MATHEMATICS

S. A. TELYAKOVSKII

APPROXIMATION OF FUNCTIONS DIFFERENTIABLE IN THE WEYL SENSE BY VALLEE-POUSSIN SUMS

(Presented by Academician A. N. Kolmogorov on 27 XI 1959)

1. Let the function \(f(x)\) be summable and

\[ f(x)\sim \frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k\cos kx+b_k\sin kx). \]

Let \(r>0\) and let \(\alpha\) be a real number. If the series

\[ \sum_{k=1}^{\infty} k^r\left[a_k\cos\left(kx+\frac{\alpha\pi}{2}\right)+b_k\sin\left(kx+\frac{\alpha\pi}{2}\right)\right] \]

is the Fourier series of some summable function, which we shall denote by \(f_\alpha^r(x)\), and if \(\lVert f_\alpha^r(x)\rVert\leq 1\) almost everywhere, then we shall say that \(f(x)\in W_\alpha^r\) (see \((^1)\)). For \(\alpha=r\), \(f_\alpha^r(x)\) is the derivative (in the Weyl sense) of order \(r\) of the function \(f(x)\), and we obtain the class \(W^r\); for \(\alpha=r-1\) we obtain the class \(\overline W^r\) of functions conjugate to functions of the class \(W^r\).

With the aid of the matrix \(\{\lambda_{n,k}\}\), \(n,k=1,2,\ldots\), \(\lambda_{n,k}=0\) for \(k\geq n\), we associate with each function \(f(x)\in W_\alpha^r\) the polynomial

\[ u_n(f,x)=\frac{a_0}{2}+\sum_{k=1}^{n-1}\lambda_{n,k}(a_k\cos kx+b_k\sin kx) \]

and, as \(n\to\infty\), find asymptotic formulas for the upper bounds

\[ U_n(W_\alpha^r)=\sup_{f\in W_\alpha^r}\lVert f(x)-u_n(f,x)\rVert_C . \tag{1} \]

If the function \(\tau(u)=\tau(u,n)\) is defined for \(u=k/n\) by the equalities

\[ \tau\left(\frac{k}{n}\right)=(1-\lambda_{n,k})\left(\frac{k}{n}\right)^{-r}\qquad (k=1,2,\ldots), \]

is continuous, and

\[ A(\tau)=\frac{1}{\pi}\int_{-\infty}^{\infty}\left|\int_0^\infty \tau(u)\cos\left(ut+\frac{\alpha\pi}{2}\right)\,du\right|dt<\infty, \tag{2} \]

then, for \(f(x)\in W_\alpha^r\),

\[ f(x)-u_n(f,x)=\frac{1}{\pi n^r}\int_{-\infty}^{\infty} f_\alpha^r\left(x+\frac{t}{n}\right)\int_0^\infty \tau(u)\cos\left(ut+\frac{\alpha\pi}{2}\right)\,du\,dt . \tag{3} \]

Representation (3) for integral \(\alpha\) was obtained by B. Nadem (2).
From (3) we find that, as \(n\to\infty\),

\[ U_n(W_\alpha^r)=A(\tau)\frac{1}{n^r}+O\left(\frac{1}{n^r}a_n(\tau)\right), \tag{4} \]

where

\[ a_n(\tau)=\int_{I_n}\left|\int_0^\infty \tau(u)\cos\left(ut+\frac{\alpha\pi}{2}\right)\,du\right|\,dt, \tag{5} \]

\[ I_n=\left(-\infty,-\frac{n\pi}{2}\right)\cup\left(\frac{n\pi}{2},\infty\right). \tag{6} \]

1. Let \(s_n(f,x)\), \(n=0,1,2,\ldots\), be the partial sums of order \(n\) of the Fourier series of the function \(f(x)\in W_\alpha^r\). The polynomials

\[ v_{n,m}(f,x)=\frac{1}{m}\sum_{k=n-m}^{n-1}s_k(f,x)\qquad (m=1,2,\ldots,n;\ n=1,2,\ldots) \tag{7} \]

are called the de la Vallée-Poussin sums of the function \(f(x)\).
The upper bounds (1) in approximation by de la Vallée-Poussin sums will be denoted by \(V_{n,m}(W_\alpha^r)\). The asymptotic behavior of \(V_{n,m}(W_\alpha^r)\) as \(n\to\infty\) is determined by us under the assumption that \(\lim \frac{m}{n}\) exists and is equal to \(\theta\), \(0\leqslant \theta\leqslant 1\).

Theorem 1. For \(V_{n,m}(W_\alpha^r)\), as \(n\to\infty\), the following asymptotic formulas hold:

1) If \(\theta=0\), then

\[ V_{n,m}(W_\alpha^r)=\frac{4}{\pi^2}\frac{1}{n^r}\log\frac{n}{m}+O\left(\frac{1}{n^r}\right). \tag{8} \]

2) If \(0<\theta<1\), then

\[ V_{n,m}(W_\alpha^r)=A(\tau_{1-\theta})\frac{1}{n^r} +O\left(\frac{1}{n^{r+1}}\right)+O\left(\frac{\varepsilon_n}{n^r}\right), \tag{9} \]

where

\[ \tau_{1-\theta}(u)= \begin{cases} 0, & \text{for } 0\leqslant u\leqslant 1-\theta,\\[4pt] \dfrac{u-(1-\theta)}{\theta}\,u^{-r}, & \text{for } 1-\theta\leqslant u\leqslant 1,\\[6pt] u^{-r}, & \text{for } 1\leqslant u<\infty, \end{cases} \]

and

\[ \varepsilon_n=\left|\frac{m}{n}-\theta\right|\log\frac{1}{|m/n-\theta|} \quad \text{for } \frac{m}{n}\ne\theta;\qquad \varepsilon_n=0 \quad \text{for } \frac{m}{n}=\theta. \]

3) If \(\theta=1\) and \(0<r<1\), then

\[ V_{n,m}(W_\alpha^r)=A(\tau_{1,r})\frac{1}{n^r} +O\left(\frac{(n-m+1)^{1-r}}{n}\right), \tag{10} \]

where

\[ \tau_{1,r}(u)= \begin{cases} u^{1-r}, & \text{for } 0\leqslant u\leqslant 1,\\ u^{-r}, & \text{for } 1\leqslant u<\infty. \end{cases} \]

4) If \(\theta=1\) and \(r=1\), then

\[ V_{n,m}(\overline W_\alpha^1)=\frac{2}{\pi}\left|\sin\frac{\alpha\pi}{2}\right|\frac{1}{n}\log\frac{n}{\,n-m+1\,}+O\left(\frac{1}{n}\right). \tag{11} \]

If, however, \(\left|\sin\frac{\alpha\pi}{2}\right|=0\), then for \(m=n\)

\[ V_{n,n}(\overline W^1)=A(\tau_{1,1})\frac{1}{n}+O\left(\frac{1}{n^2}\right), \tag{12} \]

and for \(n-m\to\infty\)

\[ V_{n,m}(\overline W^1)=2A(\tau_{1,1})\frac{1}{n} +O\left(\frac{1}{n}\sqrt{\frac{n}{\,n-m\,}\log\frac{n}{\,n-m\,}}\right) +O\left(\frac{1}{n(n-m)}\right), \tag{13} \]

where

\[ \tau_{1,1}(u)= \begin{cases} 1, & \text{for } 0\leq u\leq 1,\\ u^{-1}, & \text{for } 1\leq u<\infty . \end{cases} \]

5) If \(\theta=1\) and \(r>1\), then:

in the case \(n-m=p\to\infty\),

\[ V_{n,m}(W_\alpha^r)=A(\tau_{1,r})\left[\frac{1}{n}+\frac{p}{n^2}+\cdots+\frac{p^{r-2}}{n^{r-1}}\right] +O\left(\frac{1}{n^r}\right)+O\left(\frac{1}{np^r}\right), \tag{14} \]

where

\[ \tau_{1,r}(u)= \begin{cases} 0, & \text{for } 0\leq u\leq 1,\\ (u-1)u^{-r}, & \text{for } 1\leq u<\infty; \end{cases} \]

in the case \(n-m=p\) fixed, \(p\geq 1\),

\[ V_{n,m}(W_\alpha^r)=\frac{1}{\pi}\sup_{f\in W_\alpha^r} \left|\int_{-\infty}^{\infty} f_\alpha^r(t)\int_p^\infty \frac{u-p}{u^r} \cos\left(ut+\frac{\alpha\pi}{2}\right)\,du\,dt\right| \times \]

\[ {}\times\left[\frac{1}{n}+\frac{p}{n^2}+\cdots+\frac{p^{r-2}}{n^{r-1}}\right] +O\left(\frac{1}{n^r}\right); \tag{15} \]

in the case \(m=n\)

\[ V_{n,n}(W_\alpha^r)=\sup_{f\in W_\alpha^r}|\widetilde f'(x)|\frac{1}{n} +O\left(\frac{1}{n^r}\right). \tag{16} \]

For approximation by Fourier sums \((m=1)\), formula (8) for the classes \(W^r\) was obtained by A. N. Kolmogorov \((^3)\) (for integer \(r\)) and by V. T. Pinkevich \((^4)\) (for all \(r>0\)); for the classes \(\overline W^r\), by S. M. Nikol’skii \((^{5,6})\); for the classes \(W_\alpha^r\), by A. V. Efimov \((^7)\). For \(m=o(n)\), formula (8) for the classes \(W^r\) and \(\overline W^r\) was obtained by A. F. Timan \((^8)\).

For approximation by Fejér sums \((m=n)\), formula (11) for the class \(W^1\) belongs to S. M. Nikol’skii \((^9)\), and formula (12) to S. B. Stechkin (see \((^{10})\)). Formula (16) for the classes \(W^r\), \(\overline W^r\), and \(W_\alpha^r\) for integer \(\alpha\) was obtained by S. M. Nikol’skii \((^{11,6})\) and B. Nagy \((^{12,2})\). For the value of the upper bound appearing on the right-hand side of formula (16), see \((^{1,13})\).

The order of decrease of the quantities \(V_{n,m}(W_\alpha^r)\) for \(0<\theta<1\) and \(V_{n,n}(W_\alpha^r)\) for \(0<r\leq 1\) was also known.

For the classes \(W^r\) and \(\overline W^r\) with integer \(r\), Theorem 1 (without formula (16)) was published by the author \((^{10})\); here the integrals entering the constants \(A(\tau)\) are given in \((^{10})\) in transformed form. There an asymptotic formula for \(V_{n,m}(\overline W^1)\) in the case \(n-m\) fixed, not considered in Theorem 1, is also indicated.

1. If in the definition of \(W_\alpha^r\) given above we put \(r=0\), then we obtain the classes of functions \(W_\alpha^0\).

If, for the continuous function \(\varphi(u)=\varphi(u,n)\),

\[ \varphi\left(\frac{k}{n}\right)=\lambda_{n,k}\qquad (k=1,2,\ldots) \tag{17} \]

and \(A(\varphi)<\infty\), then for \(f(x)\in W_\alpha^0\)

\[ u_n(f,x)=\frac{1}{\pi}\int_{-\infty}^{\infty} f_\alpha^0\left(x+\frac{t}{n}\right) \int_0^\infty \varphi(u)\cos\left(ut+\frac{\alpha\pi}{2}\right)\,du\,dt, \tag{18} \]

\[ U_n(W_\alpha^0)=\sup_{f\in W_\alpha^0}\|u_n(f,x)\|_C =A(\varphi)+O(a_n(\varphi)). \tag{19} \]

Representation (18) for integral \(\alpha\) was obtained by B. Nagy \({}^{(2)}\).
For the de la Vallée Poussin sums, the upper bounds (19) will be denoted by \(V_{n,m}(W_\alpha^0)\).

Theorem 2. If \(|\alpha|\leqslant 1\) and \(n\to\infty\),

\[ V_{n,m}(W_\alpha^0)= \left(\frac{4}{\pi^2}\cos\frac{\alpha\pi}{2} +\frac{2}{\pi}\alpha\sin\frac{\alpha\pi}{2}\right) \log\frac{n}{m} +\frac{2}{\pi}\left|\sin\frac{\alpha\pi}{2}\right|\log m+O(1). \tag{20} \]

If \(\alpha=0\) and \(\dfrac{m}{n}\to\theta,\ 0<\theta\leqslant 1,\)

\[ V_{n,m}(W_0^0)=A(\varphi_{1-\theta})+O(\varepsilon_n), \tag{21} \]

where

\[ \varphi_{1-\theta}(u)= \begin{cases} 1, & \text{for } 0\leqslant u\leqslant 1-\theta,\\[4pt] \dfrac{u-(1-\theta)}{\theta}, & \text{for } 1-\theta\leqslant u\leqslant 1,\\[6pt] 0, & \text{for } 1\leqslant u<\infty. \end{cases} \]

Formula (20) for \(\alpha=1\) was communicated to the author by S. B. Stechkin. Formula (21) belongs to S. M. Nikol’skii \({}^{(14)}\) (see also \({}^{(10)}\)).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
27 XI 1959

CITED LITERATURE

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