Abstract Generated abstract
This paper studies maximal functions formed from the partial sums of cosine and sine trigonometric series whose coefficients tend to zero and form a quasiconvex sequence. It proves asymptotic estimates for the integrals, away from the origin, of the majorants of these partial sums in terms of tail maxima of the coefficients and, for the sine series, the coefficients themselves. As a consequence, the maximal functions for both the cosine and sine series are integrable on \([0,\pi]\) exactly when \(\sum_{k\ge1} k^{-1}\max_{n\ge k}|a_n|\) converges, and the result is compared with known estimates for the sine series and for monotone coefficients.
Full Text
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).