For every $p > (1 + \sqrt{ 5})/2$ we construct a uniformly discrete real sequence $\{ \lambda_n\}_{ n=1}^\infty$ satisfying $|\lambda_n| = n + o(1)$, a function $g \in L^p(\mathbb{ R})$, and continuous linear functionals $\{ g^*_n\}_{ n=1}^\infty$ on $L^p(\mathbb{ R})$, such that every $f \in L^p(\mathbb{ R})$ admits a series expansion \[ f(x) = \sum_{ n=1}^{ \infty} g_n^*(f) g(x-\lambda_n) \] convergent in the $L^p(\mathbb{ R})$ norm. We moreover show that $g$ can be chosen nonnegative.
Categories: math.CA math.FA