Basic Fourier type systems in L₂ spaces of even-dimensional vector functions

Abstract

This chapter is similar to Chapter 7. Here we pass from the interpolation Theorem 8.2-1 (the case p = 2), relating to the classes $W_{s,\sigma }^{2,\omega }$ (s ≥ 1) of entire functions, to theorems on the basis property of some systems of even-dimensional vector functions which are biorthogonal in a Hilbert space $L_2^{2s}$ (0, σ)of 2s-dimensional vector functions defined on (0, σ). First we construct the mentioned biorthogonal systems. Then, using Theorem 8.2-1 and Theorem 2.4-2 on parametric representations of the classes $W_{s,\sigma }^{2,\omega }$ (s ≥ 1) we establish the completeness and the basis property in the Riesz sense of these systems in the space $L_2^{2s}$ (0, σ) Note that a reformulation of the results of this chapter leads later in Chapter 12 to an explicit and complete apparatus of Fourier type systems of entire functions. These systems prove to be bases of the weighted space L₂ considered over the sum of 2s (s ≥ 1) segments of the same length having a common endpoint at the origin and forming equal angles of the opening π/s in the complex plane.