The Method of Boundary Value Problems in the Study of the Basis Properties of Perturbed System of Exponents in Banach Function Spaces

This paper considers the method of Riemann boundary value problems of the theory of analytic functions to study basis properties of perturbed systems of exponents in rearrangement invariant Banach function spaces. This method is demonstrated by the example of a system of exponents with a linear phase, depending on a complex parameter. The study of the basis properties of this system has a deep history starting with the classical results of Paley–Wiener and Levinson. The basis properties of this system in Lebesgue spaces were finally established in the works of Sedletskii and Moiseev. As special cases of rearrangement invariant Banach function spaces (r.i.s. for short), we can mention Lebesgue, Orlicz, Lorentz, Lorentz–Orlicz, Marcinkiewicz, grand-Lebesgue and other classical and non-standard spaces. A subspace of r.i.s. is considered where continuous functions are dense and the conditions on phase parameter are found, depending on the Boyd indices, which are sufficient for basicity of the system of exponents for this subspace. In the special case, where the Boyd indices coincide with each other, these conditions become a basicity criterion. A new proof method, different from the previously known ones, is proposed. In particular, previously known results are obtained for some Banach function spaces.