Fourier Series and Right Invertible Operators

Abstract

In several papers [S, 9, 101 M. TASCHE has applied the algebraic theory of right invertible operators [ . ; I to problems of interpolat.ion and approximation. In a recent paper [2] we used the method of parametric extension to construct right invertible operators in spaces of bivariate continuous functions. right invertible operator in the linear space C(J2) of bivariate continuous functions on the square JZ (J=[O, %I) which is related to the expansion of bivariate functions into double FOURIER series. In particuIar, we will derive a decomposition formula for smooth functions in C'J(J2) which yields as D special application the absolute convergence of the double FOURIER series of (t periodic function in C'.'(J2).It is the objective of this paper to construct Constriiction of right invertible operatorsIn this section we will describe a method for constructing right invertible operators in a real or complex linear space X (see also [2], section 3). Assume that P nncl R are linear operators on X such that (1.1) ker (R) = (0) , (1.2) p2 = P , (1.3) P R = O . The subset X, defined by is a linear subspace of X. For any f EX, condition (1.1) guarantees the existence of a unique Bf in Y such that (1.4) The linearity of P, R, XI and (1.1) imply that B is a linear operator in X with X1={fcX : f-Pfcim (R)} f = P ( f ) +R(Bf) * dom @)=XI. 7.