Complex approximation in some weighted function spaces

In this thesis, we study some problems in complex approximation theory. In particular, the incompleteness of some systems of rational functions and the completeness of some systems of generalized polynomials in some weighted function spaces are considered.; Chapters 1 and 2 form the first part of this thesis. In those Chapters, we consider the weighted Hardy spaces Hp+w and Hpw (D) of functions defined on the upper half-plane {lcub} z : Im(z) > 0{rcub} and on the unit disc D, respectively, where the weight functions w are assumed to satisfy Muckenhoupt's (Ap) condition, p > 1. For two special systems of rational functions which are incomplete and biorthogonal in these weighted Hardy spaces, we study properties of the subspaces spanned by these systems, and biorthogonal expansions with respect to the systems in questions. We also pursue some connections with interpolation, moment and expansions problems in Hp+w and Hpw (D). The properties of (Ap) weights play an essential role in our study.; Two related problems are also studied: In Chapter 1, using the Fourier transform as a "bridge" between H2+ and L2(0, infinity), corresponding expansions with respect to two related systems in L 2(0, infinity) are obtained, one of them being an exponential system; while in Chapter 2, we consider the problem of approximating in Hpw (D) by polynomials and by linear combinations of a system of rational functions. In each case, we give an estimate of the degree of the approximation.; Part II of this thesis consists of Chapters 3 and 4. The problem studied in Chapter 3 is the completeness of the system of generalized polynomials {lcub} xnk {rcub} in the weighted space C0&sqbl0;0,infinity&parr0; ,1Fx of continuous functions f on the positive real axis [0, infinity) with fxF x → 0 as x → infinity, where the weight function is 1Fx with F satisfying some convexity condition. We generalize some existing results from the case where the sequence of exponents {lcub}nu k{rcub} is real to the case where it is complex. In Chapter 4, we study the completeness of the system {lcub}f(lambda nz){rcub} in C0&sqbl0;0,infinity&parr0; ,e-xs (s > 0), where f is an entire function (thus including the exponential system {lcub}elambdanz{rcub}) or an analytic function on the Riemann surface of the logarithm.