Constructive Function Theory: I. Orthogonal Series

Abstract

Let us consider the family of measurable functions defined on a Lebesgue measurable subset E of finite or infinite measure of the real line $\mathbb{R}: = \left( { - \infty ,\infty } \right)$. The functions may take real or complex values. The function space L ² (E) consists of all measurable functions f whose squares |f|² are integrable in the Lebesgue sense. By the Schwarz inequality, f will then be integrable on the subsets of finite measure. Let us endow L ² (E) with the inner product and norm

$$ \left( {f|g} \right): = \int_E {f\left( x \right)\overline {g\left( x \right)} dx} and \left\| f \right\|: = \sqrt {\left( {f|f} \right)} , $$

respectively. Then L ² (E) becomes a normed linear space whose norm is derived from the inner product. We say that a sequence (f _n : n=1, 2, ...) of functions in L ² (E) converges in the mean to a function f in L ² (E) if

$$ \mathop {\lim }\limits_{n \to \infty } \left\| {f_n - f} \right\| = 0. $$

This survey paper was partially supported by the Hungarian National Foundation for Scientific Research under Grants TS 044 782 and T 046 192.