Expansions in Legendre polynomials and Lagrange interpolation (Q1314840)

The \(n\)th partial sums \(S_ n\) of the Fourier-Legendre expansion of a function \(f\in L[-1,1]\) are known to converge for almost every \(x\in[- 1,1]\) when \(f\in L^ p[-1,1]\) and \(p>4/3\). In the present paper the case \(p=4/3\) is considered in detail and it is shown that for \(f\) in the Lorentz space \(L^{4/3,1}[-1,1]\subset L^{4/3}[-1,1]\) the partial sums also converge almost everywhere, and that there are functions in \(L^{4/3,r}[-1,1]\) with \(r>1\) such that \(S_ n f(x)\) is unbounded for every \(x\in [-1,1]\). Another problem considered in this paper is the convergence of Lagrange interpolation at the zeros of Legendre polynomials. If \(f\) is continuous on \([-1,1]\) then the Lagrange interpolation polynomial converges in \(L^ p[-1,1]\) for every \(p<4\) but this is no longer true for \(p=4\). The author shows that for \(p=4\) the Lagrange interpolation polynomials converge in the norm of the Lorentz space \(L^{4,\infty}[-1,1]\), but there exist continuous functions \(f\) on \([-1,1]\) such that the \(L^{4,3}\)-norm of the Lagrange interpolation polynomials is unbounded.