Analysis and Filtering
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In this chapter we present, primarily, the analysis of the Gibbs phenomenon in the Fourier series approximation of (periodic) functions with jump discontinuities. The parallel analysis of the Gibbs phenomenon in the truncated Fourier integral representation was, for all practical purposes, covered in Section 1.3. Still, we start with a brief summary of the latter section for the sake of completeness as we draw on the parallels with its analysis. This is followed by discussion and illustration in Section 2.3 of the two basic methods of eliminating or reducing the Gibbs phenomenon; namely, the Fejer averaging and the Lanczos’ local averaging methods. A review of the foundations of these methods were laid in Sections 1.4 and 1.5, respectively. In Sections 2.4 and 2.5, we cover recent transform methods and some optimization-type methods of filtering the Gibbs phenomenon. Section 2.6 covers a rather different aspect, which is the possibility of using the Gibbs phenomenon to an advantage, such as in edge detection for locating shock waves, or in scanning the muscles of the heart. In Section 2.7 we present a close to complete but brief historical account of the Gibbs-Wilbraham phenomenon in the Fourier analysis, and some orthogonal expansions of functions with jump discontinuities.
KeywordsFourier Series Jump Discontinuity Gibbs Phenomenon Truncate Fourier Series Regularization Error
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Additional References (From Appendix A.)