Abstract
We report on the inverse problem for the truncated Fourier series representation of fx∈BV-L,L in a form with a quadratic degeneracy, revealing the existence of the Gibbs-Wilbraham phenomenon. A new distribution-theoretic proof is proposed for this phenomenon. The paper studies moreover the iterative numerical solvability and solution of this inverse problem near discontinuities of fx.
Highlights
IntroductionWith all of this, is that the Gibbs-Wilbraham effect is generic and is present for any periodic signal f(x) ∈ BV(−L, L) with isolated discontinuities
This paper reinvestigates the Fourier series [1, 2] representation g (x) = a0 2 ∞+ ∑ am m=1 cos m π L x + bm sin x, (1)of a piecewise continuous 2L –periodic signal f(x) ∈ BV(−L, L), with a0 1 L L
We report on the inverse problem for the truncated Fourier series representation of f(x) ∈ BV(−L, L) in a form with a quadratic degeneracy, revealing the existence of the Gibbs-Wilbraham phenomenon
Summary
With all of this, is that the Gibbs-Wilbraham effect is generic and is present for any periodic signal f(x) ∈ BV(−L, L) with isolated discontinuities The presence of this effect can lead to quite negative consequences when single infinite Fourier series, multiple infinite Fourier series, or even infinite wavelet series are employed to approximate signals of various dimensions, in many fields such as radio engineering and signal transmission.
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