Pointwise and norm convergence of a class of biorthogonal expansions

Abstract:

Let $\{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1$, be sequences of functions in ${L^\infty }(0,1)$, such that $({u_k},{v_j}) = {\delta _{kj}}$. Let ${\phi _k}(x) = \exp \;2k\pi ix$. It is shown that if for a given p, $1 < p < \infty$, the sequence $\{ {u_k}\}$ is complete in ${L^p}(0,1)$, and $\{ {v_k}\}$ is complete in ${L^q}(0,1),pq = p + q$, and if the ${u_k}$’s, ${v_j}$’s are asymptotically related to the ${\phi _k}$’s, in a sense to be made precise, then $\{ {u_k}\}$ is a basis for ${L^p}(0,1)$, equivalent to the basis $\{ {\phi _k}\}$, and for every f in ${L^p}(0,1)$ a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.