Order estimates of best orthogonal trigonometric approximations of classes of infinitely differentiable functions

TL;DR

This paper derives precise order estimates for the best uniform orthogonal trigonometric approximations of classes of periodic functions with specific smoothness properties, focusing on cases where the derivative sequence decreases faster than any power but slower than geometric progression.

Contribution

It provides exact order estimates for approximation errors of classes of infinitely differentiable functions with specific derivative decay rates, extending previous results to new decay regimes.

Findings

Established exact order estimates in uniform norm for certain function classes.

Derived similar estimates in L_s metrics for differentiable functions.

Extended approximation theory to functions with derivatives decreasing faster than any power.

Abstract

In this paper we establish exact order estimates for the best uniform orthogonal trigonometric approximations of the classes of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{p}$, $1\leq p<\infty$, in the case, when the sequence $\psi(k)$ tends to zero faster, than any power function, but slower than geometric progression. Similar estimates are also established in the $L_{s}$-metric, $1<s\leq\infty$ for the classes of differentiable functions, which $(\psi,\beta)$-derivatives belong to unit ball of space $L_{1}$.

Equations

$$\sum _{k=-\infty }^{\infty }\hat{f}(k)e^{ikx},$$

$$\sum _{k\in \mathbb{Z}\setminus \{0\}}\frac{\hat{f}(k)}{\psi (|k|)}{e}^{i(kx+\frac{\beta \pi }{2}\mathrm{sign}k)}$$

$$B_p^0:=\left\{\phi \in L_p:||\phi |{|}_p\le 1,\phi \perp 1\right\},1\le p\le \infty .$$

$$\mu (t)=\mu (\psi ;t):=\frac{t}{\eta (t)-t},$$

$${\mathfrak{M}}_{\infty }^{+}=\left\{\psi \in \mathfrak{M}:\mu (\psi ;t)\uparrow \infty \right\}.$$

$${\mathfrak{M}}_{\infty }^{{}^{\prime \prime }}=\left\{\psi \in {\mathfrak{M}}_{\infty }^{+}:\exists K>0\eta (\psi ;t)-t\ge Kt\ge 1\right\}.$$

$$\Vert {\rho }_n(f;\cdot ){\Vert }_s=\Vert f(\cdot )-S_{n-1}(f;\cdot ){\Vert }_s,1\le s\le \infty ,$$

$$S_{n-1}(f;x)=\sum _{k=-n+1}^{n-1}\hat{f}(k)e^{ikx};$$

$$e_m^{\perp }(f{)}_s=\underset{{\gamma }_m}{\inf }\Vert f(\cdot )-S_{{\gamma }_m}(f;\cdot ){\Vert }_s,1\le s\le \infty ,$$

$$S_{{\gamma }_m}(f;x)=\sum _{k\in {\gamma }_m}\hat{f}(k)e^{ikx}.$$

$${\mathcal{E}}_n(L_{\beta ,p}^{\psi }{)}_s=\underset{f\in L_{\beta ,p}^{\psi }}{\sup }\Vert {\rho }_n(f;\cdot ){\Vert }_s,1\le p,s\le \infty ,$$

$$e_n^{\perp }(L_{\beta ,p}^{\psi }{)}_s=\underset{f\in L_{\beta ,p}^{\psi }}{\sup }{e}_n^{\perp }(f{)}_s,1\le p,s\le \infty .$$

$$e_{2n}^{\perp }(L_{\beta ,p}^{\psi }{)}_s\le e_{2n-1}^{\perp }(L_{\beta ,p}^{\psi }{)}_s\le {\mathcal{E}}_n(L_{\beta ,p}^{\psi }{)}_s,1\le p,s\le \infty .$$

$$e_{2n-1}^{\perp }(L_{\beta ,p}^{\psi }{)}_{\infty }\asymp e_{2n}^{\perp }(L_{\beta ,p}^{\psi }{)}_{\infty }\asymp \psi (n)(\eta (n)-n{)}^{\frac{1}{p}}.$$

$${\mathcal{E}}_n(L_{\beta ,p}^{\psi }{)}_{\infty }\le K_{a,b}(2p{)}^{1-\frac{1}{p}}\psi (n)(\eta (n)-n{)}^{\frac{1}{p}},$$

$$K_{a,b}=\frac{1}{\pi }\max \left\{\frac{2b}{b-2}+\frac{1}{a},2\pi \right\}.$$

$$e_{2n}^{\perp }(L_{\beta ,p}^{\psi }{)}_{\infty }\le e_{2n-1}^{\perp }(L_{\beta ,p}^{\psi }{)}_{\infty }\le K_{a,b}(2p{)}^{1-\frac{1}{p}}\psi (n)(\eta (n)-n{)}^{\frac{1}{p}}.$$

$$\displaystyle f^{*}_{p,n}(t)=f^{*}_{p,n}(\psi;t):=\frac{\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\bigg{(}\frac{1}{2}\psi(1)\psi(2n)+$$

$$+\sum\limits_{k=1}^{n-1}\psi(k)\psi(2n-k)\cos kt+\sum\limits_{k=n}^{2n}\psi^{2}(k)\cos kt\bigg{)},\ \ \frac{1}{p}+\frac{1}{p^{\prime}}=1.$$

$$(f^{*}_{p,n}(t))^{\psi}_{\beta}=\frac{\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\bigg{(}\sum\limits_{k=1}^{n-1}\psi(2n-k)\cos\Big{(}kt+\frac{\beta\pi}{2}\Big{)}\\ +\sum\limits_{k=n}^{2n}\psi(k)\cos\Big{(}kt+\frac{\beta\pi}{2}\Big{)}\bigg{)}.$$

$$\displaystyle\big{|}(f^{*}_{p,n}(t))^{\psi}_{\beta}\big{|}\leq\frac{\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\bigg{(}\sum\limits_{k=1}^{n-1}\psi(2n-k)+\sum\limits_{k=n}^{2n}\psi(k)\bigg{)}<$$

$$<\frac{2\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\sum\limits_{k=n}^{2n}\psi(k)\leq\frac{2\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\bigg{(}\psi(n)+\int\limits_{n}^{\infty}\psi(u)du\bigg{)}.$$

$$\underset{m}{\overset{\infty }{\int }}\psi (u)du\le \frac{2}{1-\frac{2}{\mu (m)}}\psi (m)(\eta (m)-m).$$

$$\displaystyle\big{|}(f^{*}_{p,n}(t))^{\psi}_{\beta}\big{|}\leq\frac{2\lambda_{p}}{\psi(n)(\eta(n)-n)^{\frac{1}{p^{\prime}}}}\bigg{(}\psi(n)+\frac{2b}{b-2}\psi(n)(\eta(n)-n)\bigg{)}<$$

$$<\frac{5{\lambda }_{p}b}{b-2}(\eta (n)-n{)}^{\frac{1}{p}}.$$

$$D_{k,\beta}(t):=\frac{1}{2}\cos\frac{\beta\pi}{2}+\sum\limits_{j=1}^{k}\cos\Big{(}jt+\frac{\beta\pi}{2}\Big{)}.$$

$$\sum\limits_{k=1}^{n-1}\psi(2n-k)\cos\Big{(}kt+\frac{\beta\pi}{2}\Big{)}=\sum\limits_{k=1}^{n-2}(\psi(2n-k+1)-\psi(2n-k))D_{k,\beta}(t)\\ +\psi(n+1)D_{n-1,\beta}(t)-\psi(2n-1)\frac{1}{2}\cos\frac{\beta\pi}{2}$$

$$\sum\limits_{k=n}^{2n}\psi(k)\cos\Big{(}kt+\frac{\beta\pi}{2}\Big{)}=\sum\limits_{k=n}^{2n-1}(\psi(k)-\psi(k+1))D_{k,\beta}(t)\\ +\psi(2n)D_{2n,\beta}(t)-\psi(n)D_{n-1,\beta}(t).$$

$$\sum\limits_{k=0}^{N-1}\sin(\gamma+kt)=\sin\Big{(}\gamma+\frac{N-1}{2}t\Big{)}\sin\frac{Ny}{2}\frac{1}{\sin\frac{t}{2}}$$

$$\displaystyle|D_{k,\beta}(t)|=\left|\frac{\cos\big{(}\frac{kt}{2}+\frac{\beta\pi}{2}\big{)}\sin\frac{k+1}{2}t}{\sin\frac{t}{2}}-\frac{1}{2}\cos\frac{\beta\pi}{2}\right|$$

Full text

1 Introduction

Let $L_{p}$, $1\leq p<\infty$, be the space of $2\pi$–periodic functions $f$ summable to the power $p$ on $[0,2\pi)$, with the norm \|f\|_{p}=\Big{(}\int\limits_{0}^{2\pi}|f(t)|^{p}dt\Big{)}^{\frac{1}{p}}; $L_{\infty}$ be the space of $2\pi$–periodic functions $f$, which are Lebesque measurable and essentially bounded with the norm $\|f\|_{\infty}=\mathop{\rm{ess}\sup}\limits_{t}|f(t)|$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be the function from $L_{1}$, whose Fourier series is given by

[TABLE]

where $\hat{f}(k)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(t)e^{-ikt}dt$ are the Fourier coefficients of the function $f$, $\psi(k)$ is an arbitrary fixed sequence of real numbers and $\beta$ is a fixed real number. Then, if the series

[TABLE]

is the Fourier series of some function $\varphi$ from $L_{1}$, then this function is called the $(\psi,\beta)$–derivative of the function $f$ and is denoted by $f_{\beta}^{\psi}$. A set of functions $f$, whose $(\psi,\beta)$–derivatives exist, is denoted by $L_{\beta}^{\psi}$ (see Stepanets1 ).

Let

[TABLE]

If $f\in L^{\psi}_{\beta}$, and, at the same time $f^{\psi}_{\beta}\in B_{p}^{0}$, then we say that the function $f$ belongs to the class $L^{\psi}_{\beta,p}$.

By ${\mathfrak{M}}$ we denote the set of all convex (downward) continuous functions $\psi(t)$, $t\geq 1,$ such that $\lim\limits_{t\rightarrow\infty}\psi(t)=0$. Assume that the sequence $\psi(k),\ k\in\mathbb{N}$, specifying the class $L^{\psi}_{\beta,p},\ 1\leq p\leq\infty$, is the restriction of the functions $\psi(t)$ from ${\mathfrak{M}}$ to the set of natural numbers.

Following Stepanets (see, e.g., Stepanets1 ), by using the characteristic $\mu(\psi;t)$ of functions $\psi$ from $\in{\mathfrak{M}}$ of the form

[TABLE]

where $\eta(t)=\eta(\psi;t):=\psi^{-1}\left(\psi(t)/2\right)$, $\psi^{-1}$ is the function inverse to $\psi$, we select the following subsets of the set ${\mathfrak{M}}$:

[TABLE]

[TABLE]

The functions $\psi_{r,\alpha}(t)=\exp(-\alpha t^{r})$ are typical representatives of the set $\mathfrak{M}^{+}_{\infty}$. Moreover, if $r\in(0,1]$, then $\psi_{r,\alpha}\in\mathfrak{M}^{{}^{\prime\prime}}_{\infty}$. The classes $L^{\psi}_{\beta,p}$, generated by the functions $\psi=\psi_{r,\alpha}$ are denoted by $L^{\alpha,r}_{\beta,p}$.

For functions $f$ from classes $L^{\psi}_{\beta,p}$ we consider: $L_{s}$–norms of deviations of the functions $f$ from their partial Fourier sums of order $n-1$, i.e., the quantities

[TABLE]

where

[TABLE]

and the best orthogonal trigonometric approximations of the functions $f$ in metric of space $L_{s}$, i.e., the quantities of the form

[TABLE]

where $\gamma_{m}$, $m\in\mathbb{N}$, is an arbitrary collection of $m$ integer numbers, and

[TABLE]

We set

[TABLE]

[TABLE]

The following inequalities follow from given above definitions (4) and (5)

[TABLE]

In the case when $\psi(k)=k^{-r}$, $r>0$, the classes $L^{\psi}_{\beta,p}$, $1\leq p\leq\infty$, $\beta\in\mathbb{R}$ are well-known Weyl–Nagy classes $W^{r}_{\beta,p}$. For these classes, the order estimates of quantities $e^{\bot}_{n}(L_{\beta,p}^{\psi})_{s}$ are known for $1<p,s<\infty$ (see Romanyuk2002 , Romanyuk2007 ), for $1\leq p<\infty$, $s=\infty$, $r>\frac{1}{p}$ and also for $p=1$, $1<s<\infty$, $r>\frac{1}{s^{\prime}}$, $\frac{1}{s}+\frac{1}{s^{\prime}}=1$ (see Romanyuk2007 , Romanyuk2012 ).

In the case, when $\psi(k)$ tends to zero not faster than some power function, order estimates for quantities (5) were established in Fedorenko1999 , S_S2015 , Shkapa2014_no2 and Shkapa2014_no3 . In the case, when $\psi(k)$ tends to zero not slower than geometric progression, exact order estimates for $e^{\bot}_{n}(L_{\beta,p}^{\psi})_{s}$ were found in S_S_Dopovidi2015 for all $1\leq p,s\leq\infty$.

Our aim is to establish the exact-order estimates of $e^{\bot}_{n}(L_{\beta,p}^{\psi})_{\infty}$, $1\leq p<\infty$, and $e^{\bot}_{n}(L_{\beta,1}^{\psi})_{s}$, $1<s<\infty$, in the case, when $\psi$ decreases faster than any power function, but slower than geometric progression ($\psi\in\mathfrak{M}^{{}^{\prime\prime}}_{\infty}$).

2 Best orthogonal trigonometric approximations of the classes $L^{\psi}_{\beta,p}$, $1<p<\infty$, in the uniform metric

We write $a_{n}\asymp b_{n}$ to mean that there exist positive constants $C_{1}$ and $C_{2}$ independent of $n$ such that $C_{1}a_{n}\leq b_{n}\leq C_{2}a_{n}$ for all $n$.

Theorem 2.1

Let ${1<p<\infty}$, $\psi\in\mathfrak{M}^{{}^{\prime\prime}}_{\infty}$ and the function $\frac{\psi(t)}{|\psi^{\prime}(t)|}\uparrow\infty$ as $t\rightarrow\infty$. Then, for all $\beta\in\mathbb{R}$ the following order estimates hold

[TABLE]

Proof

According to Theorem 1 from S_S under conditions $\psi\in\mathfrak{M}^{+}_{\infty}$, $\beta\in\mathbb{R}$, ${1\leq p<\infty}$, for $n\in\mathbb{N}$, such that ${\eta(n)-n\geq a>2},\ {\mu(n)\geq b>2}$ the following estimate is true

[TABLE]

where

[TABLE]

Using inequalities (6) and (8), we obtain

[TABLE]

Let us find the lower estimate for the quantity $e^{\bot}_{2n}(L_{\beta,p}^{\psi})_{\infty}$. With this purpose we construct the function

[TABLE]

[TABLE]

Let us show that $f^{*}_{p,n}\in L_{\beta,p}^{\psi}$. The definition of $(\psi,\beta)$–derivative yields

[TABLE]

Obviously

[TABLE]

[TABLE]

To estimate the integral from the right part of formula (12), we use the following statement (Serdyuk2004, , p. 500).

Proposition 1

If $\psi\in\mathfrak{M}^{+}_{\infty}$, then for arbitrary ${m\in\mathbb{N}}$, such that $\mu(\psi,m)>2$ the following condition holds

[TABLE]

Formulas (12) and (13) imply that

[TABLE]

[TABLE]

We denote

[TABLE]

Applying Abel transform, we have

[TABLE]

and

[TABLE]

Since

[TABLE]

(see, e.g., (Gradshteyn, , p.43)), for $N=k+1$, $\gamma=(\beta-1)\frac{\pi}{2}$, the following inequality holds

[TABLE]

According to (11), (16), (17) and (19), we obtain

[TABLE]

So, (14) and (20) imply

[TABLE]

Hence, for

[TABLE]

the embedding $f^{*}_{p,n}\in L^{\psi}_{\beta,p}$ is true.

Let us consider the quantity

[TABLE]

where $V_{2n}$ are de la $\mathrm{Vall\acute{e}e}$-Poisson kernels of the form

[TABLE]

Proposition A1.1 from Korn implies

[TABLE]

Since (see, e.g., (Stepaniuk2014, , p.247))

[TABLE]

from (23) and (24) we can write down the estimate

[TABLE]

Notice, that

[TABLE]

where $\psi(0):=\psi(1)$

Whereas

[TABLE]

and taking into account (22), we obtain

[TABLE]

[TABLE]

The function $\phi_{n}(t):=\psi(t)\psi(2n-t)$ decreases for ${t\in[1,n]}$. Indeed

[TABLE]

because $\frac{\psi(t)}{|\psi^{\prime}(t)|}\uparrow\infty$ for large $n$.

Thus, the monotonicity of function $\phi_{n}(t)$ and (30) imply

[TABLE]

By considering (25) and (31) we can write

[TABLE]

Theorem 1 is proved.

Remark 1

Let $\psi\in\mathfrak{M}^{+}_{\infty}$, $\beta\in\mathbb{R}$, ${1<p<\infty}$, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$, and the function $\frac{\psi(t)}{|\psi^{\prime}(t)|}\uparrow\infty$ for $t\rightarrow\infty$. Then for $n\in\mathbb{N}$ the following estimates hold

[TABLE]

where

[TABLE]

[TABLE]

3 Best orthogonal trigonometric approximations of the classes $L^{\psi}_{\beta,1}$ in the uniform metric

Theorem 3.1

Let $\psi\in\mathfrak{M}^{+}_{\infty}$. Then for all $\beta\in\mathbb{R}$ order estimates are true

[TABLE]

Proof

According to formula (48) from S_S under conditions $\psi\in\mathfrak{M}$, $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$, $\beta\in\mathbb{R}$, for all $n\in\mathbb{N}$ the following estimate holds

[TABLE]

Using Proposition 1, we have

[TABLE]

Let us find the lower estimate for the quantity $e^{\bot}_{2n}(L_{\beta,1}^{\psi})_{\infty}$.

We consider the quantity

[TABLE]

where $V_{m}$ are de la $\mathrm{Vall\acute{e}e}$-Poisson kernels of the form (22), and

[TABLE]

In (Stepaniuk2014, , p. 263–265) it was shown that $\|(f^{*}_{m})^{\psi}_{\beta}\|_{1}\leq 1$, i.e., $f^{*}_{m}$ belongs to the class $L^{\psi}_{\beta,1}$ for all $m\in\mathbb{N}$.

Using Proposition A1.1 from Korn and inequality (24), we have

[TABLE]

Assuming again $\psi(0):=\psi(1)$, from (22) and (40), we derive

[TABLE]

Formulas (41) and (42) imply

[TABLE]

Theorem 3.1 is proved.

Remark 2

Let $\psi\in\mathfrak{M}^{+}_{\infty}$ and $\beta\in\mathbb{R}$. Then for $n\in\mathbb{N}$, such that ${\mu(n)\geq b>2}$ the following estimate hold

[TABLE]

Corollary 1

Let $r\in(0,1)$, $\alpha>0$, $1\leq p<\infty$ and $\beta\in\mathbb{R}$. Then for all $n\in\mathbb{N}$ the following estimates are true

[TABLE]

4 Best orthogonal trigonometric approximations of the classes $L^{\psi}_{\beta,1}$ in the metric of spaces $L_{s}$, $1<s<\infty$

Theorem 4.1

Let ${1<s<\infty}$, $\psi\in\mathfrak{M}^{{}^{\prime\prime}}_{\infty}$ and function $\frac{\psi(t)}{|\psi^{\prime}(t)|}\uparrow\infty$ as $t\rightarrow\infty$ . Then for all $\beta\in\mathbb{R}$ order estimates hold

[TABLE]

Proof

According to Theorem 2 from S_S under conditions $\psi\in\mathfrak{M}^{+}_{\infty}$, $\beta\in\mathbb{R}$, ${1<s\leq\infty}$ for $n\in\mathbb{N}$, such that ${\eta(n)-n\geq a>2},\ {\mu(n)\geq b>2}$ the following estimate holds

[TABLE]

Using inequalities (6) and (46), we get

[TABLE]

Let us find the lower estimate of the quantity $e^{\bot}_{2n}(L_{\beta,1}^{\psi})_{s}$.

We consider the quantity

[TABLE]

where

[TABLE]

and $f^{*}_{s^{\prime},n}$ is defined by formula (10).

On the basis of Proposition A1.1 from Korn we derive

[TABLE]

On other hand, using formulas (27), we write

[TABLE]

[TABLE]

Hence, formulas (49) and (50) imply

[TABLE]

Theorem 4.1 is proved.

Note, that functions

1. $e^{-\alpha t^{r}}t^{\gamma}$, $\alpha>0,\ r\in(0,1],\ \gamma\in\mathbb{R}$;

$e^{-\alpha t^{r}}\ln(t+K)$, $\alpha>0,\ r\in(0,1],\ \ K>e-1$,

etc., can be regarded as examples of functions $\psi$, which satisfy the conditions of Theorem 2.1 and Theorem 4.1.

Remark 3

Let $\psi\in\mathfrak{M}^{+}_{\infty}$, $\beta\in\mathbb{R}$, ${1\leq p<\infty}$ and function $\frac{\psi(t)}{|\psi^{\prime}(t)|}\uparrow\infty$ as $t\rightarrow\infty$. Then for all $n\in\mathbb{N}$, such tthe following estimates are true

[TABLE]

where $K_{a,b,s^{\prime}}$ and $K_{b,s^{\prime}}$ are defined by formulas (34) and (35) respectively.

Corollary 2

Let $r\in(0,1)$, $\alpha>0$, $1<s<\infty$ and $\beta\in\mathbb{R}$. Then for all $n\in\mathbb{N}$ the following estimates are true

[TABLE]

Acknowledgements

The author is supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”)