- #1

hwill205

- 11

- 0

## 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?