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Math Help - Need to find a specific 2pi-periodic function (Fourier Analysis)

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    Need to find a specific 2pi-periodic function (Fourier Analysis)

    Hi!

    I need to find a 2pi-periodic function f:R->C, which is (Riemann) integrable on [-pi,pi] that satisfies the following properties:

    1. the Fourier series of f is uniformly Cesaro summable for all x in [-pi,pi]

    2. the Fourier series of f diverges for some x in [-pi,pi].

    I mainly tried saw-tooth functions but they did not satisfy condition 1.

    Thanks!
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    Quote Originally Posted by mgarson View Post
    Hi!

    I need to find a 2pi-periodic function f:R->C, which is (Riemann) integrable on [-pi,pi] that satisfies the following properties:

    1. the Fourier series of f is uniformly Cesaro summable for all x in [-pi,pi]

    2. the Fourier series of f diverges for some x in [-pi,pi].

    I mainly tried saw-tooth functions but they did not satisfy condition 1.

    Thanks!
    To ensure condition 1., you really need the function to be continuous, so that FejÚr's theorem applies. But then the Fourier series has a strong tendency to converge pointwise. However, it is possible for the Fourier series of such a function to diverge at some points. There is an outline of a non-constructive proof of this here, but it's probably not easy to give concrete examples. This is a difficult area, dominated by Carleson's famous proof in 1966 that the Fourier series of a continuous function converges almost everywhere.
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