# Does Non-Uniform Convergence of Fourier Series Imply Discontinuity?

• torquerotates
In summary, the professor's statement can be understood as follows: If the Fourier series of a function f converges uniformly, then the sum of the series must be continuous. However, if the sum is not continuous, then the convergence cannot be uniform. This is due to the fact that even if f is not continuous, each individual term and all partial sums of its Fourier series are continuous, but the limit of the sum is not, therefore the convergence is not uniform.
torquerotates
I don't understand what my professor told me. He said that if a fourieer series of a function f is not continuous then it doesn't converge uniformly to f.

But the given theorem only states that, "if a sequence of functions is continuous and converges uniformly to f, then f is continuous."

I'm given that f is not continuous. and that the fourieer series of f is not continuous. How does this mean that the fourieer series doesn't converge uniformly to f?

This is just a application of logic here

A&B=> C

not C=> (not A) or (not B)

knowing (not A) doesn't really give me (not B)

Even if the function f is not continuous, each individual term and all partial sums of its Fourier series are still continuous (all terms are just sines and cosines, the sum of any finite number of sines and cosines is continuous everywhere).

The sequence of partial sums is continuous, but the limit is not - therefore, convergence is not uniform.

A & not C => not B

Oh i see that makes much more sense. So the sequence of functions are continuous even though the fourieer series is not?

Heres what my professor said, "If the FS series converges uniformly, then the FS must have been continuous (since the uniform limit of a sequence of continuous functions is continuous). Since the FS is not continuous , we conclude that the converge is not uniform."

I don't quite understand this statement.

torquerotates said:
Oh i see that makes much more sense. So the sequence of functions are continuous even though the fourieer series is not?

Heres what my professor said, "If the FS series converges uniformly, then the FS must have been continuous (since the uniform limit of a sequence of continuous functions is continuous). Since the FS is not continuous , we conclude that the converge is not uniform."

I don't quite understand this statement.

If f(x) has Fourier series

$$f(x)=\frac{a_0}{2}+\sum_{n=1} ^{\infty}a_n cos(\frac{n \pi x}{L})+b_n sin(\frac{n \pi x}{L})$$

f(x) is continuous everywhere for all x except at points x0.

So for all x≠x0, the Fourier series converges to f(x).

at x=x0, the series converges to the mean of the left and right limits i.e.

$$f(x)=\frac{1}{2}(\lim_{x \rightarrow x_0^-} f(x) +\lim_{x \rightarrow x_0^+} f(x))$$

This usually occurs when you have piecewise functions.

torquerotates said:
Oh i see that makes much more sense. So the sequence of functions are continuous even though the fourieer series is not?

Heres what my professor said, "If the FS series converges uniformly, then the FS must have been continuous (since the uniform limit of a sequence of continuous functions is continuous). Since the FS is not continuous , we conclude that the converge is not uniform."

I don't quite understand this statement.

I think that when your professor talks about "the FS" he means "the sum of the FS". If the convergence of the FS had been uniform, the sum would be a continuous function. Since the sum isn't, the convergence couldn't have been uniform. Is that where your confusion lies?

## What is the definition of convergence of Fourier series?

The convergence of Fourier series refers to the property that the partial sums of the Fourier series of a function approach the function itself as the number of terms in the series increases. In other words, the Fourier series converges to the original function.

## How is the convergence of Fourier series determined?

The convergence of Fourier series is typically determined using the concepts of pointwise convergence and uniform convergence. Pointwise convergence means that the Fourier series converges at each individual point, while uniform convergence means that the convergence is consistent across the entire function.

## What are some common examples of Fourier series convergence?

Some common examples of Fourier series convergence include the Fourier series of continuous functions with a finite number of discontinuities, periodic functions, and smooth functions. In general, Fourier series tend to converge well for functions that are smooth and well-behaved.

## What happens if the Fourier series does not converge?

If the Fourier series does not converge, it means that the function it is trying to approximate is not well-represented by a Fourier series. This could be due to the function having too many discontinuities or being too complex for the Fourier series to accurately capture its behavior.

## How can the convergence of Fourier series be improved?

One way to improve the convergence of Fourier series is to increase the number of terms in the series. This will result in a more accurate approximation of the original function. Additionally, using techniques such as smoothing or truncation can also help improve the convergence of Fourier series for certain types of functions.

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