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.

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15 views

What is the range on a fourier transform?

In particular, I want to know the range of the coefficients on the type-IV discrete cosine transform. Assuming no normalization factor or window is applied, what interval can I expect the coefficients ...
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1answer
18 views

Complex version of Fourier series

Let a be a positive real number and f $2\pi$ periodic function defined by: $f(x)= \begin{cases}0 & \text{if }-\pi\lt x\lt 0 \\ 1 &\text{if }0\le x\le a \\ 0 &\text{if }a\lt x\le ...
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1answer
22 views

Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$

Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $. Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| ...
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24 views

A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
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1answer
34 views

Series in hyperbolic sines.

I was looking into a problem and I arrived to something in which I want to expand some function $\varphi(x)$ in series of hyperbolic sines, something like: ...
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27 views

How to find coefficients in a power of a trig series?

Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: ...
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1answer
43 views

Is Fourier series used always for periodic signals and Fourier transform for aperiodic signals only?

I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and ...
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1answer
26 views

Obtaining $\sum_{n=1}^{\infty} a^n \cos{(n\theta)} = \frac{a \cos{\theta}-a^2}{1-2a\cos{\theta}+a^2}$

This is a homework problem. From Fourier Series and Boundary Value problems, Brown/Churchill 8th ed. I should begin with $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$, substitute with $A=n\theta$ and ...
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0answers
18 views

On converting from real to complex Fourier series

Let a real-valued function $f$ be defined as following: $$f\left ( x \right )=\left\{\begin{matrix} 2k-x, x\in\left [ 2k-1,2k \right ) & \\ x-2x , x \in \left[ 2k,2k+1\right )& ...
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2answers
30 views

Finding the particular solution of a pde.

I have solved a PDE up to the point of finding the particular solution. I am trying to find the constant $$C_n$$ I have the expression $$3x-x^2=\sum_{n=1}^{\infty} C_{n} \, \sin\left(\frac{\pi n ...
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1answer
31 views

Fourier synthesis of periodic signals

I was reading the Fourier synthesis of periodic signals But I didn't understand the sentence i.e. "Although the calculation of $a_0, a_1, b_1, a_2, b_2$, is a mathematically straightforward ...
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11 views

Proving Inverse DFT

I have trouble understanding the proof I was provided of the IDFT, here is what I have: $$ \nu_n = \frac{n}{\Delta N} \\ x(t) = \int_{-\infty}^{\infty}X(\nu)e^{i2\pi\nu_n t}d\nu \\ $$ the next step I ...
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26 views

A question on Parseval's Theorem (Fourier Transform) [on hold]

How to obtain Fourier Transform using Parseval's relation? Take for example: (sin(at))/at As this example cannot simply be solved using the actual formula for calculating fourier transform. So a ...
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1answer
32 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
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91 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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20 views

Reference on Gibbs phenomenon

I need a reference that explains the following result (also known as the Gibbs phenomenon) Let $g$ be a $2\pi$-periodic function, $C^1$ per pieces (i.e., there exists a partition $x_1 < \cdots ...
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16 views

Is there any mistake? Proof related to the Poisson summation formula.

I need to prove the following statement: Let $\varphi \in C(\Bbb R)$ with compact support. Then, $$ \Big \Vert \sum_{k\in \Bbb Z} \varphi(k) e^{ikx} \Big \Vert_{L_1(0,2\pi)} \leq C \Vert ...
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1answer
28 views

Fourier Analysis Help - $\mathcal L^2$

Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per. ...
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1answer
24 views

Finding the complex Fourier series… [on hold]

Here's is the problem: And this is my solution for part (i) of the problem. Is it correct? Can I simplify it more than I did? For part (ii), I don't know how to start. So please give me a ...
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1answer
12 views

Fourier Series Even Extension

For $f(x)=-x, 0\leq x \leq 1$, extend $f(x)$ into an even function into $-1 \leq x \leq 0$ and then regard $f(x)$ as a periodic function on $ -\infty < x < \infty $. Find the Fourier ...
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1answer
64 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
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28 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
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19 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
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2answers
25 views

Fourier series coefficient problem

I am having trouble calculating the $a_n$ coefficient for when $n=1$ for the following function. The function $f(x)$ is periodic with period 2 pi, and is defined on the interval $-\pi<x<\pi$ by ...
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0answers
24 views

Finite Fourier Transformation [closed]

I have to solve the boundary value problem described by $\frac{\partial v}{\partial t} =\frac{\partial^2 v}{\partial t^2} $ using finite Fourier Transform where, $0 < x < 6 $, $t >0$, and ...
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1answer
16 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
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1answer
69 views

If $\gamma$ is irrational, then $\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)\to \int_T f(t)\,dt$

I need to show that $$ \lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)=\int_T f(t)\,dt. $$ Here $\gamma$ is any irrational number on the real line and $f(t)$ is any continuous periodic ...
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1answer
38 views

On a property of Fourier coefficients

I need to prove the following: If $(\Phi_n)_{n\ge0}$ is an orthonormal system of integrable functions defined on some interval $[a,b]$, and $(c_n)_n$ is a sequence of reals such that $\sum c_n ...
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1answer
26 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
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An and Bn values are same in fourier series [closed]

hi friends i could't find the value of En(ie. An) using usual fourier series method. plz find the value of En.also i do not know to choose the f(x) for half range sine and cosine. please give the ...
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1answer
23 views

randomly rough surface by ifft : real output from ifft

I'm trying to generate a randomly rough isotropic surface with predefined roughness amplitudes (standard deviation of heights). Suppose I have the absolute values of fourier components of the surface ...
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1answer
31 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
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2answers
44 views

Integrate $ \int^{\pi}_{-\pi} (\pi^2-x^2)\sin nx \ dx$

Consider the function $f:(-\pi,\pi)\to\mathbb{R}$ be defined as $x \mapsto (\pi+x)(\pi-x)$ Compute the fourier series of $f$ So far, I've worked out $a_o$ by: \begin{equation} a_o = \frac{1}{\pi} ...
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1answer
14 views

fourier series notation question

Find the fourier series for the given function $$f(x)=-x \quad \text{for } -L\le x < L, f(x+2L)=f(x)$$ this is a question from my book, and im just wondering about one thing and that is what does ...
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1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
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1answer
30 views

question on Fourier Transformation

I have to find the Fourier Sine transform of $f(x)=1$ when $|x|<a$ and $f(x)=0$ when $|x|\ge a$ and hence show that $$\int_0^\infty {\sin(t)\over t} dt =\pi/2$$ and $$\int_0^\infty ...
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2answers
20 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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1answer
37 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
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3answers
31 views

Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
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1answer
28 views

Partial sum of Fourier series of square wave

Let $f$ be a $2π$ -periodic square wave function so that $$f\, = -1 \quad -π \le x<0$$ $$f=1 \qquad 0 \le x< π$$ $S_{2n-1}(x)$ is the $(2n-1)st$ Fourier polynomial of $f$. Prove ...
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1answer
30 views

Fourier series on $e^x$ periodic $[-1,1]$

I got the Fourier series as $(e-e^{-1})(\frac 1 2+\sum \limits _{n=1} ^\infty \frac {(-1)^n(\cos(n\pi x)-n\pi \sin(n\pi x)} {1+n^2\pi^2})$. Although I've seen the answer online as being $\sum \limits ...
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0answers
25 views

Fourier series of cotangent

I have found the Fourier series of $\cos(ax)$ and i get: $$\frac{ \sin(a \pi)}{a\pi}\left[ \left(\frac{1}{2a^2}\right)- \sum_1^\infty \frac{(-1)^n \cos(nx)}{n^2-a^2}\right]$$ How can I deduce the ...
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1answer
35 views

Problem with Fourier series

I'm trying to find the Fourier series of the function defined on the interval $(-2,2)$ $$ f(x)=\begin{cases} 0,& \,\,\, |x| <1 \\ 1, & \,\,\, 1<|x|<2 \end{cases} $$ This should be ...
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11 views

How to choose $f\in C_{c}^{\infty}(\mathbb R)$ so that $ \hat{g}\in \ell^{1}(\mathbb Z)$, where $g(x)=f(x+2\pi)$?

Suppose $K$ is compact proper subset of $[0, 2\pi]$ with the property $K\subset V \subset [0, 2\pi]$ where $V$ is open . My Question: Is it possible to choose $f\in C_{c}^{\infty}(\mathbb R)$ such ...
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0answers
26 views

Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
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1answer
37 views

Find the Fourier Series of the following function.

I have been given the following question. "The function $f(x)$ is odd, has a period $2\pi$ and satisfies: $$f(x)=\begin{cases} 1 & 0\lt x \lt \pi \\ -1 & \pi \lt x \lt 2\pi \end{cases}$$ ...
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0answers
42 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
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0answers
20 views

Mode of convergence for partial Fourier series in $B( L_p[-\pi; \pi ])$, $p \in [1; \infty]$

Which mode of convergence takes place, strong, weak, or in norm? If we have sequence of continuous linear operators in $L_p[-\pi; \pi]$: $(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) ...
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1answer
35 views

Convergent series of a functional series

Is there any way to prove that the functional series $$g(t)=\sum\limits_{n=1}^{\infty} b_n\,n\,a\sin(n\,c\,t)$$ is uniform convergent, given $$b_n=\int_{-\pi}^{\pi} f(x)\,\sin(nx)dx$$ $f\in ...
1
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1answer
32 views

Fourier series of $f(x)=x^2+x$ ,$x\in(-\pi,\pi)$

Could you please help me solve this problem: I need use Fourier series of $f(x)=x^2+x$ ,$x\in(-\pi,\pi)$ to prove that $\sum_{n\ge1} \frac 1{n^2}= \frac{\pi^2}6$. I calculated the Fourier series: ...