On approximation of functions by polynomials and by entire functions of exponential type

TL;DR

This paper reviews the history and developments in the approximation of functions using polynomials and entire functions of exponential type, highlighting contributions connected to V. K. Dzyadyk's work.

Contribution

It provides a historical overview and synthesis of approximation theory related to polynomials and entire functions, emphasizing Dzyadyk's influence.

Findings

Summarizes key results in polynomial and entire function approximation

Highlights the impact of Dzyadyk's publications on the field

Connects historical developments with modern approximation theory

Abstract

A brief overview of publications in approximation theory of functions known to the author and connected with scientific publications by V.~K.~Dzyadyk (1919--1998).

Equations

$$f\sim \sum _{k\in \mathbb{Z}}{\hat{f}}_k{e}_k,{\hat{f}}_k=\frac{1}{2\pi }\underset{\mathbb{T}}{\int }f(t)e^{-ikt}dt.$$

$$f(x)=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f^{(r)}(x-t)b_{r}(t)dt,\qquad b_{r}(t)=\sum\limits_{k=1}^{\infty}\frac{\cos\big{(}kt-\frac{r\pi}{2}\big{)}}{k^{r}}.$$

$$\Vert f^{\prime }{\Vert }_{\infty }\le \frac{\pi }{2}\Vert f{\Vert }_{\infty },$$

$$K_r=\frac{4}{\pi }\sum _{\nu =0}^{\infty }\frac{(-1{)}^{\nu (r+1)}}{(2\nu +1{)}^{r+1}}.$$

$$E_{n}^{T}(b_{r})_{1}=\min\limits_{\tau_{n}}\int\limits_{\mathbb{T}}\big{|}b_{r}(t)-\tau_{n}(t)\big{|}dt=\frac{\pi K_{r}}{(n+1)^{r}}.$$

$$\sup\limits_{f\in W^{r}(\mathbb{T})}E_{n}^{T}(f)_{\infty}=\sup\limits_{f\in W^{r}(\mathbb{T})}\min\limits_{\tau_{n}}\sup\limits_{x}\big{|}f(x)-\tau_{n}(x)\big{|}=\frac{K_{r}}{(n+1)^{r}}.$$

$$\varphi_{r}(x)=\frac{4}{\pi}\sum\limits_{k=0}^{\infty}\frac{\sin\big{(}(2k+1)x-\frac{r\pi}{2}\big{)}}{(2k+1)^{r+1}}.$$

$$\|f\|_{p,N}=\sup\limits_{x\in\mathbb{R}}\Big{(}\int\limits_{x-N}^{x+N}|f(t)|^{p}dt\Big{)}^{\frac{1}{p}}.$$

$$A:\text{leavevmode}E\text{leavevmode}\mapsto \text{leavevmode}E$$

$$\Vert Af{\Vert }_{p,N}\le a\Vert f{\Vert }_{p,N}.$$

$$A_{\sigma}(f)_{\infty}=\inf\limits_{g\sigma}\sup\limits_{\mathbb{R}}\big{|}f(x)-g_{\sigma}(x)\big{|}.$$

$$A_{\sigma}(f)\leq\varepsilon\Big{(}f;\frac{1}{\sigma}\Big{)}.$$

$${\rm dist}\leavevmode\nobreak\ (f,E)=\min\limits_{g\in E}\|f-g\|_{L_{1}}=\big{\|}f-g^{*}\big{\|}_{L_{1}}$$

$$|h|\leq 1,h(f-g^{*})=\big{|}f-g^{*}\big{|}$$

$$\max\limits_{x}\big{|}f(x)-\sigma_{n}(x)\big{|}=O\Big{(}\frac{1}{n^{\alpha}}\Big{)},$$

$$E_{n}(f)_{\infty}=\min\limits_{p_{n}}\max\limits_{[-1,1]}\big{|}f(x)-p_{n}(x)\big{|}=O\Big{(}\frac{1}{n^{\alpha}}\Big{)}\qquad(\alpha\in(0,1))$$

$$\big{|}f(x_{1})-f(x_{2})\big{|}\leq\gamma\Big{(}\frac{|x_{1}-x_{2}|}{|x_{1}-x_{2}|+\sqrt{1-x_{1}^{2}}+\sqrt{1-x_{2}^{2}}}\Big{)}^{\alpha}.$$

$$\big{|}f(x)-p_{n}(x)\big{|}\leq\frac{\pi}{2}\cdot\frac{\sqrt{1-x^{2}}}{n}+O\Big{(}\frac{\ln n}{n^{2}}\Big{)}.$$

$$\big{|}f(x)-p_{n}(x)\big{|}\leq c(r)\delta_{n}^{r}(x)\cdot\omega\big{(}f^{(r)};\delta_{n}(x)\big{)}.$$

$$\big{|}f(x)-p_{n}(x)\big{|}\leq K_{r}\Big{(}\frac{\sqrt{1-x^{2}}}{n+1}\Big{)}^{r}+c(r)\frac{(\sqrt{1-x^{2}})^{r-1}}{(n+1)^{r+1}}.$$

$$\begin{gathered}f(x)-\sum\limits_{\nu=0}^{r}\frac{f^{(\nu)}(1)}{\nu!}(x-1)^{\nu}=\frac{(-1)^{r+1}}{r!}\int\limits_{x}^{1}(y-x)^{r}df^{(r)}(y)=\int\limits_{-1}^{1}h_{r,y}(x)df^{(r)}(y),\\ h_{r,y}(x)=\frac{(x-y)^{r}}{2(r!)}\big{(}{\rm sign}\leavevmode\nobreak\ (x-y)-1\big{)}.\end{gathered}$$

$$\begin{gathered}\big{|}{\rm sign}\leavevmode\nobreak\ (x-y)-p_{n,y}(x)\big{|}\leq c(r,s)\Big{(}\frac{1-x^{2}}{1-x^{2}+1-y^{2}+\frac{1}{n^{2}}|{\rm sign}\leavevmode\nobreak\ x-{\rm sign}\leavevmode\nobreak\ y|}\Big{)}^{r}\Big{(}\frac{\delta_{n}(y)}{|x-y|+\delta_{n}(y)}\Big{)}^{s},\\ \delta_{n}(y)=\frac{\sqrt{1-y^{2}}}{n}+\frac{1}{n^{2}}\end{gathered}$$

$$z=u(x,y),z=v(x,y),z=\sqrt{u^2(x,y)+v^2(x,y)}.$$

$$\big{\|}f-\tau_{n}(f)\big{\|}_{\infty}\asymp\omega_{r}\Big{(}f;\frac{1}{n}\Big{)}\qquad(n\in\mathbb{N})$$

$$\max\limits_{x}\frac{1}{n+1}\sum\limits_{k=0}^{n}\big{|}f(x)-S_{n}(f;x)\big{|}\asymp\omega^{*}(\varepsilon_{n}).$$

Full text

517.5

R. M. Trigub

**ON APPROXIMATION OF FUNCTIONS BY POLYNOMIALS

AND BY ENTIRE FUNCTIONS OF EXPONENTIAL TYPE**

**.  .  **

A brief overview of publications in approximation theory of functions known to the author and connected with scientific publications by V. K. Dzyadyk (1919–1998).

( ) 1951  . . . 1953 . , $r\in(0,1)$ .  ( . . 1). , . . . . 4 .

, 2019 100 , , [1] – , .  .  , .  (K. Weierstrass), .  .  , .  (H. Lebesgue), .  (L. Fejér), .  (D. Jackson), .  .  .

1. $W^{r}(\mathbb{T})$ $W^{r}(\mathbb{R})$ ( ).

$f^{(r)}$ ( $r$ — , $r\in\mathbb{N}$ — , $f^{(r-1)}\in{\rm AC_{loc}}$, \big{|}f^{(r)}(x)\big{|}\leq 1 ).

$2\pi$– $L_{p}(\mathbb{T})$, $\mathbb{T}=[-\pi,\pi]$, $p\in[1,+\infty]$ $C(\mathbb{T})$, \big{(}e_{k}=e^{ikx},\ k\in\mathbb{Z}\big{)}

[TABLE]

, ($b_{r}$ — )

[TABLE]

$r$ $b_{r}$ \displaystyle b_{1}(t)=\frac{1}{2}\Big{(}1-\frac{|t|}{\pi}\Big{)}{\rm sign}\leavevmode\nobreak\ t ($|t|\leq\pi$).

.  (H. Bohr) 1935 . , $r=1$ $\widehat{f_{0}}=0$,

[TABLE]

. .  .  (1935) , $r\in\mathbb{N}$ $\widehat{f_{0}}=0$ $\|f^{(r)}\|_{\infty}\leq K_{r}\|f\|_{\infty}$,

[TABLE]

.  (J. Favard) [2], , $L_{1}(\mathbb{T})$ $\tau_{n}$ $n$ \big{(}\tau_{n}=\sum\limits_{|k|\leq n}c_{k}e_{k}\big{)}:

[TABLE]

[3] - [4] ,

[TABLE]

( )

[TABLE]

$r$- $\varphi_{0}(x)={\rm sign}\sin x$. [4] $\widetilde{W_{r}}(\mathbb{T})$ \big{(}\big{\|}\widetilde{f}^{(r)}\big{\|}_{\infty}\leq 1, $\widetilde{f}$ — $f$ \big{)} $\widetilde{K_{r}}$.

, , $K_{r}$ $r$ e $r$. , , $K_{r}$ . , $r$ ( [5, . 171]).

, ( ) .  .  (1935, [6]). .  .  (1905-2012) .

(1937) $W^{r}(\mathbb{T})$ $r>0$. , $b_{r}$ , . $r\in(0,1)$ . . [7]. .  .  – (Sun Yung-sheng). $r>0$ . . , [8] ( $L_{\infty}$ $L_{1}$).

1957 . . . 1959 , . . , , . , , .  .  ( ) , $\tau_{n}$ $W^{r}(\mathbb{T})$ $2n+1$ \big{(} \displaystyle d_{2n+1}\big{(}W^{r}(\mathbb{T})\big{)}=\frac{K_{r}}{(n+1)^{r}}\big{)}.

$L_{p}$, $p\in(1,+\infty)$, , , , $p=\infty$. .

, , $(N>0)$

[TABLE]

$f\in L_{p}(\mathbb{R})$, $\|f\|_{p}=\lim\limits_{N\rightarrow\infty}\|f\|_{p,N}$, $f$ — $2\pi$– $f\in L_{p}(\mathbb{T})$, $\|f\|_{p}=\|f\|_{p,\pi}$. .

([9, 1.2.7])

• $E$ – $\mathbb{R}$ , ,*

[TABLE]

$\|Af\|_{\infty}\leq a\|f\|_{\infty}$. $f$ $f^{t}=f(\cdot+t)\in E$ $t\in\mathbb{R}$ $A$ , $p\in[1,+\infty)$, $N>0$ $f\in E$

[TABLE]

.

I. $L_{p}$- $p=\infty$ . $L_{2}$, , – . . [9, . 89].

II. ( . . 2) $L_{p}$ $p=\infty$.

, , . , $\mathbb{R}$, , $\mathbb{R}$ ( .  .  ). ., , [10] – [13], [9, 4.2.8]. .

$g_{\sigma}$, $\sigma>0$, — $\mathbb{R}$ $\leq\sigma$. $g_{\sigma}$ $2\pi$, $g_{\sigma}=\tau_{[\sigma]}$ ( ).

[TABLE]

([9, 5.5.9]) * $W$ – $\mathbb{R}$,*

( $f\in W$, $f_{\lambda}\in W$ $\lambda>0$, $f_{\lambda}(x)=f(\lambda x)$). $f\in W$ $W$. , , $\varepsilon:W\times(0,+\infty)\mapsto(0,+\infty)$ : $\varepsilon(f_{\lambda},h)=\varepsilon(f,\lambda h)$ $(h>0)$ , $f_{m}\rightarrow f$ $m\rightarrow\infty$, , $\varepsilon(f_{m},h)\rightarrow\varepsilon(f,h)$, \displaystyle E_{n}^{T}(f)_{\infty}\leq\varepsilon\Big{(}f;\frac{1}{n}\Big{)} $(n\in\mathbb{N})$ $2\pi$– $W$ , $f\in W$ $\sigma>0$

[TABLE]

.

I. \displaystyle A_{\sigma}\big{(}W^{r}(\mathbb{R})\big{)}_{\infty}=\frac{K_{r}}{\sigma^{r}}\quad(\sigma>0,\ r\in\mathbb{N}).

II. \displaystyle A_{\sigma}(C_{\omega})_{\infty}=\frac{1}{2}\omega\Big{(}\frac{\pi}{\sigma}\Big{)}\quad(\sigma>0).

. [10]. . . , I $r>0$ ( $K_{r}$ $r$).

, $\omega$ , .  .  (1920-2003) ( , .  . 1961 .  .  ). – . . ( . [14]).

. .  .  ( , ., 2003, 160  .; Mathnet.ru).

, .  .  .  .  $L_{1}$ :

([9, 5.2.5]) * $E$ – $L_{1}$ $f\in L_{1}\setminus E$. *

[TABLE]

, $h$ ,

[TABLE]

$\int hg=0$* $g\in E$. , ${\rm dist}\leavevmode\nobreak\ (f,E)=\int hg$. $f-g^{*}\neq 0$ , $h=\overline{{\rm sign}\leavevmode\nobreak\ (f-g^{*})}$.*

2. .

.

, $2\pi$– ${\rm Lip}\leavevmode\nobreak\ \alpha$ $\alpha\in(0,1)$ ( $\sigma_{n}$)

[TABLE]

$\alpha=1$ — \displaystyle O\Big{(}\frac{\ln n}{n}\Big{)} ( . [5, . 89] [13, . 236]).

.  , $\sigma_{n}$ $D_{n}^{2}$ $D_{n}^{4}$ ${\rm Lip}\leavevmode\nobreak\ 1$, , \displaystyle O\Big{(}\frac{1}{n}\Big{)} ( . [10] – [13]). , , – $\omega_{r}$ ( ., , [9]).

– ( . [9, . 8]). [15] , \displaystyle\omega_{2}\Big{(}f;\frac{1}{n}\Big{)} $D_{n}^{3}$ $D_{n}^{4}$.

( . [12], [16]). ([17]).

($[-1,1]$, ), $x=\cos t$ $2\pi$– ( , ). ( ) $p_{n}$ $n$ $x=\cos t$ $\tau_{n}$. , ,

[TABLE]

, $f(\cos t)\in{\rm Lip}\leavevmode\nobreak\ \alpha$ $x_{1}$ $x_{2}$ $[-1,1]$

[TABLE]

[18].

.  .  (1946) , $f\in W^{1}[-1,1]$ $p_{n}$ , $x\in[-1,1]$

[TABLE]

( ) $\displaystyle\frac{\pi}{2}=K_{1}$.

.  .  (1951) \omega\big{(}f^{(r)};h\big{)}

\displaystyle\big{(}\delta_{n}(x)=\frac{\sqrt{1-x^{2}}}{n}+\frac{1}{n^{2}}\big{)}:

[TABLE]

.  .  (1956) $\omega(h)=h^{\alpha}$ $(\alpha\in(0,1))$. $p_{n}$ – .

$\omega_{2}$ .  .  .  (G. Freud) 1959 . . [11, 5.2.3].

(1962). $\omega_{r}$ .  .  (1974, [19], . [12], [13], [9]).

$\gamma W^{r}[-1,1]$, $r\in\mathbb{N}$, . [20].

60– . . ,

,

( [12] . 334–490). . [21].

.  .  (1946) .  .  (1981) $\ln n$ .

([22]) * $r\in\mathbb{N}$ $c(r)$ , $f\in W^{r}[-1,1]$ $n\geq r-1$ $p_{n}$ , $x\in[-1,1]$*

[TABLE]

$K_{r}$ , $c(r)\geq ce^{r}$ $(c>0,r\in\mathbb{N})$.

.  .  [23] $r>0$ $\ln n$ , . ( ., , [9, . 248–249]).

. [24].

. , .

( , . .) ( . . 7 [25]).

, $f\in C^{r}[-1,1]$ $(r\in\mathbb{Z}_{+})$

[TABLE]

. . $h_{r,y}$ ([12], . VII, 4], $r=0$). De Vore Yu ( . [26] [25]).

([27]) * $x\in[-1,1]$ $y\in(-1,1)$, $r$ $s\in\mathbb{Z}_{+}$ $n\geq 2r+1$ $n$ $x$ , *

[TABLE]

( $r=0$ . .).

: ; , ; , . ( . [28]).

3. .

, ( )

• : $\mathbb{C}$– , – , – , , ( ).

*.  .  ** [29] * $\mathbb{R}^{3}$ ( )

[TABLE]

$f=u+iv$ $\overline{f}=u-iv$ $D$ $u$ $v$ , $D$ .

– . , ( , ). ( . [30]).

. Mathnet.ru

4. (I–III).

I. (1962)

. . [31].

$\mathbb{C}$ ( .  .  , 1964). , $0\leq{\rm Re}\leavevmode\nobreak\ z,{\rm Im}\leavevmode\nobreak\ z\leq 1$ ([32]).

II. (1965) $f\mapsto\tau_{n}$ , $r\in\mathbb{N}$ $f\in C(\mathbb{T})$

[TABLE]

( , $r$).

$r=1$ – , $r=2$ – , $r\geq 3$ –

• ( ., , [12] [25]). $r\geq 3$ – , [16] .

$\omega^{*}$ , $f\in C(\mathbb{T})$ $\varepsilon_{n}\searrow 0$, ,

[TABLE]

( , $\delta_{n}(x)$) , ([33], . [13]) – – (Z. Ditzian – V. Totik).

III. [34] $c(r)$ - . .  .  .  .  ( . . 2). , .

Список литературы

• [1] C.J. de la Vallée Poussin Lecons sur l’approximation des fonctions d’une variable réelle. Paris, Gautier–Villars, 1919, 363 p.
• [2] J. Favard Application de la formule sommatoire d’Euler a la démonstration de quelques propriètes extremales des integrales des fonctions periodiques ou presquepérivdiques. Matematik Tidskrift København, B. H. 4 (1936), 81–94.
• [3] J. Favard Sur les meilleurs procedes d’approximation de certaines classes des fonctions par des polynomes trigonométriques. Bull. Sci. Math. 61 (1937), 207–224, 243–256.
• [4]

• .  .  , .  .  * . . – 1937. – $\bf{15}$. – . 107–111.

• [5]

• .  .  * , . II. .: – , 1954, 627  .

• [6] A. N. Kolmogorov Zur Grössen ordrung des Restgriedes Fourierischer Reichen differenzierbarer Funktionen. Ann. of Math. $\bf{36}$ (1935), 321–326.
• [7]

• .  .  * , $s$– $(0<s<1)$. . , .  . 17 (1953), 135–162.

• [8]

• .  .  * , . . . 16 (1973), 691–701.

• [9] R. M. Trigub, E. S. Belinsky Fourier Analysis and Approximation of Functions. Kluwer–Springer, 2004, 585 .
• [10]

• .  .  * . .: . 1965, 408  .

• [11]

• .  .  * . .: . 1960, 624  .

• [12]

• .  .  * . .: . 1977, 512  .

• [13] R. A. De Vore, G. G. Lorentz Constructive Approximation. Springer, Berlin, New York. 1993, 452 p.
• [14]

• .  .  * , . i i , .  , 1975,  589–592.

• [15] R. M. Trigub Exact order of approximation of periodic functions by linear polynomial operators. East J. Appr. 15 (2009), 31–56.
• [16]

• .  .  * – . . . 204 (2013), 127–146.

• [17]

• .  .  * , . . , . . 44 (1980), 1378–1409.

• [18]

• .  .  , .  .  * . . . 75 (1968),  502–557.

• [19]

• .  .  * . . , . . 32 (1968), 780–787.

• [20]

• .  .  * . . , . , . 18 (1973), 63–70.

• [21] V. V. Andrievskii, V. I. Belyi, V. K. Dzyadyk Conformal invariants in constructive theory of functions of complex variable. Atlanta: World Federation Publishers. 1995, 199 p.
• [22]

• .  .  * . . . 54 (1993), 113–121.

• [23]

• .  .  * . . . . (1999), 54,  603–613 54, 940–951.

• [24]

• .  .  * . . . 69 (2001), 934–943.

• [25] V. K. Dzyadyk, I. A. Shevchuk Theory of uniform approximation of functions by polinomials. Walter de Gruyter, Berlin, New York. 2008, 480 p.
• [26] R. A. De Vore, X. M. Yu Pointwise estimates for monotone polynomial approximation. Constr. Appr. 1 (1985), 323–331.
• [27]

• .  .  * . , . . 4 (1999), 186–194.

• [28] R. M. Trigub Approximation of functions by polynomials with various constraints. Izv. NAN Armenii. Matematika (2009), No.4,  35–52 and J. Contemporary Math. Analysis 44 (2009), 230–242.
• [29]

• .  .  * . . 15 (1960), 191–194.

• [30] V. V. Volchkov Integral geometry and convolution equations. Kluwer–Springer. 2003, 454 p.
• [31]

• .  .  * . . . 70 (2001), 123–136.

• [32]

• .  .  * . . . 59 (1996), 182–186.

• [33] V. Totik Approximation by Bernstein Polynomials. American J. Math. 116 (1994), 995-1018.
• [34]

• .  .  , .  .  * - . , . 303 (2018), 26–38.