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The Fourier Series of a Sequence of Functions
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Let fk(x)εLr, r≥1, in 0≤2π, k=1, 2... . Let an(k)→An, bn(k)→Bn, as k→∞ for each n. A necessary and sufficient condition that (An, Bn) should be the Fourier series of a function F(x)εLr which is the limit in the mean of {fk(x)} is that
lim lim x(k)n=0.
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