A class of Riesz-Fischer sequences

Abstract:

It is proved that if $\{ {\lambda _n}\}$ is a sequence of real numbers whose differences are nondecreasing and satisfy $\sum {{{({\lambda _{k + 1}} - {\lambda _k})}^{ - 2}} < \infty }$, then the set of complex exponentials $\{ {e^{i{\lambda _n}x}}\}$ is a Riesz-Fischer sequence in ${L_2}[ - A,A]$ for every $A > 0$, which is to say that for any positive A, the equations $\smallint _{ - A}^Af(x){e^{i{\lambda _n}x}}dx = {c_n}$ admit a solution f in ${L_2}[ - A,A]$ for every sequence $\{ {c_n}\}$ in ${\ell _2}$. In particular, if ${\lambda _n} = {n^p}$, then $\{ {e^{i{\lambda _n}x}}\}$ is a Riesz-Fischer sequence when $p > \frac {1}{2}$.