Uniform Convergence of Fourier sine and cosine series

[hwill205]

Homework Statement

f(x)= {1, ‐1/2<x≤1/2}
{0, ‐1<x≤ ‐1/2 or 1/2<x≤1}

State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞

Homework Equations

The Attempt at a Solution

Basically, my solution to this problem is that the function's Fourier sine series will converge to the odd extension on 1≤x≤1 where it is continuous and the average of the limits where the odd extension has a jump discontinuity. Since we only have to consider the half interval, 0≤x≤1, and the odd extension is the same as f(x) for this interval; the Fourier sine series will converge in the same manner as the regular Fourier series (which converges pointwise, but not uniformly).

You can make a similar argument for the Fourier cosine series.

Does this appear to be correct? Also, does the condition about it being true for the entire real line ‐∞<x<∞ make a difference for the answer?

 

[Dick]

No sequence of continuous functions can converge uniformly to a discontinuous function. Can it? What the definition of uniform convergence?

 

[hwill205]

Yes I understand. But if f(x) was continuous in the interval, would my explanation make sense?

 

[Dick]

Yes I understand. But if f(x) was continuous in the interval, would my explanation make sense?

I'm having a hard time making out what your explanation is actually saying. f(x) is even. There are only going to be cosine terms in the Fourier expansion. If you are saying that the convergence won't be uniform because of the discontinuities, I'd agree with that.