:Briefly, Zygmund's proof starts by proving the statement for continuous periodic functions.
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However, F is a continuous periodic function, so by the previous special case F is uniformly zero.
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With \ displaystyle A ( t ) a piecewise continuous periodic function with period T and defines the state of the stability of solutions.
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Showed that for any set of measure 0 there is a continuous periodic function whose Fourier series diverges at all points of the set ( and possibly elsewhere ).
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Finally the fact that the sequence has a dense algebraic span, in the " inner product norm ", follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on [-\ pi, \ pi ] with the uniform norm.
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