if a trigonometric series uniformly converge to a function, is it the Fourier series of the function? I understand the Uniqueness of Fourier Series, but that one is saying if I have 2 continuous functions and they have the same Fourier Coefficients, then these 2 functions are the same. But is it related to my question?

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I have seen an an example of a series which pointwisely converge at every point but it is not a fourier series of any integrable function. I don’t know whether it would happen for the uniformly convergence.

– Haiyi Tan

2 days ago

Let pnp_n be the nn-th partial sum of the series, and ff the limit function. Consider the sequence ∫2π0pn(t)cos(kt)dt\int_0^{2\pi} p_n(t)\cos (kt)\,dt.

– Daniel Fischer♦

2 days ago

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