Can Fourier Coefficient Bounds Under Hölder Conditions Be Improved?

In summary, the conversation discusses a mathematical problem involving a 2\pi-periodic and Riemann integrable function, f(x), that satisfies the Hölder condition of order \alpha. The participants discuss the upper limit of the Fourier coefficient, {\hat f\left( n \right)}, and how it relates to the function g(x) = \sum\limits_{k = 0}^\infty {2^{ - k\alpha } e^{i2^k x} } which also satisfies the condition. It is concluded that the upper limit cannot be improved based on the similarity of the Fourier coefficients of both functions.
  • #1
Zaare
54
0
I need help with the last part of the following problem:

Let [tex]f(x)[/tex] be a [tex]2\pi[/tex]-periodic and Riemann integrable on [tex][-\pi,\pi][/tex].

(a) Assuming [tex] f(x) [/tex] satisfies the Hölder condition of order [tex] \alpha [/tex]
[tex] \left| {f\left( {x + h} \right) - f\left( x \right)} \right| \le C\left| h \right|^\alpha ,[/tex]​
for some [tex]0 < \alpha \le 1[/tex], some [tex] C > 0 [/tex] and all [tex] x, h [/tex].
Show that
[tex] {\hat f\left( n \right)} \le O ( \frac {1} { \left| n \right| ^ \alpha} ) [/tex]​
where [tex] \hat f\left( n \right) [/tex] is the Fourier koefficient.

(b) Proove that the above result cannot be improved by showing that the function
[tex] g\left( x \right) = \sum\limits_{k = 0}^\infty {2^{ - k\alpha } e^{i2^k x} } [/tex]​
also with [tex]0 < \alpha \le 1,[/tex] satisfies
[tex] \left| {g\left( {x + h} \right) - g\left( x \right)} \right| \le C\left| h \right|^\alpha .[/tex]​

I have done (a). And I was able to show that the sum satisfies the condition. What I don't understand is:
how can I use it to show that the result in (a) cannot be improved?
I thought that the sum can be iterpreted as the Fourier series of [tex]g(x)[/tex], which would say that the Fourier coefficient of [tex]g(x)[/tex] is [tex]1/{n^\alpha}[/tex], where [tex]n=2^k[/tex]. That is the resemblence I see between (a) and (b). But I don't know what to do with that.
 
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  • #2
Ok, here's the way I reason:
In (a) I showed that the Fourier coefficient has an upper limit. Then in (b) I showed that a certain function satisfying the desired condition has a Fourier coefficient which is of the same size as the upper limit in (a). Then obviously the upper limit cannot be improved.
Is it this obvious, or am I forgetting something?
 
  • #3



In order to show that the result in (a) cannot be improved, we need to show that the function g(x) given in (b) does not satisfy a stronger condition than the Hölder condition of order α. This means that the Fourier coefficients of g(x) cannot decay faster than O(1/n^α), as shown in (a).

As you correctly noted, g(x) can be interpreted as the Fourier series of itself, with the Fourier coefficients being 1/n^α where n = 2^k. This means that the sum given in (b) is actually the Fourier series of g(x), and we can use this to compare the Fourier coefficients of g(x) with those of f(x).

Since f(x) satisfies the Hölder condition of order α, we know that its Fourier coefficients decay at least as fast as O(1/n^α). However, for g(x), we can see that the Fourier coefficients decay exactly at the rate of 1/n^α. This means that the Hölder condition of order α is the strongest possible condition that g(x) can satisfy, and it cannot be improved upon.

In other words, g(x) is an example of a function that satisfies the Hölder condition of order α, but does not satisfy any stronger condition. Therefore, the result in (a) cannot be improved, as g(x) provides a counterexample to any potential stronger condition.
 

Related to Can Fourier Coefficient Bounds Under Hölder Conditions Be Improved?

1. What is the Fourier coefficient problem?

The Fourier coefficient problem is a mathematical problem that involves finding the coefficients of a Fourier series, which is a representation of a periodic function as a sum of sine and cosine functions. These coefficients determine the amplitudes and frequencies of the sine and cosine functions that make up the Fourier series.

2. How is the Fourier coefficient problem used in science?

The Fourier coefficient problem has many applications in science, particularly in the fields of physics and engineering. It is used to analyze and model periodic phenomena, such as sound waves, electrical signals, and quantum mechanical systems. It also plays a crucial role in signal processing and data analysis.

3. What are the main challenges of solving the Fourier coefficient problem?

One of the main challenges of solving the Fourier coefficient problem is determining the appropriate range of frequencies and amplitudes to use in the Fourier series. Another challenge is dealing with discontinuities or non-periodic behavior in the function being represented, which can lead to inaccuracies in the coefficients.

4. Are there any techniques or algorithms for solving the Fourier coefficient problem?

Yes, there are several techniques and algorithms for solving the Fourier coefficient problem. One of the most common methods is the discrete Fourier transform, which uses numerical integration to approximate the coefficients. Other techniques include the fast Fourier transform, least-squares fitting, and complex analysis methods.

5. How do errors in the Fourier coefficient problem affect the overall accuracy of the Fourier series?

Errors in the Fourier coefficient problem can significantly affect the accuracy of the Fourier series. Inaccuracies in the coefficients can result in a poor representation of the original function, especially for functions with sharp changes or non-periodic behavior. Therefore, it is crucial to carefully consider the range of frequencies and amplitudes used and to use appropriate techniques and algorithms to minimize errors.

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