UDC 517.512.7

MATHEMATICS

M. I. MOROZOV

ON THE QUESTION OF UNIFORM APPROXIMATION OF FUNCTIONS BY POSITIVE LINEAR OPERATORS

(Presented by Academician V. I. Smirnov on 1 IV 1966)

Let \(a\) and \(b\), \(a<b\), be fixed numbers; let \(\omega(h)\), \(0\le h\le b-a\), \(\omega(0)=0\), be a nondecreasing function, positive for \(0<h\le b-a\); let \(H^\omega\) be the class of functions \(f(x)\), \(a\le x\le b\), such that
\[ |f(x_1)-f(x_2)|\le \omega(|x_1-x_2|) \]
for \(x_1\) and \(x_2\) from \([a,b]\); let \(\sigma(x)\), \(a\le x\le b\), be a nondecreasing function having an infinite number of points of increase and such that \(\sigma(b)-\sigma(a)=1\); let \(\{\varphi_k(x)\}_0^\infty\) be a system of continuous functions orthonormal on \([a,b]\) with integral weight \(\sigma(x)\) (see (1), pp. 134, 144), with \(\varphi_0(x)\equiv 1\), and let \(\lambda=\{\lambda_k\}_0^\infty\) be a collection of real numbers \(\lambda_k\) such that the series
\[ K(x,t,\lambda)=\sum_{k=0}^{\infty}\lambda_k\varphi_k(x)\varphi_k(t),\qquad a\le x,t\le b, \]
converges uniformly and its sum \(K(x,t,\lambda)\ge 0\), \(a\le x,t\le b\).

Let \(f(x)\in H^\omega\), and let \(a_k\) be the Fourier coefficients of \(f(x)\):
\[ a_k=\int_a^b f(t)\varphi_k(t)\,d\sigma(t),\qquad k=0,1,\ldots . \]

Introduce the positive linear operator
\[ U(x,f,\lambda)=\sum_{k=0}^{\infty}\lambda_k a_k\varphi_k(x) =\int_a^b f(t)K(x,t,\lambda)\,d\sigma(t). \]

In what follows we put \(\lambda_0=1\). Then
\[ f(x)-U(x,f;\lambda)=\int_a^b [f(x)-f(t)]K(x,t,\lambda)\,d\sigma(t). \]

Hence, for
\[ E(x,\lambda)=\sup_{f\in H^\omega}|f(x)-U(x,f,\lambda)| \]
we obtain the estimate
\[ E(x,\lambda)\le \int_a^b \omega(|x-t|)K(x,t,\lambda)\,d\sigma(t) =\sum_{k=0}^{\infty}\lambda_k a_k(x)\varphi_k(x), \]
\[ a_k(x)=\int_a^b \omega(|x-t|)\varphi_k(t)\,d\sigma(t). \tag{1} \]

If, for some \(x=x_0\), \(a\le x_0\le b\), the function \(\omega(|x_0-x|)\in H^\omega\), then in estimate (1) at \(x=x_0\) equality holds. In particular, if \(\omega(x)\) is a modulus of continuity, then equality in (1) holds for every \(x\), \(a\le x\le b\).

Example. Let \(a=-1,\ b=1;\ \omega(h)=h;\ \sigma(x)=1/\pi\arcsin x;\)
\(\varphi_k(x)=\sqrt{2}\cos k\arccos x=\sqrt{2}T_k(x),\ k=1,2,\ldots\). In this case

\[ K(\cos\theta,\cos\varphi,\lambda)=1+2\sum_{k=1}^{\infty}\lambda_k\cos k\theta\cos k\varphi= \]

\[ =\frac{1}{2}K(\theta-\varphi,\lambda)+\frac{1}{2}K(\theta+\varphi,\lambda), \]

\[ K(\theta,\lambda)=1+2\sum_{k=1}^{\infty}\lambda_k\cos k\theta, \]

and, consequently, in order that the function \(K(x,t,\lambda)\) be nonnegative for \(-1\le x,t\le 1\), it is necessary and sufficient that the function \(K(\theta,\lambda)\) be nonnegative for \(0\le \theta<\pi\).

Let us find \(E(x,\lambda)\) for this case. Putting \(x=\cos\theta,\ t=\cos\varphi,\ 0\le \theta,\varphi\le \pi\), we have

\[ a_0(x)=\frac{1}{\pi}\int_a^b |x-t|\frac{dt}{\sqrt{1-t^2}} =\frac{1}{\pi}\int_0^\pi |\cos\theta-\cos\varphi|\,d\varphi= \]

\[ =\left(1-\frac{2\theta}{\pi}\right)\cos\theta+\frac{2}{\pi}\sin\theta, \]

\[ a_1(x)=-\frac{1}{\sqrt{2}}\left(1-\frac{2\theta}{\pi}\right)-\frac{1}{\sqrt{2}\pi}\sin 2\theta, \]

\[ a_k(x)=\frac{\sqrt{2}}{\pi}\left[\frac{1}{k(k-1)}\sin(k-1)\theta-\frac{1}{k(k+1)}\sin(k+1)\theta\right],\qquad k=2,3,\ldots, \]

and formula (1) gives

\[ E(x,\lambda)=(1-\lambda_1)\left(1-\frac{2}{\pi}\theta\right)\cos\theta+ \]

\[ +\frac{1}{\pi}\left[2\left(1-\frac{\lambda_1}{4}-\sum_{k=2}^{\infty}\frac{\lambda_k}{k^2-1}\right) -\sum_{k=1}^{\infty}\frac{\lambda_k-\lambda_{k+1}}{k(k+1)}\sin(2k+1)\theta\right]. \]

Returning to the variable \(x\), we finally obtain

\[ E(x,\lambda)=(1-\lambda_1)x\left(1-\frac{2}{\pi}\arccos x\right)+ \]

\[ +\frac{1}{\pi}\sqrt{1-x^2}\left[2\left(1-\frac{\lambda_1}{4}-\sum_{k=2}^{\infty}\frac{\lambda_k}{k^2-1}\right) -\sum_{k=1}^{\infty}\frac{\lambda_k-\lambda_{k+1}}{k(k+1)(2k+1)}T'_{2k+1}(x)\right]. \]

If here we put \(\lambda_k=(n-k)/n,\ k=1,2,\ldots,n-1,\ \lambda_k=0,\ k=n,n+1,\) then we arrive at the result of A. F. Timan \({}^{2}\) (obtained by him by another method) for the arithmetic means of the Fourier–Chebyshev series.

In conclusion I express my deep gratitude to N. A. Lebedev for useful advice.

Received
18 III 1966

REFERENCES

\({}^{1}\) V. L. Goncharov, Theory of Interpolation and Approximation of Functions, 1954.
\({}^{2}\) A. F. Timan, DAN, 77, No. 6, 969 (1951).