UDC 517.512.6

MATHEMATICS

N. P. KORNEICHUK, A. I. POLOVINA

ON THE APPROXIMATION OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS BY ALGEBRAIC POLYNOMIALS ON AN INTERVAL

(Presented by Academician A. N. Kolmogorov, May 15, 1965)

1. Let \(E_n(f)\) be the best uniform approximation of a function \(f(x)\), continuous on the interval \([-1,1]\), by algebraic polynomials of degree not exceeding \(n\). If \(\mathfrak M\) is some class of functions, then we set

\[ E_n[\mathfrak M]=\sup_{f\in\mathfrak M} E_n(f) \qquad (n=0,1,2,\ldots). \]

In the case of approximation of \(2\pi\)-periodic functions by trigonometric polynomials of order \(\leq n\) on the whole axis, we shall write correspondingly \(E_n^*(f)\), \(E_n^*[\mathfrak M]\).

We denote by \(W^{(r)}H_\omega\) \((r=0,1,2,\ldots,\; W^{(0)}H_\omega=H_\omega)\) the class of functions \(f(x)\), defined on the interval \([-1,1]\), for which the modulus of continuity \(\omega(f^{(r)};t)\) of the \(r\)-th derivative does not exceed a prescribed modulus of continuity \(\omega(t)\). If \(\omega(t)=Kt^\alpha\) \((0<\alpha\leq 1)\), we shall write \(W^{(r)}KH^{(\alpha)}\). The corresponding classes of \(2\pi\)-periodic functions will be denoted by \(W^{(r)}H_\omega^*\), \(W^{(r)}KH_*^{(\alpha)}\).

Exact estimates of the quantities \(E_n^*[W^{(r)}H_\omega^*]\), obtained in papers \((^{1,2})\) for the case \(\omega(t)=Kt\), \(r=0,1,2\ldots\), and in papers \((^{3-6})\) for an arbitrary concave upward modulus of continuity \(\omega(t)\) and \(r=0,1,2,3\), can be written in the form

\[ E_{n-1}^*[W^{(r)}H_\omega^*] = E_{n-1}^*(f_{nr}) = \max_x |f_{nr}(x)| \qquad (n=1,2,\ldots), \tag{1} \]

where \(f_{nr}(x)=f_{nr}(\omega;x)\) is a function of period \(2\pi/n\) with mean value over a period equal to zero, whose derivative of order \(r\) \((r=0,1,2,\ldots)\) is defined by the equalities

\[ f_{nr}^{(r)}(x)= \begin{cases} \frac12\,\omega(2x), & (0\leq x\leq \pi/2n),\\ -\frac12\,\omega(-2x), & (-\pi/2n\leq x\leq 0), \end{cases} \]

\[ f_{nr}^{(r)}(x+\pi/n)=-f_{nr}^{(r)}(x). \]

Theorem 1. For any concave upward modulus of continuity \(\omega(t)\) and for each \(r=0,1,2,\ldots\), there exists a sequence of functions \(g_n(x)\in W^{(r)}H_\omega\) such that

\[ E_{n-1}(g_n)\geq (1-\varepsilon_n)\max |f_{nr}(x)|, \]

where \(\varepsilon_n\geq 0\), \(\varepsilon_n=O(1/\ln n)\).

For the class \(KH^{(1)}\) this assertion was proved earlier by S. M. Nikol’skii \((^7)\). By means of Theorem 1 one can show that, for arbitrary concave \(\omega(t)\) and \(r=0,1,2,3\), the upper bound \(E_{n-1}[W^{(r)}H_\omega]\) is asymptotically equal to the right-hand side of equality (1). The fact of asymptotic coincidence of the quantities \(E_n[W^{(r)}KH^{(\alpha)}]\) and \(E_n^*[W^{(r)}KH_*^{(\alpha)}]\) \((r=0,1,2,\ldots)\) was established in papers \((^{7-9})\).

2. S. M. Nikol’skii, in 1946, was the first to discover one important feature of approximation of functions on an interval. He showed \((^7)\) that for any function \(f \in KH^{(1)}\) one can construct a sequence of algebraic polynomials \(P_n(f;x)\) of degree \(\leq n\) (depending linearly on \(f\)) such that, for all \(x \in [-1,1]\),

\[ |f(x)-P_{n-1}(f;x)| \leq \frac{K\pi}{2n}\sqrt{1-x^2}+|x|\,O(\ln n/n^2) \]
\[ =E_{n-1}[KH^{(1)}]\{\sqrt{1-x^2}+o(1)\}, \]

and this relation holds uniformly with respect to \(x\). Thus the polynomials \(P_n(f;x)\), while giving asymptotically best approximation on the class \(KH^{(1)}\), give a significantly smaller deviation at the ends of the interval \([-1,1]\). Later this result was generalized \((^{10})\) to the classes \(W^{(r)}KH^{(1)}\) \((r=1,2,\ldots)\).

It turns out that an analogous fact also holds for other, broader, classes of functions.

Theorem 2. For any function \(f \in KH^{(\alpha)}\) \((0<\alpha<1)\) there exists a sequence of algebraic polynomials \(P_n(f;x)\) of degree \(\leq n\) such that, uniformly in \(x \in [-1,1]\), the inequality

\[ |f(x)-P_{n-1}(f;x)| \leq \frac{K}{2}\left(\frac{\pi}{n}\sqrt{1-x^2}\right)^\alpha+O(n^{-3/2\alpha}) \]
\[ =E_{n-1}[KH^{(\alpha)}]\{(1-x^2)^{\alpha/2}+o(1)\}. \tag{2} \]

Theorem 3. Whatever convex modulus of continuity \(\omega(t)\) may be, for any function \(f \in H_\omega\) there exists a sequence of algebraic polynomials \(P_n(f;x)\) of degree \(\leq n\) such that, uniformly in \(x \in [-1,1]\),

\[ |f(x)-P_{n-1}(f;x)| \leq \frac{1}{2}\omega\left(\frac{\pi}{n}\sqrt{1-x^2}\right)+o\left(\omega\left(\frac{1}{n}\right)\right). \tag{3} \]

The constant \(1/2\) on the right-hand side cannot be decreased.

Theorem 4. Whatever convex modulus of continuity \(\omega(t)\) may be, for any function \(f \in W^{(1)}H_\omega\) there exists a sequence of algebraic polynomials \(P_n(f;x)\) of degree \(\leq n\) such that, uniformly in \(x \in [-1,1]\),

\[ |f(x)-P_{n-1}(f;x)| \leq \frac{1}{4}\int_0^{\frac{\pi}{n}\sqrt{1-x^2}}\omega(t)\,dt +o\left(\frac{1}{n}\omega\left(\frac{1}{n}\right)\right). \tag{4} \]

The constant \(1/4\) on the right-hand side cannot be decreased.

Let us note that the polynomials \(P_n(f;x)\) referred to in Theorems 2–4 depend on the function \(f(x)\) nonlinearly.

The proof of Theorems 2–4 is based on the idea of intermediate approximation of functions of the classes \(H_\omega\) (respectively \(W^{(1)}H_\omega\)) by continuous functions having almost everywhere a first (respectively, second) derivative bounded by a certain majorant depending on \(x\). In this way asymptotically exact estimates are also obtained. For example, the following assertion holds.

Theorem 5. If \(\omega(t)\) is an arbitrary convex modulus of continuity, then for any function \(f \in H_\omega\) one can specify a sequence of polygonal functions \(\varphi_n(x)\) \((n=1,2,\ldots)\) such that:

1) for almost all \(x \in [-1,1]\),

\[ |\varphi'_n(x)| \leq M_n(x) =\frac{1}{2}\left\{\omega'_+\left(\frac{\pi}{n}\sqrt{1-x^2}\right) +\omega'_-\left(\frac{\pi}{n}\sqrt{1-x^2}\right)\right\} \quad (n=1,2,\ldots); \]

* The right-hand sides of inequalities (3) and (4) asymptotically coincide respectively with \(E_{n-1}[H_\omega]\) and \(E_{n-1}[W^{(1)}H_\omega]\) for \(x=0\).

2) at each point \(x\) of the interval \([-1,1]\), uniformly in \(x\),

\[ |f(x)-\varphi_n(x)| \leqslant \frac12 \max_{0\leqslant t\leqslant 2} [\omega(t)-tM_n(x)] + o\left(\omega\left(\frac1n\right)\right). \]

1. In paper (11) it is proved that, for any function \(f(x)\) continuous on the interval \([-1,1]\), the inequalities

\[ E_{n-1}(f) \leqslant \omega(f;\pi/n) \qquad (n=1,2,\ldots). \tag{5} \]

hold. From analogous considerations and Theorem 3 it follows that there exists a sequence of polynomials \(P_n(f;x)\) such that

\[ |f(x)-P_{n-1}(f;x)| \leqslant \omega\left(f;\frac{\pi}{n}\sqrt{1-x^2}\right) + o\left(\omega\left(f;\frac1n\right)\right) \qquad (-1\leqslant x\leqslant 1). \tag{6} \]

Estimates (5) and (6) are, as the following theorem shows, in a certain sense unimprovable.

Theorem 6. Whatever the number \(\delta>0\), there exists a sequence \(f_n(x)\) of functions continuous on \([-1,1]\) for which

\[ E_{n-1}(f_n) \geqslant (1-\varepsilon_n)\,\omega\left(f_n;\frac{\pi-\delta}{n}\right), \]

where \(\varepsilon_n \geqslant 0,\ \varepsilon_n = O(1/\ln n)\).

Dnepropetrovsk State
University

Received
2 III 1965

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