MATHEMATICS

L. Ya. SHALASHOVA

AN APPROXIMATION THEOREM OF A. F. TIMAN FOR FUNCTIONS HAVING A CONTINUOUS DERIVATIVE OF FRACTIONAL ORDER

(Presented by Academician S. L. Sobolev on March 17, 1969)

A. F. Timan \((^{1})\) established the following theorem:

If a function \(f(x)\), defined on the interval \([-1,1]\), has there an \(r\)-th continuous derivative (\(r\) an integer), then for every \(n \ge r\) there exists an ordinary polynomial \(P_n(x)\) of degree \(\le n\) such that

\[ |f(x)-P_n(x)| \le C_r \left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right)^r \omega\left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right), \tag{1} \]

where \(C_r\) is a constant independent of \(n\) and \(f\);

\[ \omega(t)=\omega(f^{(r)},t)= \sup_{|x_1-x_2|\le t}|f^{(r)}(x_1)-f^{(r)}(x_2)| \quad (x_1,x_2\in[-1,1]). \]

Later I. E. Gopengauz \((^{2})\) and S. A. Telyakovskii \((^{3})\) showed that inequality (1) can be replaced by the inequality

\[ |f(x)-P_n(x)| \le C_r \left(\frac{\sqrt{1-x^2}}{n}\right)^r \omega\left(\frac{\sqrt{1-x^2}}{n}\right). \tag{2} \]

In the present note we give a generalization of these results to the case where the function \(f(x)\) has a continuous derivative of arbitrary fractional order \(r\).

Theorem. For every function \(f(x)\), defined on \([0,1]\), for which there exists a continuous derivative of fractional order \(r\) \((r=r' + \alpha,\ r'\) an integer, \(0<\alpha<1)\), for every natural \(n \ge r-1\) one can specify an algebraic polynomial \(P_n(x)\) of degree \(\le n\) such that

\[ |f(x)-P_n(x)-cx^r| \le C_r \left(\frac{\sqrt{x(1-x)}}{n}\right)^r \omega\left(f^{(r)};\frac{\sqrt{x(1-x)}}{n}\right), \tag{3} \]

where \(C_r\) is a constant independent of \(n\) and \(f\); \(c\) is a constant depending only on \(f\) and \(r\).

The proof of this theorem is based on the following auxiliary proposition.

Lemma. If

\[ F_\alpha(x)=\frac{1}{\Gamma(\alpha)}\int_0^x (x-t)^{\alpha-1}\varphi(t)\,dt \quad (0<\alpha<1), \tag{4} \]

where \(\varphi(t)\) is some continuous function and \(\varphi(0)=0\), then

\[ \omega_2(F_\alpha,h)\le C(\alpha)h^\alpha\omega(\varphi,h) \quad (h>0), \tag{5} \]

where

\[ \omega_2(F_\alpha,h)= \sup_{|x_1-x_2|\le h} \left|F_\alpha(x_1)-2F_\alpha\left(\frac{x_1+x_2}{2}\right)+F_\alpha(x_2)\right|, \quad x_1,x_2\in[0,1]. \]

Inequality (4), after extending the function \(\varphi(t)\) so that \(\varphi(t)=0\) for \(-\infty\le t\le 0\), is verified directly.

Representing now, in accordance with the known definition of the fractional derivative (see (4), § 9.8), the function \(f(x)\) by the equality

\[ f(x)=\sum_{k=0}^{r'-1} c_k x^k+cx^r+\frac{1}{(r'-1)!}\int_0^x (x-t)^{r'-1}F_\alpha(t)\,dt, \]

where \(F_\alpha(x)\) is expressed by equality (4), and \(\varphi(t)=f^{(r)}(t)-f^{(r)}(0)\), and applying to the function

\[ \Phi(x)=\frac{1}{(r'-1)!}\int_0^x (x-t)^{r'-1}F_\alpha(t)\,dt =f(x)-\sum_{k=0}^{r'-1}c_kx^k-cx^r \]

the theorem of I. E. Gopengauz \((^2)\), we find a polynomial \(Q_n(x)\) of degree \(\leq n\) \((n\geq r)\) such that

\[ |\Phi(x)-Q_n(x)|\leq C_{r'}\left(\frac{\sqrt{x(1-x)}}{n}\right)^{r'} \omega_2\left(\Phi^{(r')},\frac{\sqrt{x(1-x)}}{n}\right). \]

If now

\[ R_n(x)=Q_n(x)+\sum_{j=0}^{r'-1}c_kx^k, \]

then we have

\[ |f(x)-R_n(x)-cx^r|\leq C_{r'}\left(\frac{\sqrt{x(1-x)}}{n}\right)^{r'} \omega_2\left(F_\alpha,\frac{\sqrt{x(1-x)}}{n}\right). \]

After applying the lemma to the right-hand side of the last inequality, we obtain the assertion of the theorem.

Let us note that in the left-hand side of inequality (3), when \(r\) is fractional, the term \(cx^r\) cannot be omitted. In order to see this, it suffices to consider the function \(f(x)=x^r\).

Dnepropetrovsk Agricultural
Institute

Received
25 II 1969

CITED LITERATURE

\(^1\) A. F. Timan, DAN, 78, No. 1 (1951).
\(^2\) I. E. Gopengauz, in: Theory of Functions and Functional Analysis and Their Applications, vol. 1, Kharkov, 1967.
\(^3\) S. A. Telyakovskii, Mat. sbornik, 70 (112), No. 2 (1966).
\(^4\) A. Zygmund, Trigonometric Series, 1939.