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

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244 views

Lie algebra of the bounded continuous functions

I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the ...
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0answers
7 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 ...
2
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1answer
40 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 ...
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1answer
104 views
+50

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 ...
4
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1answer
75 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
3
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1answer
103 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
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0answers
42 views

Are there any new research results on approximating Riemann $\Xi(z)$ by a Fourier transformation

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
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0answers
26 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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0answers
25 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 ...
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0answers
37 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
3
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1answer
150 views

Fourier series of f(x)

I want to find the Fourier series of $f(x)$ defined by $f(x)=\begin{cases} 1 , -L\le x<0\\ 0, 0\le x<L. \end{cases} $ Well, to find $a_0$ I do this integral: $$a_0=1/L \int _{-L}^0 dx +1/L ...
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1answer
36 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 ...
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1answer
18 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 ...
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2answers
4k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
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0answers
29 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 ...
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0answers
63 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 ...
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3answers
32 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 ...
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2answers
60 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 $\theta \in \mathbb{R}$?
3
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1answer
30 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 ...
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1answer
57 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. ...
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1answer
37 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 ...
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3answers
204 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 ...
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1answer
203 views

Fourier series of square wave with DC component (mean component) - amplitude question

Do I subtract the DC component (mean value) from the amplitude in my sine terms? $f(t)=\left\{ \begin{array}{l l} 0 & \quad -5\le t\leq 0\\ 1 & \quad 0< t\leq 5 \end{array} ...
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0answers
38 views

Monotonically decreasing Fourier transform

What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both ...
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1answer
58 views

Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
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1answer
54 views

Find the first four terms in the Fourier series for a solution of the wave equation

The question is to find the first four terms in the Fourier series for $u(x,t), t>0$. It is for a plucked string of length L, has zero initial displacement (I.e. $u(x,0)=0, 0<x<L$) and ...
4
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4answers
81 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 ...
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1answer
35 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 ...
2
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1answer
35 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 ...
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1answer
36 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 ...
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1answer
30 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 ...
4
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1answer
68 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 ...
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1answer
28 views

Convolution theorem for product of functions

I like to Fourier transform the following product of functions: $$g(\vec{r})f(\vec{r}).$$ So I like to calculate the following: $$\int g(\vec{r})f(\vec{r}) e^{-i\vec{k}\cdot\vec{r}}d^3r.$$ ...
19
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1answer
525 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
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0answers
20 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
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4answers
188 views

Evaluate $\int_{-\pi}^\pi \big|\sum^\infty_{n=1} \frac{1}{2^n} e^{inx}\big|^2 \operatorname d\!x$

I am trying to solve exercises for the coming exam, and I am stuck on this exercise: Evaluate $$\int_{-\pi}^\pi \Big|\sum^\infty_{n=1} \frac{1}{2^n} \mathrm{e}^{inx}\,\Big|^2 \operatorname d\!x$$ ...
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0answers
13 views

Fourier series solution

Show that Summation (1/n^2) = pi^2 /6. Using fourier series. This is the question in BS grewal for fourier series. I dont understand how to apply fourier series without a limit. (-pi to pi ) or to 2pi ...
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1answer
52 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
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1answer
28 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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1answer
87 views

Fourier Series $\sin(\sin(x))$

Can anyone find the Fourier Series of $ \sin(\sin(x))$? I have tried evaluating the integrals to determine the coefficients of each of the coefficients of the sine waves, but have no idea where to ...
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1answer
18 views

How to visualize projection of a function onto fourier basis?

I wonder if there are any notes on how one would visualize a projection of function f(x) onto cos(x) and sin(x) in the same way that you would for two vectors. Is there a picture, or a figure ...
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1answer
23 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 ...
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1answer
26 views

Show solution to ODE's fourier series is a series of sines only

This question was given in an exam in applied mathematics, on the subject of Fourier series: Observe the following ODE: $u\left ( x \right) ^{\prime \prime}+Q \left ( x \right) u\left ( x \right) ...
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1answer
23 views

Shortcut to sine series using regular expansion?

If we're given the Fourier series of $e^x$ on the interval $(0,2\pi)$, I'm wondering if there's a nicer way to extract the sine series of $e^x$ on the same interval other than getting the coefficients ...
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1answer
40 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
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1answer
75 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
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1answer
24 views

Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
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0answers
39 views

Real-valued Fourier series representation

I have got stuck on the following task: Find the value of the series $${4\over \pi^2}\sum_{k=1}^\infty {1\over k^2}-{1\over \pi^2}\sum_{k=1}^\infty{(-1)^k\over k^2}$$ using real-valued Fourier ...
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0answers
81 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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1answer
32 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...