\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=4.10.0.2345} %TCIDATA{Created=Saturday, September 28, 2002 09:28:35} %TCIDATA{LastRevised=Wednesday, November 06, 2002 21:49:16} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=40 LaTeX article.cst} %TCIDATA{PageSetup=108,108,108,108,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36,\PARA{038

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \thepage } %F=36 %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \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.  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  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.  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 : 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 $00.$ $\qquad \qquad f(x)=\lim\limits_{n\rightarrow \infty }S_{n}(x)$ \qquad \qquad \qquad =$\dfrac{1}{2\sin x}\left[ \sum\limits_{k=1}^{\infty }(\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\tilde{S}% _{k}^{\alpha }(x)\right]$ Thus $\qquad f(x)-S_{n}(x)=\dfrac{1}{2\sin x}\left[ \sum\limits_{k=n-\left( \alpha +1\right) }^{\infty }(\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\tilde{S}_{k}^{\alpha }(x)\right]$ \qquad \qquad \qquad \qquad $-\dfrac{1}{2\sin x}\left[ \sum\limits_{k=0}^{% \alpha }(\triangle ^{k+1}a_{n-k-1}A_{n-k-1}^{k}\tilde{T}_{n-k-1}^{k}(x)% \right]$ \qquad \qquad \qquad \qquad $-\dfrac{1}{2\sin x}\left[ \sum\limits_{k=0}^{% \alpha }(\triangle ^{k+1}a_{n-k-2}A_{n-k-1}^{k}\tilde{T}_{n-k-1}^{k}(x)% \right]$ \qquad \qquad \qquad \qquad $-\left( a_{n-1}\dfrac{\sin nx}{2\sin x}+a_{n}% \dfrac{\sin (n+1)x}{2\sin x}\right)$ \qquad \qquad $\qquad \qquad$ $\left\Vert f(x)-S_{n}(x)\right\Vert \leq \int\limits_{0}^{\pi }\left\vert \dfrac{1}{2\sin x}\left[ \sum\limits_{k=n-\left( \alpha +1\right) }^{\infty }(\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\tilde{S}% _{k}^{\alpha }(x)\right] \right\vert dx$ \qquad \qquad \qquad \qquad $+\int\limits_{0}^{\pi }\left\vert \dfrac{1}{% 2\sin x}\sum\limits_{k=0}^{\alpha }(\triangle ^{k+1}a_{n-k-1}A_{n-k-1}^{k}% \tilde{T}_{n-k-1}^{k}(x)\right\vert dx$ \qquad \qquad \qquad \qquad $+\int\limits_{0}^{\pi }\left\vert \dfrac{1}{% 2\sin x}\left[ \sum\limits_{k=0}^{\alpha }(\triangle ^{k+1}a_{n-k-2}A_{n-k-1}^{k}\tilde{T}_{n-k-1}^{k}(x)\right] \right\vert dx$ \qquad \qquad \qquad \qquad $+\int\limits_{0}^{\pi }\left\vert \left( a_{n-1}% \dfrac{\sin nx}{2\sin x}+a_{n}\dfrac{\sin (n+1)x}{2\sin x}\right) \right\vert dx$ \qquad \qquad $\qquad \qquad$ $\left\Vert f(x)-S_{n}(x)\right\Vert \leq C\left[ \sum\limits_{k=n-\left( \alpha +1\right) }^{\infty }(\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\int\limits_{0}^{\pi }\left\vert \tilde{S}_{k}^{\alpha }(x)\right\vert dx\right]$ \qquad \qquad \qquad \qquad $+C\left[ \sum\limits_{k=0}^{\alpha }A_{n-k-1}^{k}\left\vert \triangle ^{k+1}a_{n-k-1}\right\vert \int\limits_{0}^{\pi }\left\vert \tilde{T}_{n-k-1}^{k}(x)\right\vert dx% \right]$ $\qquad \qquad \qquad \qquad +C\left[ \sum\limits_{k=0}^{\alpha }A_{n-k-1}^{k}\left\vert \triangle ^{k+1}a_{n-k-2}\right\vert \int\limits_{0}^{\pi }\left\vert \tilde{T}_{n-k-1}^{k}(x)\right\vert dx% \right]$ \qquad \qquad \qquad $\qquad +\int\limits_{0}^{\pi }\left\vert \left( a_{n-1}% \dfrac{\sin nx}{2\sin x}+a_{n}\dfrac{\sin (n+1)x}{2\sin x}\right) \right\vert dx$ \qquad \qquad \qquad $\qquad \leq C\left[ \sum\limits_{k=n-\left( \alpha +1\right) }^{\infty }A_{k}^{\alpha }\left\vert (\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\right\vert \int\limits_{0}^{\pi }\left\vert \tilde{T}_{k}^{\alpha }(x)\right\vert dx\right]$ \qquad \qquad \qquad $\qquad +C\left[ \sum\limits_{k=0}^{\alpha }A_{n-k}^{k}\left\vert \triangle ^{k+1}a_{n-k-1}\right\vert \int\limits_{0}^{\pi }\left\vert \tilde{T}_{n-k-1}^{k}(x)\right\vert dx% \right]$ \qquad \qquad \qquad $\qquad +C\left[ \sum\limits_{k=0}^{\alpha }A_{n-k}^{\alpha }\left\vert \triangle ^{k}a_{n-k-2}\right\vert \int\limits_{0}^{\pi }\left\vert \tilde{T}_{n-k-1}^{k}(x)\right\vert dx% \right]$ \qquad \qquad \qquad $\qquad +\int\limits_{0}^{\pi }\left\vert \left( a_{n-1}% \dfrac{\sin nx}{2\sin x}+a_{n}\dfrac{\sin (n+1)x}{2\sin x}\right) \right\vert dx$ \qquad \qquad \qquad $\qquad \qquad \leq C_{1}\sum\limits_{k=n-\left( \alpha +1\right) }^{\infty }A_{k}^{\alpha +1}\left\vert (\triangle ^{\alpha +2}a_{k-1}+\triangle ^{\alpha +2}a_{k})\right\vert$ $\qquad \qquad \qquad \qquad \qquad +C_{1}\sum\limits_{k=0}^{\alpha }A_{n-k+1}^{k}\left\vert \triangle ^{k+1}a_{n-k-1}\right\vert$ \qquad \qquad \qquad $\qquad \qquad +C_{1}\sum\limits_{k=0}^{\alpha }A_{n-k+1}^{\alpha }\left\vert \triangle ^{k}a_{n-k-2}\right\vert$ (3.1)\qquad \qquad \qquad $\qquad +O(a_{n-1}\log n)\qquad \qquad$($C_{1}$ is an absolute constant) The first three terms of the above inequality are of $o(1)$ by the Lemma and the hypothesis of theorem. Because, \qquad \qquad $\int\limits_{0}^{\pi }\left\vert \left( a_{n-1}\dfrac{\sin nx% }{2\sin x}+a_{n}\dfrac{\sin (n+1)x}{2\sin x}\right) \right\vert dx$ \qquad \qquad $\leq \left\vert a_{n-1}\right\vert \int\limits_{0}^{\pi }\left\vert D_{n}(x)\right\vert dx$ \qquad \qquad $=O(a_{n-1}\log n)\qquad$as $\int\limits_{0}^{\pi }\left\vert D_{n}(x)\right\vert dx\sim \log n.$ $\int\limits_{0}^{\pi }\left\vert f(x)-S_{n}(x)\right\vert dx\rightarrow 0$ if and only if $\left\vert a_{n-1}\right\vert \log n\rightarrow 0$ as $% n\rightarrow \infty .$ This completes the proof of the theorem. \bigskip \begin{center} \textrm{RFFERENCES} \end{center} \begin{enumerate} \item Bala R. and Ram B., Trigonometric series with semi-convex coefficients, Tamkang J. Math, 18(1), (1987), 75-84. \item Bosanquet L.S. , Note on convergence and summability factors (III), Proc. London Math. Soc., (1949) 482-496. \item Kolmogorov A.N., Sur 1' ordere de grandeur des coefficients de la series de Fourier -- Lebesgue, Bull. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1923), 83-86. \item Zygmund A., Trigonometric Series, Volume 1, Cambridge University Press, Vol. I (1968). \end{enumerate} \bigskip \bigskip {\small School of Mathematics and Computer Applications,} {\small Thapar Institute of Engineering and Technology,} {\small Post Box No. 32, Patiala (Pb.)-147004. INDIA} \bigskip {\small *Department of Mathematics,} {\small Maharshi Dyanand University,} {\small Rohtak.INDIA.} {\small E-mail address: mathkk@hotmail.com} \end{document}