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
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
This survey paper was partially supported by the Hungarian National Foundation for Scientific Research under Grants TS 044 782 and T 046 192.
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Móicz, F. (2006). Constructive Function Theory: I. Orthogonal Series. In: Horváth, J. (eds) A Panorama of Hungarian Mathematics in the Twentieth Century I. Bolyai Society Mathematical Studies, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30721-1_2
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