Abstract
This chapter discusses the convergence behavior of orthogonal series by methods belonging to the general theory of series. The chapter reviews the Abel-Poisson summation process, and the Abel transform. The classical Abel transform of the partial sums of a series is very useful in the theory of orthogonal series. In the theory of series, those theorems are called Tauberian that afford criteria enabling one to conclude the convergence of a weaker summation process from the convergence of a stronger one. The fundamental theorem concerning the convergence of orthogonal series is discussed. Generalities on the Cesàro summation of orthogonal series are reviewed. The chapter presents the coefficient tests for the Cesàro summability of orthogonal series, and reviews the summability of lacunary orthogonal series. Riesz summation of orthogonal series is reviewed. The chapter also discusses Abel non-summable orthogonal series with monotone coefficients, and the Menchoff's summation theorem.
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