# Tagged Questions

Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

129 views

### Completeness condition for periodic function

I know that for a real-valued function set $\{f_n(x)\}$, its completeness condition is $\Sigma_n f_n(x)=\delta(x-x')$. That is, this condition guarantees that a well-behaved function can be write as a ...
68 views

### Why does the Fourier sine series of $x^2$ on $[0,l]$ converge to 0?

When expanding, for example, $x^2$ on $[0,l]$ as a sine series, we get $f(x) = \sum_1^{\infty}b_n sin(\frac{n\pi x}{l})$ If we plug in $x=l$ to this expansion, we get $f(x)=0$. Why aren't we getting ...
108 views

### Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
161 views

### Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
55 views

### Fourier Transform, Laplace Transform, but what about…

I have a question regarding the fourier and laplace transform. First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of ...
66 views

49 views

### Discrete Fourier Series

I have a series of discrete values that are periodic and I am looking to calculate the Fourier series of it. I learnt all of this in college but I can't for the life of me remember now. The discrete ...
40 views

### characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
125 views

### Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
413 views

### Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
69 views

### What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
68 views

### Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$\sum_{n \geq 1} \dfrac{\sin(n\theta)}{n}$$ converges for all $\theta \in \mathbb{R}$?
177 views

### How to compute Fourier coefficients using a cubic spline-corrected FFT?

I'm not particularly experienced in numerical analysis, and so I recently had quite a massive shock when I discovered that sampling a smooth function and computing the FFT of the result does not ...
66 views

### Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
91 views

### When do the sine components of a Fourier series vanish?

A Fourier series is given by: $$s_N(x) = \sum c_n \cdot e^{i \frac{2\pi n x}{P}}$$ With Euler's identity, the exponential can be converted to a sums of sines and cosines. When do the sine ...
410 views

### Why is $\sum_{n=-\infty}^{\infty}\exp(-(x+n)^2)$ “almost” constant?

I did some numerical approximation of $$\sum_{n=-\infty}^\infty \exp(-(x+n)^2)$$ and found that this function is "almost" constant ($\approx 1.772$). Why does the sum fluctuate little? Is there a ...
59 views

### What is the Fourier series of $\frac1T\sum^{\infty}_{m=-\infty}\delta(f-\frac mT)$?

As the title mentioned, I've not known exactly about Fourier series and when I was reading an digital communication textbook, I wondered about below equation derivation of Fourier series like ...
120 views

### a question how to prove:$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos(nx)}\over {n}}=\ln(2\cos(x/2))$

I found a complicated question in my textbook, I can't solve it? How to prove $$\sum_{n=1}^{\infty}{{(-1)}^{n-1}{\cos nx}\over {n}}=\ln(2\cos(x/2))$$ where $x\in(-\pi,\pi)$. My tried method: I tried ...
104 views

### Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
55 views

### A problem concerning finite number of Fourier coefficients

Is there a smooth, non-zero $2\pi$-periodic function $f,$ with support of $f$ contained in an interval $[a,b]\subset[0,2\pi],$ such that $b-a<2\pi$ and only finitely many Fourier coefficients of ...
61 views

### Prove the uniform convergence of a Fourier series

Suppose that $f$ is a $2\pi$-periodic function that satisfies the estimate $$|f(x)-f(y)|\leq M|x-y|^\alpha$$ for an $0<\alpha<1,$ and let ...
89 views

### Show $\lim_{n\to\infty} n^p f(nx) = 0$ exists in the distributional sense

Let $f\in C^\infty(\mathbb R)$ be periodic, with period $2\pi$ and have mean zero ($\int^{2\pi}_0 f(x)dx =0$). Show that for any positive integer $p$ the following limit is valid in the ...
40 views

### The bandwidth of the signal $x(t)$.
The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values ​​different from $0$. Very often $X(\omega)$ is different from ...