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\begin{document}
\begin{center}
\textbf{L}$^{\mathbf{1}}$\textbf{-CONVERGENCE OF COSINE SERIES WITH HYPER
SEMI-CONVEX COEFFICIENTS\bigskip }
KULWINDER KAUR \ *BABU\ RAM
and
S.S.BHATIA
\bigskip
\end{center}
\textbf{Abstract.} In this paper we obtain a necessary and sufficient
condition for $L^{1}$-convergence of the Fourier cosine series with hyper
semi-convex coefficients.\ Results of Bala R. and Ram B. [1] have been
obtained as a special case.
2000 Mathematics Subject Classification: 42A20. 42A32
KEY WORDS AND PHRASES. Ces\`{a}ro means, $L^{1}$-convergence, hyper
semi-convexity.\bigskip
\textbf{1. Introduction. }Consider
$(1.1)\qquad \qquad g(x)=\dfrac{a_{0}}{2}+\sum\limits_{k=1}^{\infty
}a_{k}\cos kx$
to be the cosine series with partial sums defined by
$\qquad \qquad \qquad S_{n}(x)=\dfrac{a_{0}}{2}+\sum\limits_{k=1}^{n}a_{k}%
\cos kx$ \qquad \qquad
and let $g(x)=\lim\limits_{n\rightarrow \infty }S_{n}(x)$
\qquad Concerning the $L^{1}$-convergence of cosine series $(1.1)$
Kolmogorov [3] proved his well known theorem:
\textbf{Theorem A}. If $\{a_{n}\}$ is a quasi-convex null sequence, then for
the $L^{1}$-convergence of the cosine series ($1.1)$ it is necessary and
sufficient \ that \ $\lim\limits_{n\rightarrow \infty }a_{n}\log n=0.$
\textbf{Definition.} \ A sequence $\{a_{n}\}$ is said to be semi-convex if $%
\{a_{n}\}\rightarrow 0$ as $n\rightarrow \infty $,
and
\qquad \qquad $\sum\limits_{n=1}^{\infty }n\left\vert \triangle
^{2}a_{n-1}+\triangle ^{2}a_{n}\right\vert <\infty ,\qquad (a_{0}=0)$
where
$\qquad \qquad \triangle ^{2}a_{n}=\triangle a_{n}-\triangle a_{n+1}$
It may be remarked here that every quasi-convex null sequence is semi-
convex.
Bala R. and Ram B. [1] have proved that Theorem A holds true for cosine
series with semi-convex null coefficients in the following form:
\textbf{Theorem B.} If \ $\{a_{k}\}$ is a semi--convex null sequence, then
for the convergence of the cosine series in the metric space $L$, it is
necessary and sufficient that $a_{k-1}\log k=o(1).$
We define $\{a_{n}\}$ to be hyper semi-convex of order $\alpha ,$in the
following way:
\textbf{Definition. }A sequence $\{a_{n}\}$ is said to be hyper semi-convex,
if
$\qquad \qquad \{a_{n}\}\rightarrow 0$ as $n\rightarrow \infty $,
\qquad \qquad $\sum\limits_{n=1}^{\infty }n^{\alpha +1}\left\vert (\triangle
^{\alpha +2}a_{n-1}+\triangle ^{\alpha +2}a_{n})\right\vert <\infty ,\qquad $%
for $\alpha =0,1,2..........$,
$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad (a_{0}=0).$
By definition, hyper semi-convexity of order zero is same as semi-convexity.
\qquad The purpose of this paper is to generalize the Theorem B for the
cosine series with hyper semi-convex null coefficients.
\textbf{2. Notation and Formulae. }In what follows, we use the following
notation [4]:
Given a sequence $S_{0},$ $S_{1},$ $S_{2},.............$,we define for every
$\alpha =0,1,2..........,$
the sequence $S_{0}^{\alpha },$ $S_{1}^{\alpha },$ $S_{2}^{\alpha
},.........,$ by the coditions
$S_{n}^{0}=S_{n},\qquad $
$S_{n}^{\alpha }=S_{0}^{\alpha -1}+S_{1}^{\alpha -1}+S_{2}^{\alpha
-1}+.....+S_{n}^{\alpha -1}\qquad (\alpha =1,2......,n=0,1,2.......).$
Similarly for $\alpha =0,1,2...,$we define the sequence of numbers
$A_{0,}^{\alpha }$ $A_{1,}^{\alpha }$ $A_{2,}^{\alpha }.......$.by the
conditions
$A_{n}^{0}=1,\qquad$
$A_{n}^{\alpha }=A_{0}^{\alpha -1}+A_{1}^{\alpha -1}+A_{2}^{\alpha
-1}+.....+A_{n}^{\alpha -1}\qquad (\alpha =1,2......,$ $n=0,1,2.......).$
where $A_{p}^{\alpha }$ denotes the binomial coefficients and are given by
the following relations.
\qquad\qquad$\sum\limits_{p=0}^{\infty}A_{p}^{\alpha}x^{p}=(1-x)^{-\alpha-1}$
and $\tilde{S}_{n\text{ }}^{,}s$ are given by
\qquad \qquad $\sum\limits_{p=0}^{\infty }S_{p}^{\alpha
}x^{p}=(1-x)^{-\alpha }\sum\limits_{p=0}^{\infty }S_{p}x^{p}$
Also
\qquad \qquad $A_{n}^{\alpha }=\sum\limits_{p=0}^{n}A_{p}^{\alpha -1},\qquad
A_{n}^{\alpha }-A_{n-1}^{\alpha }=A_{n}^{\alpha -1}$
\qquad \qquad $A_{n}^{\alpha }=\dbinom{n+\alpha }{n}\simeq \dfrac{n^{\alpha }%
}{\Gamma \alpha +1}\qquad \qquad (\alpha \neq -1,-2,-3.......)$
Also for $0