Abstract
We begin the present section with some simple definitions (probably already known to most readers). For m, n integers the so-called Kronecker delta δ mn is defined by δ mn = 1 if m = n and δ mn = 0 if m ≠ n. For our second definition, let (f n : n = 0, ±1, ±2,...) be a set of real or complex functions, defined on the subset Δ of ℝk. The set (f n ) is called an orthonormal set (or orthonormal system) on Δ if
is the complex conjugate of f n and dx stands for dx1… dx k Of course, the definition makes sense only if the integral of f m
exists for all m, n. The definition is analogous if m and n are restricted to 0,1,2,… or to 1,2,… . If it is only given that
then (fn) is said to be an orthogonal system on Δ. We immediately mention an example. For n = 0, ±1, ±2,…, let e n (x) = (2π)-1/2einx on ℝ. The system (e n : n = 0, ±1, ±2,…) is orthonormal on any interval [a, a + 2π], i.e., on any interval of length 2π in ℝ. The proof is immediate by observing that
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© 1989 Springer-Verlag Berlin Heidelberg
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Zaanen, A.C. (1989). Fourier Series of Continuous Functions. In: Continuity, Integration and Fourier Theory. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73885-2_3
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DOI: https://doi.org/10.1007/978-3-642-73885-2_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50017-9
Online ISBN: 978-3-642-73885-2
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