The hilbert inequality for a one-parameter family of functions

Abstract

etc. Particular systems of this kind were considered by numerous authors (e.g., see [1-5]), and it always turned out that the set of values of c~ for which the system is a basis (or a conditional basis) in L2(0, 1) is an open set. Since the investigation of the unconditional basis property of a system in L2 can be reduced to the analysis of the Hilbert and Bessel properties and the minimality for a normalized system in L2, we naturally encounter the following problem: what is the set of values of c~ for which the system {u~(x,c~)}~__t has the nilbert and Bessel properties in L2(a,b) [here II" II is the norm in L2(a,b)]? Recall that a system {e~}~__l in a Hilbert space H is called a Hilbert system if there exists a 7 > 0 such that the Hilbert inequality