Asymptotic Behavior of Partial Sums of Fourier--legendre Series

R. Bojanić, Z. Divis

If $f$ is defined and has a derivative of bounded variation on $[-1,1]$ the main result of this paper is the asymptotic formula for the partial sums of the Fourier-Legendre expansion of $f$: $$ S_n(f,x) = f(x)+(n\pi)^{-1}\sqrt{1-x^2}(f_R'(x)-f_L'(x))+o(1/n). $$ Here $f_R'(x)$ and $f_L'(x)$ are the right and the left derivatives of $f$ at $x\in (-1,1)$.