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Old 2005-03-25, 02:38   #4
ewmayer
2ω=0
 
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Sep 2002
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Quote:
Originally Posted by Crook
Last question: does anybody know if new progresses were made in defining the necessary conditions for a function to be defined with a Fourier series?
Not sure I understand you correctly - the classic Fourier series is an example of approximation in a Hilbert space (L^2), and as such, any function defined in L^2 (i.e. with finite L^2 norm on the interval of definition) can be approximated to any desired degree of accuracy via Fourier series, though the convergence properties of the series depend on the smoothness of the function - specifically, for a function that is C^k (continuous derivatives up to kth order) the coefficients (typically written as a_n and b_n, where one set is the coefficients of the cosine terms and the other set is the coefficients of the sine terms) of the series approximation decay proportionally to 1/n^(k+2). Also note (and this is very important and frequently misunderstood) that when we say the FS approximation "converges to the function" we mean CONVERGENCE IN THE SENSE OF THE L^2 NORM, i.e. as n->oo, the norm of the difference between the function and the FS approximation vanishes. That need not not imply pointwise convergence: a classic example is the FS approximation to a step function (which is discontinuous, i.e. only C^0 smooth), where the FS exhibits oscillations around the discontinuity ("Gibbs' phenomenon") whose amplitude does not go to zero as the number of terms in the FS approximation goes to infinity - rather the width of the wiggles gets increasingly smaller, such that their L^2 integral vanishes as n->oo.
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