{aligned}$$

$\{k(n)\}^\infty _{n=1}$ is a bounded sequence of even natural numbers, then

$$\begin{aligned} \Vert f-R_{n,\alpha }(f)\Vert _{p,\lambda }\le C_2\omega _2(f,n^{-1})_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

Let $\tau _n(f)\in T_n$ be such that $\Vert f-\tau _n(f)\Vert _{p,\lambda }=E_n(f)_{p,\lambda }$. Then $S_n(\tau _n(f))=\tau _n(f)$ and we have by virtue of Corollary 1, Lemmas 6 and 10

$$\begin{aligned}{} & {} \Vert f-R_{n,\alpha }(f)\Vert _{p,\lambda }\\{} & {} \quad =\Vert f-(S_n(f-\tau _n(f))(\cdot -\alpha _n)+S_n(f-\tau _n(f)) (\cdot +\alpha _n))/2-\\{} & {} \quad \quad -(\tau _n(f)(\cdot -\alpha _n)+\tau _n(f)(\cdot +\alpha _n))/2\Vert _{p,\lambda }\le \Vert f-(f(\cdot -\alpha _n)+f(\cdot +\alpha _n))/2\Vert _{p,\lambda }+\\{} & {} \quad \quad +2^{-1}\Vert R_{n,\alpha }(f-\tau _n(f))\Vert _{p,\lambda }+ 2^{-1}\Vert f(\cdot -\alpha _n)-\tau _n(f)(\cdot -\alpha _n)\Vert _{p,\lambda }+\\{} & {} \quad \quad +2^{-1}\Vert (f-\tau _n(f))(\cdot +\alpha _n)\Vert _{p,\lambda }\le \frac{1}{2}\omega _2(f,\alpha _n)_{p,\lambda }+ C_1\Vert f-\tau _n(f)\Vert _{p,\lambda }+\\{} & {} \quad \quad + \Vert f-\tau _n(f)\Vert _{p,\lambda }\le C_2(\omega _2(f,\alpha _n)_{p,\lambda }+\omega _2(f,n^{-1})_{p,\lambda }). \end{aligned}$$

Finally, it is known that for $t_2>t_1>0$ one has $\omega _2(f,t_2)_{p,\lambda }\le (2t_2/t_1)^2\omega _2(f,t_1)_{p,\lambda }$ (see, e.g., [11, Ch. 3, (4.18)] for continuous functions). Thus, $\omega _2(f,\alpha _n)_{p,\lambda }\le C_3\omega _2(f,n^{-1})_{p,\lambda }$ and the inequality of Theorem 4 holds. □

Corollary 3

Let $\alpha _n=\pi /(2n)$, $n\in {\mathbb {N}}$, $1\le p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} \Vert f-R_{n,\alpha }(f)\Vert _{p,\lambda }\le C\omega _2(f,n^{-1})_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

Let us consider $\beta _n=\pi /n$, $n\in {\mathbb {N}}$. Then

$$\begin{aligned} \frac{\pi }{n}-\frac{2\pi }{2n+1}=\frac{2\pi }{2n(2n+1)}=O\left( \frac{1}{n\ln (n+1)}\right) \end{aligned}$$

and $\{\beta _n\}^\infty _{n=1}$ satisfies the conditions of Theorem 4. But $\alpha _n=\beta _{2n}$ and

$$\begin{aligned} \Vert f-R_{n,\alpha }(f)\Vert _{p,\lambda }= & {} \Vert f-R_{2n,\beta }(f)\Vert _{p,\lambda }\le C_1\omega (f,(2n)^{-1})_{p,\lambda }\\\le & {} C_1\omega (f,n^{-1})_{p,\lambda }. \end{aligned}$$

□

4 Inverse and equivalence theorems of Sunouchi and Zhuk–Natanson type

The counterpart of Theorem 6 was proved in [29] for variable exponent Lebesgue spaces $L^{p(\cdot )}_{2\pi }$. In [29] a general class of exponents $p(\cdot )$ was considered and an analogue of Proposition 1 is not valid in all such spaces $L^{p(\cdot )}_{2\pi }$. Here we give a more simple proof of such result than in [29] in the case $1<p<\infty$.

The content of Theorem 5 is close to one of Proposition 3. Since $S_n$ are linear operators and $S_n(S_m)=S_{\min (m,n)}$, the proof is simpler than for polynomials of best approximation.

Theorem 5

Let $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} \omega _m(f,1/n)_{p,\lambda }\le C\sum ^\infty _{k=n+1}k^{-m-1}\Vert S^{(m)}_n(f)\Vert _{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

Using again the property $\omega _m(S_n(f),\delta )_{p,\lambda }\le \Vert S^{(m)}_n(f)\Vert _{p,\lambda }\delta ^m$, $\delta \in [0,2\pi ]$, we have

$$\begin{aligned} \omega _m(f,1/n)_{p,\lambda }\le & {} \omega _m(f-S_n(f),1/n)_{p,\lambda }+\omega _m(S_n(f),1/n)_{p,\lambda }\le \nonumber \\\le & {} 2^m\Vert f-S_n(f)\Vert _{p,\lambda } +n^{-m}\Vert S^{(m)}_n(f)\Vert _{p,\lambda }. \end{aligned}$$

(13)

Due to Lemma 8 and Proposition 1 (iii) we write for $k\in {\mathbb {Z}}_+$

$$\begin{aligned}{} & {} \Vert f-S_{2^kn}(f)\Vert _{p,\lambda }-\Vert f-S_{2^{k+1}n}(f)\Vert _{p,\lambda }\\{} & {} \quad \le \Vert S_{2^{k+1}n}(f)-S_{2^kn}(f)\Vert _{p,\lambda }=\\{} & {} \quad =\Vert S_{2^{k+1}n}(f)-S_{2^kn}(S_{2^{k+1}n}(f))\Vert _{p,\lambda }\le C_1E_{2^kn}(S_{2^{k+1}n}(f))_{p,\lambda }\le \\{} & {} \quad \le C_2(2^kn)^{-m}\Vert S^{(m)}_{2^{k+1}n}(f)\Vert _{p,\lambda }. \end{aligned}$$

We note that for $f\in L^{p,\lambda }_{2\pi }$ and $m,n\in {\mathbb {N}}$, $m<n$, by Proposition 1 (i)

$$\begin{aligned} \Vert S_m(f)\Vert _{p,\lambda }=\Vert S_m(S_n(f))\Vert _{p,\lambda }\le C_3\Vert S_n(f)\Vert _{p,\lambda }. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert f-S_{2^kn}(f)\Vert _{p,\lambda }-\Vert f-S_{2^{k+1}n}(f)\Vert _{p,\lambda }\le C_4\sum ^{2^{k+2}n}_{j=2^{k+1}n+1}\frac{\Vert S^{(m)}_j(f)\Vert _{p,\lambda }}{j^{m+1}}. \end{aligned}$$

Summing up these inequalities over $k=0,1,\dots$, we obtain

$$\begin{aligned} \Vert f-S_{n}(f)\Vert _{p,\lambda }\le C_5\sum ^{\infty }_{j=2n+1}j^{-m-1}\Vert S^{(m)}_j(f)\Vert _{p,\lambda }. \end{aligned}$$

(14)

If we substitute (14) into (13), then we find that

$$\begin{aligned} \omega _m(f,1/n)_{p,\lambda }\le & {} C_4\sum ^{\infty }_{j=2n+1}j^{-m-1}\Vert S^{(m)}_j(f)\Vert _{p,\lambda }+\\{} & {} +2^{m+1}C_3\sum ^{2n}_{j=n+1}j^{-m-1}\Vert S^{(m)}_j(f)\Vert _{p,\lambda }\le C_6\sum ^{\infty }_{j=n+1}j^{-m-1}\Vert S^{(m)}_j(f)\Vert _{p,\lambda }. \end{aligned}$$

□

By virtue of Lemma 6 (i) Theorem 6 sharpens the result of Theorem ??.

Theorem 6

Let $r,m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$. Then

$$\begin{aligned} \omega _m(f,1/n)_{p,\lambda }\le C\sum ^{\infty }_{j=n+1}j^{-m-1}\Vert (Z^r_j(f))^{(m)}\Vert _{p,\lambda }. \end{aligned}$$

Proof

Let us put $\mu ^{(n)}_k=(2n+1)^r/((2n+1)^r-k^r)$, $k=0,1,\dots ,n$, and $\mu ^{(n)}_k=0$ for $k>n$. Then $\mu ^{(n)}_k$ increases for $k=0,1,\dots ,n$ and

$$\begin{aligned} |\mu ^{(n)}_k|=\frac{(2n+1)^r}{(2n+1)^r-n^r}\le \frac{(3n)^r}{(2n)^r-n^r}=\frac{3^r}{2^r-1}. \end{aligned}$$

On the other hand,

$$\begin{aligned} \sum ^\infty _{k=0}|\mu ^{(n)}_k-\mu ^{(n)}_{k+1}|=\sum ^{n-1}_{k=0}(\mu ^{(n)}_{k+1}-\mu ^{(n)}_k)+\mu ^{(n)}_n-0 \le 2\mu ^{(n)}_n=2\frac{3^r}{2^r-1}. \end{aligned}$$

For the operator $F_n(f)=\sum ^\infty _{k=0}\mu ^{(n)}_k A_k(f)$ we have the equality $F_n(Z^r_{2n}(f))=S_n(f)$. We obtain $\Vert S_n(f)\Vert _{p,\lambda }\le C_1\Vert Z^r_{2n}(f)\Vert _{p,\lambda }$, $n\in {\mathbb {N}}$, where $C_1$ does not depend on n by Lemma 3. By Theorem 5 we have for $n\in {\mathbb {N}}$

$$\begin{aligned} \omega _m(f,1/n)_{p,\lambda }\le & {} C_2\sum ^\infty _{j=n+1}j^{-m-1}\Vert S^{(m)}_j(f)\Vert _{p,\lambda } \le \\\le & {} C_2C_1\sum ^{\infty }_{j=n+1}\frac{\Vert (Z^r_{2j}(f))^{(m)}\Vert _{p,\lambda }}{j^{m+1}} \le 2^{m+1}C_1C_2\sum ^{\infty }_{j=n+1}\frac{\Vert (Z^r_j(f))^{(m)}\Vert _{p,\lambda }}{j^{m+1}}. \end{aligned}$$

□

From Theorem 6 and Lemma 5 (i) we deduce

Corollary 4

Under conditions of Theorem 6 we have the inequality

$$\begin{aligned} E_n(f)_{p,\lambda }\le C\sum ^{\infty }_{j=n+1}j^{-m-1}\Vert (Z^r_j(f))^{(m)}\Vert _{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Theorem 7 is a counterpart of Proposition 2.

Theorem 7

Suppose that $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$ and $\omega \in B\cap \Delta _2$. Then the conditions $f\in H^{m,\omega }_{p,\lambda }$ and

$$\begin{aligned} \Vert (Z^m_n(f))^{(m)}\Vert _{p,\lambda }=O(n^m\omega (n^{-1})), \quad n\in {\mathbb {N}}, \end{aligned}$$

(15)

are equivalent.

Proof

If (15) holds, then by Theorem 5 we obtain

$$\begin{aligned} \omega _m(f,n^{-1})_{p,\lambda }\le & {} C_1\sum ^\infty _{k=n+1}k^{-m-1}\Vert (Z^m_k(f))^{(m)}\Vert _{p,\lambda }\le \\\le & {} C_2\sum ^\infty _{k=n+1}k^{-1}\omega (k^{-1})\le C_3\omega (n^{-1}), \quad n\in {\mathbb {N}}, \end{aligned}$$

due to the condition $\omega \in B$. Since $\omega$ satisfies the $\Delta _2$-condition $\omega (2t)\le C_4\omega (t)$, $t\in [0,\pi ]$, we have for $n\in {\mathbb {N}}$ and $\delta \in ((n+1)^{-1},n^{-1}]$

$$\begin{aligned} \omega _m(f,\delta )_{p,\lambda }\le & {} \omega _m(f,n^{-1})_{p,\lambda }\le C_3\omega (n^{-1})\le \\\le & {} C_3\omega (2(n+1)^{-1})\le C_3C_4\omega ((n+1)^{-1})\le C_3C_4\omega (\delta ). \end{aligned}$$

For $\delta \in (1,2\pi ]$ a similar inequality follows from monotonicity and boundedness of $\omega _m(f,\delta )_{p,\lambda }$ on $[0,2\pi ]$. Thus, $f\in H^{m,\omega }_{p,\lambda }$.

Conversely, let $f\in H^{m,\omega }_{p,\lambda }$. By Lemma 7 (ii) and Theorem 1

$$\begin{aligned} \Vert (Z^m_n(f))^{(m)}\Vert _{p,\lambda }\le & {} C_5n^m\omega _m(Z^m_n(f),1/n)_{p,\lambda }\le \\\le & {} C_6n^m\omega _m(f,1/n)_{p,\lambda }\le C_7n^m\omega (1/n) \end{aligned}$$

and (15) is proved. □

The statement below sharpens the Proposition 2 in Morrey setting (the case $\alpha =m$ is included).

Corollary 5

Let $m\in {\mathbb {N}}$, $1<p<\infty$, $0<\lambda \le 1$, $f\in L^{p,\lambda }_{2\pi ,0}$ and $0<\alpha \le m$. Then the conditions $\omega _m(f,\delta )_{p,\lambda }=O(\delta ^\alpha )$, $\delta \in [0,2\pi ]$, and $\Vert (Z^m_n(f))^{(m)}\Vert _{p,\lambda }=O(n^{m-\alpha })$, $n\in {\mathbb {N}}$, are equivalent.

In particular, the conditions $\omega _1(f,\delta )_{p,\lambda }=O(\delta ^\alpha )$, $\delta \in [0,2\pi ]$, and $\Vert \sigma '_n(f)\Vert _{p,\lambda }=O(n^{1-\alpha })$, $n\in {\mathbb {N}}$, are equivalent for $0<\alpha \le 1$.

5 Approximation in Hölder type spaces

We give an application of Lemma 7 to problems of approximation in Hölder type spaces. The following Theorem 8 is an analogue of the result by Telyakovskii [22] concerning uniform Hölder spaces. Let us remind that Hölder type spaces $H^{m,\omega }_{p,\lambda }$ and its norms $\Vert \cdot \Vert _{p,\lambda ,m,\omega }$ are defined in (3) and (4).

Theorem 8

Let $1\le p<\infty$, $0<\lambda \le 1$, $m\in {\mathbb {N}}$, $\omega ,\varphi \in \Phi$ and $\eta (t)=\omega (t)/\varphi (t)$ be increasing on $(0,2\pi ]$. If $f\in H^{m,\omega }_{p,\lambda }$ and $t_n\in T_n$, $n\in {\mathbb {N}}$, are such that the inequality

$$\begin{aligned} \Vert f-t_n\Vert _{p,\lambda }\le C\omega _m(f,1/n)_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

(16)

holds, then

$$\begin{aligned} \Vert f-t_n\Vert _{p,\lambda ,m,\varphi }\le C\eta (1/n), \quad n\in {\mathbb {N}}. \end{aligned}$$

Proof

By the condition of Theorem 8 we have

$$\begin{aligned} \Vert f-t_n\Vert _{p,\lambda }\le C_1\omega (n^{-1})=C_1\eta (n^{-1})\varphi (n^{-1})\le C_1\varphi (2\pi )\eta (n^{-1}). \end{aligned}$$

(17)

Let us estimate $\omega _m(f-t_n,\delta )_{p,\lambda }/\varphi (\delta )$. For $\delta \ge 1/n$ we have

$$\begin{aligned} \frac{\omega _m(f-t_n,\delta )_{p,\lambda }}{\varphi (\delta )}\le \frac{C_2\Vert f-t_n\Vert _{p,\lambda }}{\varphi (n^{-1})}\le C_3\frac{\omega (n^{-1})}{\varphi (n^{-1})}=C_3\eta (n^{-1}). \end{aligned}$$

(18)

For $0<\delta <1/n$ by virtue of Lemma 7 (ii) we find that

$$\begin{aligned} \frac{\omega _m(f-t_n,\delta )_{p,\lambda }}{\varphi (\delta )}\le \frac{\omega _m(f,\delta )_{p,\lambda }+\omega _m(t_n,\delta )_{p,\lambda }}{\varphi (\delta )}\le C_4\frac{\omega (\delta )}{\varphi (\delta )}\le C_4\eta (n^{-1}). \end{aligned}$$

(19)

From (17), (18) and (19) we deduce the statement of Theorem 8. $\square$

Now we give some applications of Theorem 8 to approximation by famous means of Fourier series.

Corollary 6

Let $1< p<\infty$, $0<\lambda \le 1$, $m\in {\mathbb {N}}$, $\omega ,\varphi \in \Phi$ and $\eta (t)=\omega (t)/\varphi (t)$ be increasing on $(0,2\pi ]$. If $f\in H^{m,\omega }_{p,\lambda }$, then

$$\begin{aligned} \Vert f-S_n(f)\Vert _{p,\lambda ,m,\varphi }\le C\eta (1/n), \quad n\in {\mathbb {N}}. \end{aligned}$$

(20)

Proof

By Proposition 1 (iii) and Lemma 6 (i) we have

$$\begin{aligned} \Vert f-S_n(f)\Vert _{p,\lambda }\le C_1\omega _m(f,1/n)_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Using Theorem 8 we obtain (20). □

Now we consider the Vallée-Poussin means

$$\begin{aligned} v_n(f)(x)=2Z^1_{2n-1}(f)(x)-Z^1_{n-1}(f)(x)=2\sigma _{2n-1}(f)-\sigma _{n-1}(f). \end{aligned}$$

Corollary 7

Let $1\le p<\infty$, $0<\lambda \le 1$, $m\in {\mathbb {N}}$, $\omega ,\varphi \in \Phi$ and $\eta (t)=\omega (t)/\varphi (t)$ be increasing on $(0,2\pi ]$. If $f\in H^{m,\omega }_{p,\lambda }$, then

$$\begin{aligned} \Vert f-v_n(f)\Vert _{p, \lambda ,m,\varphi }\le C\eta (1/n), \quad n\in {\mathbb {N}}. \end{aligned}$$

(21)

Proof

Let$\tau _n(f)\in T_n$ be such that $\Vert f-\tau _n(f)\Vert _{p,\lambda }= E_n(f)_{p,\lambda }$. It is known that $v_n(t_n)=t_n$ for $t_n\in T_n$, $n\in {\mathbb {N}}$, and that $Z^1_n(f)=F^1_n*f$, where $\Vert F^1_n\Vert _1=1$ (see [4, Ch. 1, § 47,(47.10)]). By Corollary 1 we have $\Vert v_n(f)\Vert _{p,\lambda }\le 3\Vert f\Vert _{p,\lambda }$, while by the previous inequality and Lemma 6 (i) we find that

$$\begin{aligned} \Vert f-v_n(f)\Vert _{p,\lambda }\le & {} \Vert f-\tau _n(f)\Vert _{p,\lambda }+\Vert \tau _n(f)-v_n(\tau _n(f))\Vert _{p,\lambda }+\\{} & {} +\Vert v_n(f-\tau _n(f))\Vert _{p,\lambda }\le 4\Vert f-\tau _n(f)\Vert _{p,\lambda }\\\le & {} C_1\omega _m(f,1/n)_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

Using Theorem 8 we obtain (21). □

Corollary 8

Let $1< p<\infty$, $0<\lambda \le 1$, $m\in {\mathbb {N}}$, $\omega ,\varphi \in \Phi$ and $\eta (t)=\omega (t)/\varphi (t)$ be increasing on $(0,2\pi ]$. If $f\in H^{m,\omega }_{p,\lambda }$, then

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{p,\lambda ,m,\varphi }\le C\lambda (1/n), \quad n\in {\mathbb {N}}. \end{aligned}$$

(22)

If $p=1$, m is even and other conditions above are valid, then (22) also holds. Finally, if $p=1$, m is odd and $\omega \in B_m$, then (22) is valid.

Proof

By Theorem 1 we have for $1<p<\infty$ and $m\in {\mathbb {N}}$

$$\begin{aligned} \Vert f-Z^m_n(f)\Vert _{p,\lambda }\le C_1\omega _m(f,n^{-1})_{p,\lambda }, \quad n\in {\mathbb {N}}. \end{aligned}$$

(23)

In turn, (26) is valid for $p=1$ and even m by Theorem 3. Applying Theorem 8 we prove (22) in these cases.

If $p=1$, $\omega \in B_m$ and $f\in H^{m,\omega }_{p,\lambda }$, the inequality of Lemma 5 together with Lemma 6 (i) gives us

$$\begin{aligned} \Vert f-Z^r_n(f)\Vert _{p,\lambda }\le \frac{C_2}{(n+1)^{m}}\sum ^n_{k=0}(k+1)^{r-1}\omega (k^{-1})\le C_3\omega (n^{-1}). \end{aligned}$$

Repeating the proof of Theorem 8 we obtain

$$\begin{aligned} \frac{\omega _m(f-Z^m_n(f),\delta )_{p,\lambda }}{\varphi (\delta )}\le \frac{C_3\Vert f-Z^m_n(f)\Vert _{p,\lambda }}{\varphi (n^{-1})}\le C_3\frac{\omega (n^{-1})}{\varphi (n^{-1})}=C_3\eta (n^{-1}) \end{aligned}$$

(24)

for $\delta \ge n^{-1}$. Since the translation and the convolution commute, we obtain $\Delta ^m_h(f*F^m_n)=\Delta ^m_hf*F^m_n$ (the definition of $F^m_n$ see in the proof of Lemma 5) and $\omega _m(Z^m_n(f),\delta )_{p,\lambda }\le C_4\omega _m(f,\delta )_{p,\lambda }$. Now we have

$$\begin{aligned} \frac{\omega _m(f-Z^m_n(f),\delta )_{p,\lambda }}{\varphi (\delta )}\le \frac{\omega _m(f,\delta )_{p,\lambda }+\omega _m(Z^m_n(f),\delta )_{p,\lambda }}{\varphi (\delta )}\le \end{aligned}$$

$$\begin{aligned} C_5\frac{\omega (\delta )}{\varphi (\delta )}\le C_5\eta (n^{-1}) \end{aligned}$$

(25)

for $0<\delta <n^{-1}$. From (24), (25) and obvious inequality $\Vert f-Z^r_n(f)\Vert _{p,\lambda }\le C_3\varphi (1)\eta (n^{-1})$ we deduce (22). □

Corollary 9

Let $1\le p<\infty$, $0<\lambda \le 1$, $\omega ,\varphi \in \Phi$ and $\eta (t)=\omega (t)/\varphi (t)$ be increasing on $(0,2\pi ]$. If $\{\alpha _n\}^\infty _{n=1}$ satisfies the conditions of Theorem 4, $f\in H^{2,\omega }_{p,\lambda }$, then

$$\begin{aligned} \Vert f-B_{n,\alpha }(f)\Vert _{p,\lambda ,2,\varphi }\le C\lambda (n^{-1}), \quad n\in {\mathbb {N}}. \end{aligned}$$

(26)

Proof

By Theorem 4 we have the inequality $\Vert f-B_{n,\alpha }(f)\Vert _{p,\lambda }\le C_1\omega _2(f,n^{-1})_{p,\lambda }$. Applying Theorem 8, we obtain (26). □