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Convergence of Fourier series
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1. Let f be a continuous function on the circle. Then the Fourier series of f converges uniformly.
2. Let f be an integrable function. Then its Fourier coefficients goes to zero when n go to infinity.
3. If the derivative of f exists and is integrable, then the Fourier coefficients of f is o(1/n).
4. Let f be a differentiable function on the circle. Then the Fourier series of f converges everywhere to f.
5. If f is continuously differentiable on the circle, then its Fourier series converges uniformly.
6. If the Fourier series converges, then the Cesaro means converges.
7. If the Cesaro means converges, then the Fourier series converges.
8. The Cesaro mean of a continuous function converges uniformly
9. The Fourier series of a continuous function always converges pointwise.
10. The Fourier series of an integrable function converges in L2 to the function.
11. The Cesaro mean of an integrable function converges in L1 to the function.
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