On the Muckenhoupt condition
A system \((\phi_ n)_{n\in {\mathbb{Z}}}\) is called a Riesz basis of a Hilbert space H if it is complete and for finite sequences \((c_ n)\) of complex numbers \(\| \sum c_ n\phi_ n\|^ 2\) is of the same order as \(\sum | c_ n|^ 2\). Systems of the form \((E^{i\lambda_ nx},xe^{i\lambda_ nx},...,x^{m_ n- 1}e^{i\lambda_ nx})_{n\in {\mathbb{Z}}}\), where \(\lambda_ n\) is complex, \(m_ n\) is a positive integer (which arise when solving systems of linear differential equations) are of interest for \(H=L^ 2(0,a)\). Several results are quoted which ensure such a system is a Riesz basis. The aim of the paper is to prove a converse of one of these results.
scientific article; zbMATH DE number 4050272
scientific article