FOURIER ANALYSIS: LECTURE 6

FOURIER analysis : LECTURE Convergence of FOURIER seriesFourier series (real or complex) are very good ways of approximating functions in a finite range, bywhich we mean that we can get a good approximation to the function by using only the first fewmodes ( truncating the sum overnafter some low valuen=N).This is how music compression works in MP3 players, or how digital images are compressed inJPEG form: we can get a good approximation to the true waveform by using only a limited numberof modes, and so all the modes below a certain amplitude are simply saw a related example of this in our approximation to using Eqn. ( ) and Table examinable:Mathematically, this translates as the FOURIER components converging to zero ,bn!0asn!1,providedf(x) is bounded ( has no divergences). But how quickly do the high ordercoe cients vanish? There are two common function and its firstp 1derivatives(f(x),f0(x).)

FOURIER ANALYSIS: LECTURE 6 2.11.1 Convergence of Fourier series Fourier series (real or complex) are very good ways of approximating functions in a finite range, by

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