On the convergence of Legendre series with coefficients of bounded variation

In this paper, we study pointwise, uniform, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>r</m:mi> </m:msup> </m:math> {Lʳ} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>r</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<r<1} ) convergence of the Legendre series with coefficients of bounded variation. We also show that “coefficients being of generalized bounded variation” gives a Tauberian theorem for Legendre series, that is, if the coefficients of Legendre series is of generalized bounded variation, then the Legendre series is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>C</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(C,1)} summable if and only if the series is convergent.