Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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527 views

Should I learn Fourier Analysis or Complex Analysis first?

Are the two subjects highly interrelated? Which draws more heavily from the other? Which do you recommend I learn first? Thank you.
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2answers
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How to remove high frequncies having fft?

What are the units of measure of FFT elements? Can I just set higher elements to zero to filter out higher frequencies? It looks like no. ...
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1answer
59 views

Do you need to zeropad an image of 1920*1080 to 2048*2048 when using the Cooley-Tukey FFT?

User @Paul_R wrote that you need to zeropad an image of 1920*1080 = 2^20,984 to 2048*2048 = 2^22 when using the Cooley-Tukey FFT? Why don't we just zeropad it to 2^21=2048*1024?
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52 views

How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
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1answer
57 views

How to define the Fourier transform on arbitrary Hilbert spaces?

The Fourier transform is a unitary operator from $L^2$ to $L^2$. But all infinite-dimensional Hilbert spaces are isometrically isomorphic to $L^2$. So that means we can define the Fourier transform ...
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2answers
76 views

Use Laplace transforms to solve the integral equation.

enter image description here I know I have to use convolutions but I'm not sure how. Any help is appreciated. https://i.stack.imgur.com/rFwnk.png
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1answer
34 views

get the n knowing Fourier coefficients [closed]

Given a generic Fourier series and knowing that $a = \dfrac{-2}{3}$ and $b = \dfrac{-2}{3}$ and that $n\ge 1$, how to obtain the exponential representation, i.e. complex coefficients and the power of $...
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1answer
36 views

i want to prove that the forier transform of sinc(100t) is a pulse by integration ..

I am trying to prove that the foreir transform of sinc(100t) is a pulse . I know that the forier transform of sinc is a normal pulse but am trying prove that by integration.. any help .....?
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2answers
58 views

Eigenfunction/Fourier Series Relationship

I'm having difficulty formulating a proof detailing that every eigenfunction of $D^2$ is either a constant or of the form $a$cos$(nx)$ + $b$sin($nx$) for some value $n$. I understand that the ...
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1answer
58 views

Proving $\frac{π^2}8=\sum\limits_{n=1}^\infty\frac1{(2n - 1)^2}$ using Fourier series

I found the Fourier series of $x^2$, which is $$x^2 = \frac{π^2}{3} + 4 \sum_{n = 1}^\infty \frac{(-1)^n}{n^2} \cos(nx).$$ So now, how can I prove that $$\frac{π^2}{8} = \sum_{n = 1}^\infty \frac{1}{(...
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1answer
30 views

Condition about equality between a function and his Fourier series

What are the conditions about the equality between a function and his Fourier series? In case of pointewise convergence of the series, I know that the series converges pointewise at the value $$\frac{...
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1answer
21 views

How to calculate this Fourier inverse? [closed]

$$ F(t) = (1-t^{2})^{3} \mathbf{1}_{\{|t|<1\}}(t) $$ Then $$ \frac{1}{2\pi} \int F(t) e^{-itx} dx = ? $$ Thanks!
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1answer
49 views

conjugate of fourier transform [closed]

Let $\mathscr F$ be the Fourier transform. I am wondering if $$ \overline{\mathscr F^{-1} \{e^{-i|\xi|^2} \mathscr F f }\}= \mathscr F^{-1} \{e^{+i|\xi|^2}\overline{ \mathscr F f}\} $$
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2answers
80 views

Use the Table of Fourier Series with suitable values of $x$ to evaluate the following series $\sum \frac{1}{n^2}$ [closed]

I have the answer but not the procedure. Using the Table of Fourier Series: $ \frac{L^2}{3} + \frac{4L^2}{\pi^2}\cdot∑\frac{(-1)^n}{n^2}\cdot\cos(\frac{n\cdot\pi\cdot x}{L}) ;\\ f(x) = x^2 ; -L<x&...
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1answer
59 views

Simple proof for Fourier transformation (DFT)

The question is in the image below: Image
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1answer
399 views

Fourier series and floor function

It is easy to see that the the floor function on $[0,1]$ has Fourier expansion: $$[x] = x - {1 \over 2} + {1 \over \pi }\sum\limits_{n > 0} {{{\sin (2\pi x)} \over n}} $$. Let $a$ be a positive ...
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1answer
158 views

Is the fourier transform of a function in two different variables, the same?

Is the Fourier transform of $f(x)$ the exact same as the Fourier transform of $f(t)$ when $t$ is related to $x$? For example if $t = x + 5$, would their Fourier transforms be the exact same? Isn't ...
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1answer
78 views

Fourier Transform of $f'(x)$

If the Fourier transform of the function $f(x)$ is $F(k)$, find the Fourier transform of function $f'(x) = [f(-x)]^*$ I am unsure if $f'(x)$ in this case is the derivative function of $f(x)$ or just ...
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1answer
61 views

I want to show fourier series by using matlab. help me. [closed]

I want to show basic fourier series by using matlab. I don't know where did I wrong please fix my code... this code keeps making error in matlab. ...
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1answer
37 views

Finding a Fourier Serie from uniform converge

Question: Let $f(x)=x-x^2$ in $[0,1]$. Define the Fourier Serie in that interval. My Try: I define the Fourier Serie to $g(x)=x$ in $[0,1]$ as $\mathcal{F(g)}$. As $f$ is continuous, differentiable ...
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1answer
805 views

Two ways of writing the Fourier transform of the mod function

I feel there are two ways of writing the Fourier transform of $\vert x \vert$ and they are, that it is $\pm i \sqrt{2\pi} \delta'(t)$ for $x \geq0$ or $<0$ that it is $-\sqrt{\frac{2}{\pi}} \...
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1answer
941 views

How to integrate (1/t) *( e^t )? [duplicate]

I've tried solving it by parts but ended with a loop. And I can´t remember another method. Thanks.
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1answer
40 views

Find a function by its Fourier coefficients

Suppose $x[n]$ ($n$ integers) is periodic with period 8 and its Fourier coefficients are $$ a_k = \cos(k\pi /4) + \sin(3k\pi /4). $$ Prove $x[n] = 4\delta[n-1] + 4\delta[n-7] + 4j\delta[n-3] - 4j\...
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1answer
59 views

Show that if $h \in C^1([a,b]\times[-ε,ε])$ with $ε>0$, then the function $s \in [-ε,ε] \mapsto \int_a^b h(t,s) \,dt$ is $C^1$

Show that if $h \in C^1([a,b]\times[-ε,ε])$ with $ε>0$, then the function $$s \in [-ε,ε] \mapsto \int_a^b h(t,s) \,dt$$ is $C^1$ and $${d\over ds}\int_a^b h(t,s) \,dt=\int_a^b {\partial h \over \...
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1answer
59 views

Nash's equilibrium

I'm struggling with this question and was hoping someone could help. There are 4 parts which I think lead on from one another. We have Nash's inequality for $f\in\mathcal{S}(\mathbb{R})$ of the ...
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1answer
44 views

How can I show that there is M>0 for all positive a<A s.t $|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}| <= M $? [closed]

Let f be $L^1(R)$ and odd function. Then, for any positive $a < A$, there is $M>0$ such that $$ \left|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}\right| \leq M $$ ($\hat{f}$ is the ...
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1answer
48 views

$h = \sum_{n=0}^\infty (ae^{j\omega})^n$ , how is the approximation of this equal to $\frac {1}{1-h}$

the question in the title. im working on a z- transform problem. to find the Z - transform of $x(n) = a^ncos(\omega n)u(n)$, u(n) being the step unit function essentially i come down to the answer ...
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1answer
36 views

PDE, separation of variables

I need some help here. It's regarding question 5a. I am pretty lost as i've got no clue regarding how I should use the boundary conditions(i would've known if they weren't derivatives)
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1answer
57 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period 4....
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1answer
104 views

Aperiodic signals fourier transform short question?

What is the fourier transform of the aperiodic signals with infinite sequence? How about the transform of aperiodic fourier signals with finite sequence?
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1answer
65 views

Given the Fourier transform pair $h(t) \leftrightarrow H(\omega)$, what is the counterpart of $H(-\omega)$?

Given that $H(\omega)$ is the Fourier transform of $h(t)$, what is $H(-\omega)$ the Fourier transform of? Any help will be much appreciated. Thank you.
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1answer
422 views

The maximum absolute value of DFT of window vector

Let x=[1, ⋯ ,1, 0, ⋯ ,0] be a window vector of length N, which consists of B consecutive 1s and the remaining N-B consecutive 0s. I took the N-point DFT on x and got X=[X_0, X_1, ⋯, X_(N-1)] which is ...
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1answer
92 views

Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ &=\int\mathrm{sinc}(...
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1answer
96 views

Discussion on a beautiful function, $\sin(nx) / \sin x$ [closed]

In this post, I may need help with your broad knowledge on a function $$ \frac{\sin{Nx}}{\sin{x}} $$ where $N$ is an positive integer. The questions are as follows. What is the name of this ...
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1answer
50 views

Fourier transform of $\cos(\omega)$ [closed]

We have $$\begin{align}f(t) &= \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\cos(\omega)e^{j\omega t} \ dt \\ &=\frac{e^{j\omega t}}{2\pi(1-t^2)}\{jt\cos(\omega)+\sin(\omega)\}_{-\pi/2}^{\pi/2} \\ &=...
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1answer
38 views

Fourier transform of following equation

Say I have the following equation $$ f(x,y) = \left\{ \begin{array}{lr} 1 & \text{if} \;|x|,|y| \leq 1 \\ 0 & \text{otherwise} \end{array} \right. $$ What is the ...
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1answer
734 views

1 complex addition = 2 real additions, 1 complex multiplication = 4 multiplications + 2 additions.

Could you show that, when talking about Fourier transforms, that one complex addition requires 2 real additions, and one complex multiplication requires 4 multiplications and 2 additions.
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1answer
45 views

Meaning om notation $g(f;x)$

I'm currently studying fourier seriers in Walter Rudin Principles of Mathematical Analysis, where the following defintion is made $$s_N(x)=s_N(f;x)= \sum _{-N}^N c_n e^{inx}$$ where $f$ is a function ...
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1answer
58 views

How to calculate $\int\limits_{-\pi}^{\pi}f(x)\cos^5(nx)\,dx$ using Fourier series?

Given Fourier series $f(x)$ in $E[-\pi,\pi]$ $f(x) \approx \frac {a_0}2 + \sum\limits_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$ Calculate $\int\limits_{-\pi}^{\pi}f(x)\cos^5(nx)\,dx$ Is there an ...
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1answer
50 views

Estimate for L1 funcction [closed]

Suppose for every $r\in \mathbb{R}$, $\int_{\mathbb{R}}\lvert f(x)\rvert e^{rx}dx<\infty$. Then is it true that there exists $C>0$ and $\alpha>1$, such that $\lvert f(x)\rvert\leq Ce^{\lvert ...
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1answer
49 views

Fourier transform for specific signal [closed]

Could someone show me how to calculate fourier transform for the following signal? s(t) = e^(-3*t^2) Thanks!
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1answer
27 views

Hermite polynomials, prove the solution [closed]

$ \text { The Hermite polynomials, } H_{n}(x) \text { , satisfy the following: } $ \begin{array}{l}{\text { i. }<H_{N}, H_{M}>=\int_{-\infty}^{\infty} e^{-x^{2}} H_{n}(x) H_{m}(x) d x=\sqrt{\...
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1answer
104 views

Meaning of the imaginary term in the Fourier transform of sin(x)

Can someone explain why there is an imaginary term when taking the fourier transform of sin(x)? I can see the math, but shouldn't the fourier transform of sin(x) be equivalent to $delta(x -1)$? I'm ...
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1answer
102 views

Integral minimization [closed]

For the following integral the parameters a and b should be found, such that the value of the integral is minimal. $$\int_{-\pi}^{\pi}dx (f(x) - a \cdot cos(3x) - b \cdot sin(4x))^2$$ How can I do ...
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2answers
17 views

Fourier Computation integral [closed]

$$F(w)=\int_{-\infty}^{\infty} e^{-|x|+ix}e^{-iwx} \, dx$$ Need help to compute.
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1answer
104 views

What is the fourier transform of $e^{-\frac{t^2}2}$ [closed]

What is the fourier transform of $e^{-\frac{t^2}2}$. I need a classical solution, I mean, straight-forward. done by hand, without using tricks and convolutions. Many thanks in advance First steps, am ...
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1answer
59 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
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1answer
120 views

Riemann-Lebesgue Lemma Theorem 11.7 Tom Apostol [closed]

I am reading page 314 of Tom Apostol's Mathematical Analysis theorem 11.7. It says that the limit on left hand side of equation (11) exists because the quotient is continuous and bounded. My questions ...
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1answer
165 views

Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
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1answer
375 views

If the function is neither even nor odd ,is its fourier series coefficient a complex? or is this wrong? [closed]

i thought if the fourier series coefficient of $f$ is neither even nor odd,$f$ must be the complex. we can know the function,$f$, is real or imaginary or complex from its FS coefficients. Now i don'...