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RESULTS ON CONVERGENCE OF FOURIER SERIES Let f : [-, ] C be a Lebesgue integrable function. Then the Fourier coeffs of f are
 

Summary: RESULTS ON CONVERGENCE OF FOURIER SERIES
Let f : [-, ] C be a Lebesgue integrable function. Then the Fourier coeffs of f are
defined by f(n) = 1
2

-
f(x)e-inx
dx, and the partial sums of the Fourier series of f are
SN f(x) =
N
n=-N
f(n)einx
. Here are the results we have proved about the convergence of SN f
to f, ordered by decresing regularity of f:
If f Ck
(T):
1. If k 2, then |f(n)| = O( 1
n2 ) as n , which implies that SN f converges to f
uniformly on T (Chapter 2, Corollary 2.4).
2. An improvement: For k 1, SN f -f L(T) = O 1

  

Source: Alfonseca-Cubero, Maria - Department of Mathematics, North Dakota State University

 

Collections: Mathematics