A biorthogonal expansion related to the zeros of an entire function

Abstract:

Let $D(\lambda ) = {a_0} + \Sigma _{k = 1}^\infty {a_k}{e^{\lambda /k}}$, where the ${a_k}$’s are given complex constants, with ${a_0}{a_1} \ne 0,\Sigma _{k = 1}^\infty |{a_k}| < \infty$. Let $\{ {\lambda _k}\} ,k = 1,2, \cdots$, denote the sequence of distinct zeros of $D(\lambda )$, labeled in order of increasing modulus, and with multiplicities ${m_k} \geq 1$. Let $\{ {\phi _l}(x)\}$ denote the sequence of functions $\{ {x^j}\exp ({\lambda _k}x):j = 0, \cdots ,{m_k} - 1;k = 1,2, \cdots ;0 < x \leq 1\}$. We show that for each $p.1 \leq p < \infty$, there is a sequence $\{ {\psi _l}(x)\}$ in ${L^q}(0,1)\;(pq = p + q)$ such that $({\phi _l},{\psi _m}) = {\delta _{lm}}$. Then we show that $\{ {\phi _l}\}$ is complete in ${L^p}(0,1),1 \leq p < \infty$, and for $1 < p < \infty$, we find a subspace of ${L^p}(0,1)$ such that the biorthogonal expansion $f = \Sigma _{k = 1}^\infty (f,{\psi _k}){\phi _k}$ is valid in the norm of ${L^p}(0,1)$.