On the convergence of Legendre series with coefficients of bounded variation

Key Points

• The aim is to explore the convergence characteristics of Legendre series with coefficients of bounded variation and their generalizations.
• Analyzed pointwise, uniform, and L^r convergence of Legendre series.
• Established conditions under which coefficients exhibit generalized bounded variation.
• Demonstrated a Tauberian theorem linking generalized bounded variation to convergence.
• Identified specific conditions under which Legendre series are (C,1) summable.
• Showed that coefficients of generalized bounded variation lead to convergence of the series.