UDC 517.512

MATHEMATICS

S. A. TELYAKOVSKII

INTEGRABILITY OF THE MAJORANT OF PARTIAL SUMS OF A TRIGONOMETRIC SERIES WITH QUASICONVEX COEFFICIENTS

(Presented by Academician I. M. Vinogradov on 3 XI 1969)

Let the numbers \(a_k,\ k=0,1,2,\ldots,\) tend to zero and form a quasiconvex sequence, i.e. the series converges

\[ \sum_{k=1}^{\infty} k\left|\Delta^2 a_{k-1}\right|,\qquad \text{where } \Delta^2 a_{k-1}=a_{k-1}-2a_k+a_{k+1}. \]

Then the series

\[ \frac{a_0}{2}+\sum_{k=1}^{\infty} a_k\cos kx,\qquad \sum_{k=1}^{\infty} a_k\sin kx \tag{1} \]

converge for all \(x\in(0,\pi]\) to functions continuous for these \(x\), which we shall denote respectively by \(f(x)\) and \(g(x)\).

It is known that under the assumptions made \(f\in L[0,\pi]\) ((\(^{1}\); (\(^{2}\), § 5.12), and for the integral of the modulus of the function \(g\) the estimate (\(^{3}\)) is valid

\[ \int_{\varepsilon}^{\pi} |g(x)|\,dx = \sum_{k=1}^{[1/\varepsilon]} \frac{|a_k|}{k}+O(1), \tag{2} \]

whence it follows that \(g\in L[0,\pi]\) if and only if the series

\[ \sum_{k=1}^{\infty}\frac{|a_k|}{k} \]

converges.

Consider the question of integrability of the majorants of the partial sums of the series (1), i.e. of the functions

\[ f^*(x)=\max_n\left|\frac{a_0}{2}+\sum_{k=1}^{n} a_k\cos kx\right|,\qquad x\ne0, \]

\[ g^*(x)=\max_n\left|\sum_{k=1}^{n} a_k\sin kx\right|. \]

Theorem. Let \(\{a_k\}\) be a quasiconvex sequence of numbers tending to zero. Then, as \(\varepsilon\to+0\), the estimates

\[ \int_{\varepsilon}^{\pi} f^*(x)\,dx = \sum_{k=1}^{[1/\varepsilon]} \frac{1}{k}\max_{n\ge k}|a_n|+O(1), \tag{3} \]

\[ \int_{\varepsilon}^{\pi} g^*(x)\,dx = \sum_{k=1}^{[1/\varepsilon]} \frac{1}{k}\left(|a_k|+\max_{n\ge k}|a_n|\right)+O(1). \tag{4} \]

In particular, each of the functions \(f^*\) and \(g^*\) belongs to \(L[0,\pi]\) if and only if the series

\[ \sum_{k=1}^{\infty} \frac{1}{k}\max_{n\ge k}|a_n| \tag{5} \]

converges.

From estimates (2) and (4) we conclude that, for series with quasiconvex coefficients,

\[ \int_{\varepsilon}^{\pi} g^*(x)\,dx \geq 2\int_{\varepsilon}^{\pi}|g(x)|\,dx + O(1), \tag{6} \]

and if one additionally assumes that \(a_k\) decrease monotonically, then

\[ \int_{\varepsilon}^{\pi} g^*(x)\,dx = 2\int_{\varepsilon}^{\pi}|g(x)|\,dx + O(1). \tag{7} \]

Let us note that from the known results for series with monotone coefficients it follows that relation (7) holds even without the assumption of quasiconvexity of \(\{a_k\}\), under the sole condition that \(a_k\) decrease monotonically.

Steklov Mathematical Institute
Academy of Sciences of the USSR
Moscow

Received
30 X 1969

References

\(^{1}\) A. Kolmogoroff, Bull. Acad. Polon., Ser. A., Sci. Math., 83 (1923).
\(^{2}\) A. Zygmund, Trigonometric Series, Moscow–Leningrad, 1939.
\(^{3}\) S. A. Telyakovskii, Matem. sbornik, 63 (105), No. 3, 426 (1964).