Talk:Dirichlet conditions

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Edit Request - Dirichlet's Theorem for 1-Dimensional Fourier Series

The section: Dirichlet's Theorem for 1-Dimensional Fourier Series, the following sentence: "The analogous statement holds..." appears right in the middle of the explanation of what the Theorem is. I am left wondering which "statement" it is referring to?? Apparently for a periodic function of any period, (even an infinite period?? doesn't this contradict the condition of boundedness??) The more I read this section the worse it appears. "For all x..." - are we supposed to know that "x" is a Real number? The Fourier Series is generally a FINITE integral, which contradicts (seemingly) the -∞ to +∞ integration in this section. "We state..." is about as pompous as you can get, but a trivial point. How does a function "oscillate" at a point?? BY DEFINITION, a function must be single valued at any point (in its domain). Why does the definition assume the given period? It either IS necessary or is NOT necessary. (Apparently it is not). Can someone either delete or move the offending sentence and possibly rewrite the entire section in a more clear fashion? e.g. "Dirichlet's theorem: If (a function [f:x ∈ R → C (?)] satisfies Dirichlet conditions, then for all x, the Fourier series, given by [insert formula here] is convergent. Where etc. etc." (talk) 05:17, 7 July 2014 (UTC)


"These three conditions are satisfied if f is a function of bounded variation over a period."

This seems to be a false statement. Here are some examples.

(1) f=0 has bounded variation, yet it does not satisfy the three conditions. Indeed, it has an infinite number of maxima and minima in each finite interval. So the thrid condition is violated.

(2) FIx a decreasing sequence of positive numbers x_i -->0. Let g be a function with g(0)=0, g(x_i) = (-1/2)^i for i=0,1,2,..., and monotone between successive x_i. Then g is BV, in fact even continuous, so the Fourier series actually converges uniformly, but g has an infinite number of maxima and minima, so again the third condition is violated.

(3) A BV function can easily have a countably infinite number of jumps, so the second condition is easy to violate. (talk) 03:32, 30 March 2016 (UTC)

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