# Questions tagged [fourier-series]

A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

5,709 questions
Filter by
Sorted by
Tagged with
16 views

### Finding out the number of minima for a fourier expansion

Suppose I have a Fourier series f(x) = $\sum_{n=1}^N t_n \cos(nx)$ defined in the domain $(-\pi,\pi]$. we need to prove that mathematically we can ' at most' have N minima points excluding the ...
56 views
+50

### Is there a compact form of $\frac{2}{n+1}\sum_{k=1}^n\frac{\sin^2(\pi jk/(n+1))}{z-2\cos(\pi k/(n+1))}$ in terms of $z$, $n$, $j$?

I would like to evaluate the following sum: $$S_j(z,n)=\frac{2}{n+1}\sum_{k=1}^n\frac{\sin ^2\left(\frac{\pi j k}{n+1}\right)}{z-2 \cos \left(\frac{\pi k}{n+1}\right)}$$ where $z\in\mathbb{C}$ is an ...
• 123
73 views

49 views

### Rate of Uniform Convergence of Fourier Series to a Smooth Function?

I'm wondering if there are any known results on the rate of uniform convergence of a Fourier partial sum to a smooth function ?. More specifically, I am wondering ...
16 views

### FFT for the Estimation of Power Spectra (Welch's Method) - DFT Definition

I was reading Peter Welch's famous paper "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Aver. aging Over Short, Modified Periodograms" the ...
88 views

### If we know the even fourier series are we able to find the odd version?

If we have a sequence, call it $a_{n}$ where $n>0$, and we take it as the fourier coeficents of an even function... $$F_{e}(x) = \sum^{\infty}_{n=1}a_{n}\cos(2\pi nx)$$ ... and we know the form ...
• 718
1 vote
85 views

### Difficulty in proof of a lemma in Katznelson's book about Harmonic Analysis chapt. 2 section 3 (divergence sets)

To explain my problem I must insert more from Katznelson's book than the part where I have a difficulty. (My comments to these copies in red.) Beginning of book quote End of book quote In the remark ...
28 views

### Series expansion involving integrals of higher derivatives and Bernoulli polynomials

Given a (smooth) function f defined on $[0, 1]$, I am looking for a series expansion of the form $$f(x) = \sum_{n = 0}^\infty \frac{c_n}{n!} P_n(x),$$ where $c_n = \int_{0}^{1} f^{(n)}(t) dt$, where ...
• 343
41 views

### Nonhomogeneus PDE function requires expansion in sines?

I'm studying about the solution to the PDE: $$\Delta u(x,y)=-f(x,y)\\ u(0,y)=u(a,y)=0 \\ u(x,0)=g(x) \ \ , \ \ u(x,b)=h(x)$$ And the first step is to start solving it like a homogeneus equation with ...
33 views

### How to find best fitting trend that fits this periodic data?

I have this data only for this month and I need to find a good trend function. I used a polynomial of degree 6 but with that you can't really extrapolate. I heard you should use Fourier series? It is ...
1 vote
36 views

### Is there uncertainty principle for Fourier series?

I know there exists many types of uncertainty principle for Fourier transform. I tried to search but I couldn't find any such principle for Fourier series
• 59