# Uniform Convergence of Fourier sine and cosine series

• hwill205
In summary, the function's Fourier sine and cosine series converge to the odd extension on 1≤x≤1 and to the same manner as the regular Fourier series on 0≤x≤1. However, since no sequence of continuous functions can converge uniformly to a discontinuous function, the convergence of the Fourier series will not be uniform. This holds true for the entire real line -∞<x<∞ and the definition of uniform convergence remains the same.
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<∞

## 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?

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

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

hwill205 said:
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.

## What is the definition of uniform convergence of Fourier sine and cosine series?

Uniform convergence of Fourier sine and cosine series is a mathematical concept that describes the behavior of a sequence of functions. It means that the series of functions approaches a limiting function uniformly, meaning that the difference between the value of the function and the limiting function is small for all values of the independent variable.

## What is the significance of uniform convergence of Fourier sine and cosine series?

Uniform convergence is important because it guarantees that the series of functions will converge to the expected limiting function, regardless of the order in which the terms are added. This allows for easier manipulation and analysis of the series, as well as the ability to extend the series to other intervals.

## How is uniform convergence of Fourier sine and cosine series different from pointwise convergence?

The main difference between uniform convergence and pointwise convergence is that in uniform convergence, the convergence occurs at the same rate for all points in the interval, while in pointwise convergence, the convergence may occur at different rates for different points in the interval. In other words, in uniform convergence, the difference between the value of the function and the limiting function is small for all values of the independent variable, while in pointwise convergence, the difference may be large for some values of the independent variable.

## What is the Cauchy criterion for uniform convergence of Fourier sine and cosine series?

The Cauchy criterion states that a series of functions is uniformly convergent if for any positive number ε, there exists a positive integer N such that for all n > N, the difference between the nth partial sum of the series and the limiting function is less than ε for all values of the independent variable.

## How is uniform convergence of Fourier sine and cosine series tested?

To test for uniform convergence of Fourier sine and cosine series, one can use the Weierstrass M-test. This test states that if there exists a sequence of positive numbers M_n such that the absolute value of each term in the series is less than or equal to M_n, and if the series of M_n converges, then the series of functions is uniformly convergent.

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