Bases of exponential functions in \(L^ p\)-spaces
The author considers systems of exponential functions, \(e(\Lambda)=\{e^{i\lambda_ nx},xe^{i\lambda_ nx},...,x^{m_ n- 1}e^{i\lambda_ nx}\},\) where \(\Lambda =(\lambda_ n)\) is the sequence of zeros of an entire function L(z), ordered in nondecreasing absolute values, and \(m_ n\) are the corresponding multiplicities. Continuing some earlier work of the author and others in the \(L^ 2\)- case, he treats here the \(L^ p\)-case, \(1<p<\infty.\) Sample result: If the \(\lambda_ n\) are separated and lie in some horizontal strip \(y\leq h\), and if off that strip we have \(| L(z)|(1+| z|)^{-b}\sim \exp(a| y|),\) then \(e(\Lambda)\) is a basis in \(L^ p(-a,a)\) provided \(-1/p<b<1-1/p\). These bounds for b are shown to be best.
bases of exponential functions in \(L^ p\)-spaces
scientific article; zbMATH DE number 3863846