UDC 517.512.6

MATHEMATICS

V. N. MALOZEMOV

ON THE SIMULTANEOUS APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY ALGEBRAIC POLYNOMIALS

(Presented by Academician V. I. Smirnov on 13 I 1966)

The following assertion generalizes A. F. Timan’s theorem on the approximation of a function given on a finite interval by algebraic polynomials (¹), to the case of simultaneous approximation of a function and its derivatives. It also strengthens the corresponding results of G. Freud (²) and A. O. Gelfond (³).

Theorem. For every function \(f(x)\) having a continuous \(r\)-th derivative on the finite interval \([-1,1]\), and for every natural number \(n \ge r\), one can indicate an algebraic polynomial \(Q_n(x)\) of degree not exceeding \(n\) such that, for all \(s=0,1,2,\ldots,r;\ -1 \le x \le 1\),

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

where \(\omega(\delta)=\omega(f^{(r)};\delta);\ \delta \in [0,\infty)\) is the modulus of continuity of \(f^{(r)}(x)\); \(C_r\) is a constant depending only on \(r\).

The polynomial \(Q_n(x)\) can be written explicitly. To this end, consider the kernels \(U_{N,s}(t)\) \((s=0,1,2,\ldots,r;\ N=1,2,\ldots)\)

\[ U_{N,s}(t)=\frac{1}{\psi_N^{(s)}}\left(\frac{\sin Nt/2}{N\sin t/2}\right)^{2s+4}, \]

where

\[ \psi_N^{(s)}=\int_{-\pi}^{\pi} \left(\frac{\sin Nt/2}{N\sin t/2}\right)^{2s+4}\,dt . \]

Next put

\[ P_{N,s}(f;x)=\int_{-\pi}^{\pi} f(\cos t)\, U_{N,s}(t-\arccos x)\,dt . \]

Let \(E\) be the identity operator, \(N=[(n-r)/(r+2)]+1\). Then

\[ Q_n(x)=f(x)-(E-P_{N,r}) \left( \int_{0}^{x} (E-P_{N,r-1})\times \right. \]

\[ \left. \times \left( \int_{0}^{x_1} \cdots \left( \int_{0}^{x_{r-2}} (E-P_{N,1}) \left( \int_{0}^{x_{r-1}} (E-P_{N,0})(f^{(r)};x_r)\,dx_r \right) dx_{r-1} \right) \cdots \right) dx_1 \right). \]

The main role in the proof of the theorem is played by the following

Lemma. Let the function \(f(x)\) have a continuous first derivative on the interval \([-1,1]\), and suppose that

\[ |f'(x)| \le \left[ \frac{1}{n}\left(\sqrt{1-x^2}+\frac{1}{n}\right) \right]^{\nu-1} \omega \left[ \frac{1}{n}\left(\sqrt{1-x^2}+\frac{1}{n}\right) \right] \]

\((\omega(\delta)\) is some modulus of continuity, \(\nu\) is a natural number). Then for \(-1 \leq x \leq 1\) the inequalities
\[ \left| f(x)-P_{n,\nu}(f;x)\right| \leq M_\nu \left[\frac{1}{n}\left(\sqrt{1-x^2}+\frac{1}{n}\right)\right]^\nu \omega\left[\frac{1}{n}\left(\sqrt{1-x^2}+\frac{1}{n}\right)\right], \]
\[ \left|P_{n,\nu}^{(s)}(f;x)\right| \leq K_\nu \left[\frac{1}{n}\left(\sqrt{1-x^2}+\frac{1}{n}\right)\right]^{\nu-s} \omega\left[\frac{1}{n}\sqrt{1-x^2}+\frac{1}{n}\right] \]
\[ (s=1,2,\ldots,\nu). \]

Omitting the proof of this lemma because of its bulkiness, we pass to the proof of the theorem.

Since for all \(n \geq r\)
\[ \frac{1}{N}=\frac{1}{n}\cdot \frac{n}{1+[(n-r)/(r+2)]}\leq 2(r+1)\frac{1}{n}, \]
it is enough to verify the relations
\[ \left| f^{(s)}(x)-Q_n^{(s)}(x)\right|\leq C_r \left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]^{r-s} \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right] \]
\[ (s=0,1,2,\ldots,r;\qquad -1\leq x\leq 1). \]

First note that, by virtue of the assertion proved in (1),
\[ \left|(E-P_{N,0})(f^{(r)};x)\right|\leq A\omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right], \]
where \(\omega(\delta)=\omega(f^{(r)};\delta)\) and \(A\) is an absolute constant. The function
\[ \Phi_1(x)=\int_0^x (E-P_{N,0})(f^{(r)};t)\,dt \]
has on the interval \([-1,1]\) a continuous first derivative, and moreover
\[ |\Phi_1'(x)|\leq A\omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]. \]

By the lemma,
\[ \left|(E-P_{N,1})(\Phi_1;x)\right|\leq AM_1\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right] \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right], \]
\[ \left|P'_{N,1}(\Phi_1;x)\right|\leq AK_1\omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]. \]
Hence, for \(s=0,1\),
\[ \left|\frac{d^s}{dx^s}(E-P_{N,1})(\Phi_1;x)\right|\leq C_1\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]^{1-s} \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]. \]

Now put
\[ \Phi_2(x)=\int_0^x (E-P_{N,1})(\Phi_1;t)\,dt. \]

Analogously to the preceding, referring to the lemma, we obtain
\[ \left|(E-P_{N,2})(\Phi_2;x)\right|\leq C_1M_2\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]^2 \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right], \]
and for \(s=1,2\)
\[ \left|P_{N,2}^{(s)}(\Phi_2;x)\right|\leq C_1K_2\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]^{2-s} \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]. \]
Hence, for \(s=0,1,2\),
\[ \left|\frac{d^s}{dx^s}(E-P_{N,2})(\Phi_2;x)\right|\leq C_2\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]^{2-s} \omega\left[\frac{1}{N}\left(\sqrt{1-x^2}+\frac{1}{N}\right)\right]. \]

Continuing this process, each time setting

\[ \Phi_\nu(x)=\int_0^x (E-P_{N,\nu-1})(\Phi_{\nu-1};t)\,dt, \]

we finally obtain

\[ \left|\frac{d^s}{dx^s}(E-P_{N,r})(\Phi_r;x)\right| \leq C_r\left[\frac1N\left(\sqrt{1-x^2}+\frac1N\right)\right]^{r-s} \omega\left[\frac1N\left(\sqrt{1-x^2}+\frac1N\right)\right] \]

\[ (s=0,1,2,\ldots,r). \]

It remains to note that

\[ (E-P_{N,r})(\Phi_r;x)=f(x)-Q_n(x). \]

Leningrad State University
named after A. A. Zhdanov

Received
2 I 1966

References Cited

¹ A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960, pp. 276–280. ² G. Szegő, Orthogonal Polynomials, Moscow, 1962, pp. 20–22. ³ A. O. Gelfond, UMN, 10, no. 1 (1955).