On the Jackson constants for algebraic approximation of continuous functions

TL;DR

This paper derives new bounds for Jackson constants in algebraic approximation of continuous functions, improving understanding of approximation quality and constants involved.

Contribution

It provides new estimates for the Jackson constants in the Brudnyi-Jackson inequality, refining previous bounds for algebraic approximation of continuous functions.

Findings

Established bounds for Jackson constants: 1/2 < J_a(2k, α) < 10.

Provided inequalities for approximation error in terms of modulus of smoothness.

Improved constants for algebraic polynomial approximation of continuous functions.

Abstract

We establish new estimates for the constant $J_a(k,\alpha)$ in the Brudnyi-Jackson inequality for approximation of $f \in C[-1,1]$ by algebraic polynomials: $$ E_{n}^a (f) \le J_a(k, \alpha) \ \omega_k (f, \alpha \pi /n ), \quad \alpha >0 $$ The main result of the paper implies the following inequalities $$ 1/2< J_a (2k, \alpha) < 10, \quad n \ge 2k(2k-1), \quad \alpha \ge 2 $$

Equations

$$E_n^a(f)\le J_a(k,\alpha ){\omega }_k(f,\alpha \pi /n),\alpha >0$$

$$1/2<J_a(2k,\alpha )<10,n\ge 2k(2k-1),\alpha \ge 2$$

$$E_n^a(f)\le J_a(k,\alpha ){\omega }_k(f,\alpha \pi /n),\alpha >0.$$

$$E_{n-1}^a(f)\le \frac{1}{2}\overline{\omega }(f,\pi /n),$$

$$E_{n-1}^a(f)\le 5{\omega }_2(f,\alpha \pi /n),\alpha =8^{-1/2}.$$

$$J_a(2k,\alpha )<10,\alpha \ge 2.$$

$${\omega }_{2k+1}(f,\delta )\le 2{\omega }_{2k}(f,\delta ),\delta >0,$$

$$\Vert f\Vert :=\Vert f{\Vert }_A:=\underset{x\in A}{\mathrm{esssup}}|f(x)|.$$

$$E_{n-1}^a(f):=\underset{p_{n-1}}{\inf }\underset{x\in \mathbb{I}}{\sup }|f(x)-p_{n-1}(x)|=\underset{p\in {\mathcal{P}}_{n-1}}{\inf }\Vert f-p\Vert .$$

$${\omega }_k(f,\delta ):=\underset{x\in (1-kh/2)I,0<h\le \delta }{\sup }|{\hat{\Delta }}_h^{k}f(x)|,{\hat{\Delta }}_h^{k}f(x):=\sum _{j=0}^k(-1{)}^{k-j}(\genfrac{}{}{0ex}{}{k}{j})f(x+jh-kh/2),$$

$$(f\ast g)(x):={\int }_{\mathbb{R}}f(t)g(x-t)dt,$$

$${\chi }_h(x):=\{\begin{array}{ll}\frac{1}{h}& x\in [-h/2,h/2],\\ 0,& x\notin [-h/2,h/2].\end{array}$$

$${\chi }_h^2(x):=({\chi }_h\ast {\chi }_h)(x)=\{\begin{array}{ll}\frac{1}{h}(1-|x|/h),& x\in [-h,h],\\ 0,& x\notin [-h,h].\end{array}$$

$$W_{2k}(f,x,{\chi }_h^2):=(-1{)}^k\frac{1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{\int }_{\mathbb{R}}{\hat{\Delta }}_t^{2k}f(x){\chi }_h^2(t)dt=(f-{\Lambda }_{2k}\ast f)(x),$$

$${\Lambda }_{2k}(x):={\Lambda }_{2k,h}(x)=2\sum _{j=1}^k(-1{)}^{j+1}{a}_{j,k}{\chi }_{jh}^2(x),a_{j,k}:=(\genfrac{}{}{0ex}{}{2k}{k+j})/(\genfrac{}{}{0ex}{}{2k}{k}).$$

$${\int }_{\mathbb{R}}|{\Lambda }_{2k}(t)|dt\le c_k-1,$$

$$c_1=2,c_2<2.18,c_3<2.26,c_4<2.31,c_k<3,k\ge 5,$$

$$|W_{2k}(f,x,{\chi }_h^2)|\le c_k\underset{t\in [x-kh,x+kh]}{\sup }|f(t)|.$$

$${\mathcal{K}}_k:=\frac{4}{\pi }\sum _{j=-\infty }^{\infty }(4j+1{)}^{-k-1},$$

$${\mathcal{K}}_0=1,{\mathcal{K}}_1=\frac{\pi }{2},{\mathcal{K}}_2=\frac{{\pi }^2}{8},{\mathcal{K}}_3=\frac{{\pi }^3}{24},{\mathcal{K}}_4=\frac{5{\pi }^4}{384},{\mathcal{K}}_6=\frac{61{\pi }^6}{46080},{\mathcal{K}}_8=\frac{277{\pi }^8}{2064384}.$$

$$E_{n-1}^a(f)\le J_a(2k,\alpha ){\omega }_{2k}(f,\alpha \pi /n),$$

$$1/2\le J_a(2k,\alpha )<3(2+e^{-2})\left(2\sec (\pi /2\alpha )-1-\frac{{\mathcal{K}}_2}{{\alpha }^2}\right),1<\alpha \le (2k-1){\pi }^{-1}.$$

$${\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^{-1}\le J(2k,\alpha )<\sec (\pi /2\alpha ){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^{-1},\alpha >1.$$

$$\frac{1}{2}\le J_a(2,\alpha )$$

$$\frac{1}{2}\le J_a(4,\alpha )$$

$$\frac{1}{2}\le J_a(6,\alpha )$$

$$\frac{1}{2}\le J_a(8,\alpha )$$

$${\beta }_k=\frac{4{\gamma }_k}{{\pi }^2{\alpha }^2},{\gamma }_2=\frac{4}{3},{\gamma }_3=\frac{68}{45},{\gamma }_4=\frac{512}{315}.$$

$$J_a(2,1)\le 0.94,J_a(4,1)\le 4.38,J_a(6,1)\le 6.71,J_a(8,1)\le 8.9,$$

$$J_a(2,2)\le 0.8,J_a(4,2)\le 2.59,J_a(6,2)\le 2.84,J_a(8,2)\le 2.97,$$

$$J_a(2k,2)\le 9.74,n\ge 2k(2k-1).$$

$$J_a(2k,2)\le 9.74\cdot (2+e^{-2}{)}^{-1}=4.57,$$

Full text

On the Jackson constants for algebraic approximation of continuous functions

A. G. Babenko,  Yu. V. Kryakin

Dedicated to Professor Igor A. Shevchuk on the occasion of his 70th birthday

11footnotetext: AMS classification: Primary 41A17, 41A44, 42A10.22footnotetext: *Key words *: Algebraic approximation, Brundyi–Jackson theorem, $k$-th modulus of smoothness, estimate of constants.

Abstract We establish new estimates for the constant $J_{a}(k,\alpha)$ in the Brudnyi–Jackson inequality for approximation of $f\in C[-1,1]$ by algebraic polynomials:

[TABLE]

The main result of the paper implies the following inequalities

[TABLE]

1. Introduction

In this note we use the relatively new approach (convolution series by C.Neumann [1] with the Boman–Shapiro integral operators [2, 3] ) for the constant problems in the following Brudnyi–Jackson theorem (see [4, 5] ) for algebraic approximation of $f\in C[-1,1]$:

[TABLE]

The case of algebraic approximation $E_{n-1}^{a}(f)$ of a continuous function $f$ by algebraic polynomials of degree $\leq n-1$ is in some sense more difficult then the case of trigonometric approximation. Usually the reduction to the trigonometric approximation is used. There are some technical problems in the case of the modulus of $\omega_{k}(f,\alpha\pi/n)$ of the order $k\geq 2.$ The problem of exact constants in this case is a difficult one and we do not have sharp results for $k>1$.

We recall here a result by Korneichuk [6], as the corollary of his remarkable theorem on constants in the case of concave modulus of continuity $\overline{\omega}$:

[TABLE]

and the new Mironenko’s result [7] for the second modulus of continuity:

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In the present paper we prove that for $n>2k(2k-1)$ the constants in (1.1) are bounded by an absolute constant:

[TABLE]

It is clear that

[TABLE]

and therefore the main result of this paper states that for $\alpha\geq 2,\ n>k(k-1)$ we can write constant in (1.1) that do not depend on $k$.

2. Notation. Auxiliary facts. Main results

In this paper $i,j,k,l,m,n$ denote the natural numbers. Let $A$ be $\mathbb{I}=[-1,1]$ or $\mathbb{R}$. We denote by $W^{k}_{*}(A)$ the space of smooth functions $f^{(j)}\in C(A),\ j=0,1\dots,k-1$, $f^{(k)}$ bounded a.e. on $A$. We will also use the notation

[TABLE]

We consider the approximation of real functions on $\mathbb{I}=[-1,1]$ by algebraic polynomials $p_{n-1}(x)=\sum_{j=0}^{n-1}a_{j}x^{j}$ of degree at most $n-1$. We will denote by $\mathcal{P}_{n-1}$ the space of such polynomials. The best approximation of $f\in C(\mathbb{I})$ by $p\in\mathcal{P}_{n-1}$ is defined by standard way

[TABLE]

Smoothness of function $f\in C(A)$ is measured by modulus of smoothness. Beside the classical $k$–th modulus of smoothness

[TABLE]

we will use the special Boman–Shapiro modulus of continuity, which measures the deviation of the function from the special linear combination of Steklov’s means (see [8, 9, 10]). We will use the following convolution notation

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and the following notation for characteristic function and the convolution square of a characteristic function:

[TABLE]

[TABLE]

Define the special difference operator for a locally integrable function $f$ in the following way (see [9, 10])

[TABLE]

where

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It was proved in [10] that

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with

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and therefore

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We will use the standard notation for the Favard constants

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[TABLE]

We are now ready to state the main results of this paper.

Theorem 1 Suppose $f\in C(\mathbb{I})\,.$ Then for $n\geq 2k(2k-1),\ k\geq 5$

[TABLE]

and

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Note that in the trigonometric case we have (see [9, 10])

[TABLE]

Theorem 2 For small $k=1,2,3,4,\ \alpha>0,$ we have the following estimates of the constants

[TABLE]

where

[TABLE]

Thus we may write the estimates for $\alpha=1,2$. The constants are increasing as $k$ increases for $\alpha=1$:

[TABLE]

but for $\alpha=2$

[TABLE]

and for $k\geq 5$ the estimates of Theorem 1 give

[TABLE]

Note that if the Sendov conjecture $w_{k}\leq 1$ is true (see Theorem B below), then we achieve better inequality

[TABLE]

which is near the results in the case of small $k$.

The proofs of the main inequalities are based on the following fundamental facts. The first important fact is the algebraic variant [15] of the Bohr–Favard–Akhiesier–Krein inequality (see [11, 12, 13, 14]):

Theorem A For $f\in W_{*}^{m}(\mathbb{I})$

[TABLE]

[TABLE]

The second important fact is the modern variant of Whitney’s theorem, with good estimates of constants (see [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]):

Theorem B * Suppose $f\in C[a,b],\ 0<a<b$. Then*

[TABLE]

with

[TABLE]

Theorems A,B are the main technical tools for proving Theorem 1 and Theorem 2. Theorem 1 and Theorem 2 follow from Proposition 1 and Proposition 2.

Proposition 1 concerns the problem of continuation of a function from $\mathbb{I}$ to $\mathbb{R}$. This continuation allows us to use the periodic–case approach. **Proposition 1 ** Let $f\in C(\mathbb{I}),\,0<h<(2k)^{-1}.$ Then there exists a function $g_{f}$ which is equal to $f$ on $\mathbb{I}$, continuous on $\mathbb{R}\setminus\mathbb{I}$ and such that

[TABLE]

[TABLE]

with

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where

[TABLE]

and

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The key-idea is the same as in the periodic case (see [9, 10]): we will use the truncated Neumann convolution series

[TABLE]

with some modification for the algebraic case.

Proposition 2 Let $f\in C(\mathbb{I})$, $g_{f}\equiv f$ on $\mathbb{I}$ and $g_{f}\in C(\mathbb{R}\setminus\mathbb{I})$. Then for $k\geq 2,\ n\geq 2k(2k-1)$

[TABLE]

For deducing the main theorems from Proposition 2 we present here a new variant of the known estimates (see [9, 10]).

Put

[TABLE]

*Lemma 1 ** * For $k\geq 2,\ j=0,\dots,k-1$

[TABLE]

Note that in comparison with [9, 10] we add new inequality here, which allows to estimate the last term in inequality (2) .

To prove Lemma 1 we represent $\Lambda_{2k}(x)$ as the special linear combination of

[TABLE]

Lemma 2   For $k\geq 2,\ h>0$ we have

[TABLE]

The last part of the paper is organized in the following way. In the next section we give the proofs of auxiliary results. The proofs of main results will appear in the last section.

3. Proofs of auxiliary results

The proofs of auxiliary results will be given in the reverse order. First, we will prove Lemma 2. Then, on the basis of Lemma 2, we will prove Lemma 1. After that, we will use the C. Neumann decomposition to give the proof of Proposition 2, and then, finally we prove Proposition 1.

Proof of Lemma 2. We decompose the characteristic function $\chi_{jh}$ in the operator

[TABLE]

in the special way

[TABLE]

In particular

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It is easy to see that

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Therefore, we have by direct calculation

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Equality (3.6) implies that this representation is equivalent to the following equalities for the Fejér kernel. For $j=2\nu+1,$

[TABLE]

and for $j=2\nu$

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The substitution of

[TABLE]

into (3.5) gives

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[TABLE]

[TABLE]

with the coefficients

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Note that $\mbox{sign}\,\alpha_{i,k}=(-1)^{i}$ (see [10], Lemma A).

By using (3.8) and the identity

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we obtain

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[TABLE]

[TABLE]

[TABLE]

∎Proof of Lemma 1 Lemma 2 implies for $j=1,\dots,k$

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Indeed, we have

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[TABLE]

[TABLE]

[TABLE]

Now, one can apply the identities

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[TABLE]

[TABLE]

[TABLE]

[TABLE]

which are true a.e. for the function $f$ continuous on $(-\infty,-1)\cup(-1,1)\cup(1,+\infty)$ and the inequalities

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to end the proof. ∎Proof of Proposition 2   By using the Neumann decomposition of $g_{f}$

[TABLE]

one can estimate the approximation of $f$ on $\mathbb{I}$ by algebraic polynomials $p\in\mathcal{P}_{n-1}$ of degree $n-1\geq 2k(2k-1)-1$:

[TABLE]

We apply Theorem A for $(g_{f}-g_{f}*\Lambda_{2k})*\Lambda_{2k}^{j}$, and the inequality

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The estimate (3.9) follows from the inequality

[TABLE]

We have

[TABLE]

∎

Proof of Propostion 1 Suppose that $p_{2k-1}^{-},p_{2k-1}^{+}$ are the polynomials of the best approximation of $f$ on $\mathbb{I}^{-}=[-1,-1+2kh]$ and $\mathbb{I}^{+}=[1-2kh,1]$ respectively. Define

[TABLE]

Theorem B implies

[TABLE]

We will prove that

[TABLE]

with

[TABLE]

By symmetry, it is sufficient to consider only the cases

1. $x\in[0,1-kh]$,

2. $x\in(1-kh,1]$,

3. $x\in(1,+\infty)$. In the case 1) we have

[TABLE]

In the second case we have $x+kh\geq 1$. The identity $W_{2k}(p^{+}_{2k-1},x,\chi_{h})\equiv 0$ yields

[TABLE]

By using the inequaltiy (2.2)

[TABLE]

and Whitney’s theorem (2.3)

[TABLE]

we deduce the estimate in the second case.

The third case is similar to the second case, when $x\in(1,1+kh]$. If $x>1+kh$, the $2k$–th difference is equal to zero. Thus the estimate

[TABLE]

is proved.

The proof of the estimate

[TABLE]

is the same as in the case $W_{2k}$. It is sufficient to consider only $x\geq 0$ and instead of the inequality

[TABLE]

to use in the cases 2) and 3) the following inequalities

[TABLE]

∎

4. Proofs of main results

Proof of Theorem 1 Put

[TABLE]

Proposition 2 and Lemma 1 imply

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Now we can apply Proposition 1 and obtain

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By choosing

[TABLE]

and by using the identity

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we deduce the estimate

[TABLE]

The last estimate follows from the fact that $2\sec\left(\frac{\pi}{2x}\right)-1-\frac{\mathcal{K}_{2}}{x^{2}}$ is a decreasing function for $x>1$. So, the upper estimate for $J(2k,\alpha)$ is proved.

At last we prove the lower estimate. Consider the function

[TABLE]

The best approximation of this function is $\geq 1/2$. But the $k$–th difference

[TABLE]

for $x=-1$ is equal to $(-1)^{k}$.

Consider now a regularization of $f_{0}$. Suppose that $k\geq 2,\ \varepsilon\in(0,1/k)$ and define

[TABLE]

We will use the representation

[TABLE]

Note that

[TABLE]

For $h>0$ the first difference

[TABLE]

is not equal to zero only if

[TABLE]

From

[TABLE]

and

[TABLE]

we obtain for $x,x+kh\in\mathbb{I}$

[TABLE]

It is clear, that for small $\varepsilon>0$

[TABLE]

In the case $k=1$ sufficient to consider the function $f_{\varepsilon,2}$. ∎

Proof of Theorem 2   In the case $k=1$ we have

[TABLE]

In this case the inequalities (see Proposition 1)

[TABLE]

are better than inequality

[TABLE]

In the case $k\geq 2$ the estimates $W_{2k}(g_{f},h)\leq d_{k}\ \omega_{2k}(f,h)$ give better results.

If $k=2,3,4,$ then  Theorem 2 follows from the inequality (see the proof of Theorem 1)

[TABLE]

and the estimates of $d_{k},d_{k}^{*},\gamma_{k}$ (see Proposition 1 and definition of $\gamma_{k}$).  ∎