A basis of weighted exponentials
The present paper begins with an interesting discussion about some basic results concerning different expansions by weighted exponentials in \(L_ 2(-\infty,\infty)\). One remarks that there exist no examples of bases of weighted exponentials. In this difficult problem, the author gives a partial answer. Using certain technical lemmas he establishes the following main result: Let \(L_ 2(-\infty,\infty)\) be the Banach space of functions that are square-integrable on \((-\infty,\infty)\) with respect to the measure \(d\mu =\exp (-2\epsilon t^ 2)dt\), endowed with the corresponding square-norm. Then, for every \(\epsilon >\), the exponential sequence \(g_ k(t)=\exp (- t^ 2/2+c_ kt),\) \(k\in Z^*\), where \(c_ k=2[\pi (k-1)]^{1/2}\) for \(k=2,3,...\), \(c_ 1=1\), \(c_{-k}=c_ k\) for \(k=1,2,...\), is a basis of \(L_ 2(-\infty,\infty)\) considered as a subspace of the Banach space \(L_ 2(-\infty,\infty)\).
different expansions by weighted exponentials in \(L_ 2(-\infty ,\infty )\)
scientific article; zbMATH DE number 4136752
scientific article