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[Let us consider the family of measurable functions defined on a Lebesgue measurable subset E of finite or infinite measure of the real line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}: = \left( { - \infty ,\infty } \right) $$\end{document}. The functions may take real or complex values. The function space L2(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 L2(E) with the inner product and norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left( {f|g} \right): = \int_E {f\left( x \right)\overline {g\left( x \right)} dx} and \left\| f \right\|: = \sqrt {\left( {f|f} \right)} , $$\end{document} respectively. Then L2(E) becomes a normed linear space whose norm is derived from the inner product. We say that a sequence (fn: n=1, 2, ...) of functions in L2(E) converges in the mean to a function f in L2(E) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathop {\lim }\limits_{n \to \infty } \left\| {f_n - f} \right\| = 0. $$\end{document}]
Published: Jun 24, 2010
Keywords: Fourier Series; Divergence Theorem; Orthogonal Series; Haar System; Unconditional Convergence
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