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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55 views

Does $|f\sin (x)|$ integrable on $\mathbb{R}$ imply that $|f|$ integrable on $\mathbb{R}$?

I guess not. Because we usually require $|f|$ to be integrable on ℝ so that it has the fourier transform. Can anyone give me an counterexample for the statement in the title? I have searched for ...
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1answer
21 views

What happens to fourier transform of the sampled output of pure sinusoidal input of 26kHz if sampled with 44.1kHz sample frequency?

Because pure sinusoidal signal only contains impulses, I was wondering what happens to the fourier transform of the sample output from the sinusoidal input of $26$kHz if the sampling is done with ...
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0answers
30 views

bessel function with Fourier transform

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
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1answer
37 views

Odd or Even for Fourier Series?

I have the function $f(x) = -x^2 + x\pi$ and $0\le x\le \pi$ and without seeing the graph I want to show if it is odd or even, but of course $f(x) = f(-x)$ doesn't show that it is even because I can't ...
3
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3answers
42 views

$f\in M(\mathbb{R})$ but $\hat{f}$ is not

I am studying Fourier analysis. I noted some problems state $f,\hat{f}\in M(\mathbb{R})$ as assumption, where $M(\mathbb{R})$ denote the collection of all continuous and of moderate decrease functions ...
3
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0answers
46 views

Riemann-Lebesgue Lemma for Spherical Harmonics expansion

Here is my question: A basic result of classical Fourier analysis is that the fourier coefficients of an $L^1$ function must tend to zero (Riemann-Lebesgue Lemma). Is there analogous result to the ...
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1answer
48 views

proving Riemann-Lebesgue lemma

I have looked at proofs of the Riemann-Lebesgue lemma on the internet; all of these proofs use the technique of Riemann integration and making step functions. ...
0
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1answer
36 views

Sharpening a curve

I have a frequency domain graph as shown. I need to "sharpen" the curve to get a better response, and computing large butterworth orders is not possible on my machine. Hence, I would like to know if ...
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2answers
45 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
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0answers
14 views

estimate on a convolution

Let $\psi$ be a non-negative Schwartz function on $\mathbb{R}$ such that supp$\hat{\psi}$ is contained in $[-0.1, 0.1]$ and $\hat{\psi}(0)=1$. Define $\psi_k(x)=2^k\psi(2^kx)$ for any integer $k$. Let ...
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1answer
8 views

Support of Auto-correlation

Suppose $f\in C_0^{\infty}(\mathbb{R}^n ),$ then clearly we have supp$(f\ast f)\subseteq$ supp$(f)+$ supp$(f)$. The question is whether supp$(f\ast f)\subseteq 2$ supp$(f)$ holds? Any counterexample?
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1answer
46 views

$L^2$ and uniform norm of $\text{sinc}\, x$ and its derivatives

Looking at the graphs of the derivatives of $\mathrm{sinc}\,x$, it appears that they all are bounded by $1/x$, with $[\mathrm{sinc}\,(x)]'$ the sole exception: A few questions: 1) With the ...
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1answer
21 views

Solving convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside step function

How does one solve convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside (unit) step function? I tried using Fourier transform of both functions to ...
4
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1answer
45 views

convolution with $C^{\infty}$ produces $C^{\infty}$

Problem: So I have the following function in ...
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0answers
37 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
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0answers
14 views

Power law in power spectrum and memory.

If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law ...
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1answer
48 views

Find the Fourier transform of $\frac1{1+t^2}$

Find the Fourier transform of $$f(t)=\frac1{1+t^2}$$ using contour integration that $$F\{f(t)\}=\int^\infty_{-\infty}\frac1{1+t^2}e^{2\pi ft}dt$$ How can I do this?
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11 views

Strategies for approximating fourier transform of $k$-th power of the $n$-th derivative of a function

For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I ...
0
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1answer
51 views

Young's inequality for convolutions

Let's assume that the convolution $f * g$ is continuous with $\lim_{|x| \to \infty}(f*g)(x) = 0$ and that $f, g \in L^2$. Then the following inequality holds $$ \| f * g \|_{\infty} \leq \| f \|_2 ...
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1answer
66 views

Fourier transform of compactly supported differentiable function

Let $K$ be the space of infinitely differentiable functions $\mathbb{R}\to\mathbb{C}$ with compact support. I read the unproved statement in Kolmogorov-Fomin's Элементы теории функций и ...
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0answers
18 views

Fast Fourier Transform and its example

I read the wikipedia and my textbook, but I can't understand the whole process of Fast Fourier Transform. Especially the book uses the Cooley-Tukey algorithm and it gives an example of 4X4 matrix like ...
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0answers
31 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ ...
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1answer
21 views

Support of tempered distribution under exponetiation and differentiation

Suppose $u$ is a tempered distibution in $\mathbb{R}^n$. How are supp$(\widehat{u})$ and support of $\sum_{|\alpha|\leq k}\frac{\partial^{\alpha}\widehat{u^n}}{\partial x^{\alpha}}$ compared , where ...
2
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1answer
54 views

The inverse Fourier transform of $\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}$

I need help to compute the following integral $$\int_{\mathbb{R}^n}\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}e^{2\pi i\xi \cdot x} \mathrm{d}\xi $$ where $\widehat{\varphi}$ is the Fourier ...
0
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1answer
45 views

using fourier method to compute this integral

Use the method of Fourier analysis to calculate the following integral: $$ \int_{0}^{\infty} \frac{\cos x}{1+4x^2} \operatorname{d} x .$$ Could someone help about this question? what ...
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20 views

Hoe can I find the Inverse FourierTransform for 1/(1+w^4)?

I have the expression $S(w)=\frac{1}{1+w^4}$. I am trying to find its inverse FourierTransform. I know that I have to get a sin-cos expression, but I haven´t found the way to do it. On the tables that ...
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0answers
23 views

Product of Fourier transform

I'm supposed to calculate : $ F[\delta(x-a)](\nu)F[e^{i2\pi \nu x}](\nu)$ Since $F[\delta(x-a)] = e^{i 2\pi \nu a}F[\delta(x)] = e^{i 2\pi \nu a}$ it leaves : $$e^{i 2\pi \nu a}F[e^{i 2\pi \nu x}] ...
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19 views

Solve this PDE, using Fourier transforms

PDE: $v_t(x, t) = kv_{xx}(x, t) + bv_x(x, t)$, $v(x, 0) = f(x)$, $\quad-\infty < x < \infty$ I can't apply the inverse Fourier in this case. If someone could help me, because i can't find a ...
2
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1answer
60 views

Prove a trigonometric series is positive

Let $f(x)= \sum_{n=-\infty}^\infty \frac {e^{inx}}{1+n^2}$ on $[-\pi,\pi]$. Prove $f(x)>0$ for $x\in[-\pi,\pi]$. This is an review question for my Fourier course. I am not sure how to approach ...
2
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0answers
28 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...
2
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2answers
74 views

Why am I allowed to set a fixed point in a fourier series?

I'm working with $f(t)=\cos(at)$, for $a\in (0,1)$, on the interval $(-\pi,\pi)$. I've calculated the fourier series on this interval. what I would want to do next is to fix $t=\pi$ and get a nice ...
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0answers
19 views

Evaluate the limit of function presented as a series

This is an additional exercise given in my Fourier analysis course. Define $F(t)=\sum_{n=-\infty}^\infty (-1)^n e^{-2n^2t^2}, \,t>0$, Prove that $\lim_{t\to \infty}F(t)=1$. ...
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1answer
39 views

Evaluate integral using Fourier analysis

$\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$ $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$ There is no hint for these two questions. I think for Q2, since it's a square, I can use Plancherel ...
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0answers
14 views

Fourier coefficients of three times differentiable functions.

I was wondering, can we determine somehow the decay of the Fourier coefficients of a function $g \in C^3\mathbb{(T)}$ /three times continuously differentiable/ as $|n|\rightarrow \infty$? Any help ...
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0answers
13 views

fourier transforms of $e^{(-kw^2+b)t}$

I was solving and PED using fourier transforms and reached this point: $$v(x,t)=f(t)*F^{-1}\big[e^{(-kw^2+b)t}\big]$$ $F^{-1}$ denotes inverse fourier transforms, and $*$ is used for convolution. I ...
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0answers
26 views

Continuity of Fourier-Stieltjes transform

I read in Kolmogorov-Fomin's (p. 419 here) that, if $F$ is a function having bounded variation on $\mathbb{R}$ then the Fourier-Stieltjes transform$$g(\lambda):=\int_{-\infty}^\infty e^{-i\lambda ...
7
votes
2answers
165 views

Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
3
votes
0answers
34 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
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0answers
13 views

Solving DFT of this array

I am told to solve the DFT of the following array (5,5,5,5) However, according to the answer sheet, it is suppose to be (5,0,0,0). I tried working it out by hand, and F(0) was correct. But my ...
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1answer
23 views

Convolution with heaviside function, argument of the heaviside carry on to the dirac function?

So I have this equation to demonstrate: $$ x(t)*u(t)= \int_{-\infty}^t x(\tau)d\tau $$ , where $u(t)=\int_{-\infty}^t \delta(\tau)d\tau$ I opened the convolution as $ \int_{-\infty}^\infty ...
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0answers
12 views

Inverse Fourier Transform with Duality

I need to find the inverse Fourier transform of the following equation using the duality property: $X(w)=\begin{cases} 2w+2 &\mbox{if} -1<w<0 \\ -2w+2 &\mbox{if }0<w<1 \\ 0 ...
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0answers
27 views

Inversion formula for $\int_{\mathbb{R}}f(x)e^{-izx}dx$

Let $f:\mathbb{R}\to\mathbb{C}$ be a measurable function such that$$\forall x\ge 0\quad|f(x)|<Ce^{\gamma_0 x}$$$$\forall x<0\quad f(x)=0$$I must specify that all the integrals I am going to ...
3
votes
1answer
80 views

Relation between Fourier components of a positive function

Here's a problem that has recently come up in my physics research: Let f be a function on [0, 2 $\pi$], which yields positive real numbers. Let the integral of $\int_0^{2\pi}f(x)= 1$. (Just for the ...
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0answers
15 views

Wiener's lemma and Hulanicki's lemma

Let $\mathcal{A}(\mathbf{T})$ be the Banach algebra of continuous complex-valued functions on the unit circle with absolutely convergent Fourier series. Then Wiener's lemma states that if $f \in ...
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16 views

Fourier series, even and odd n properties

I recently started learning about Fourier series, so I'm still kind of shaky on the topic Given $f(t)=f(t+T)$ and $$f(t)=\sum_{n=-\infty}^{\infty}F_ne^{jn \omega_0t}$$ Show that: (a) If ...
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1answer
75 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
0
votes
1answer
31 views

Support of polynomial distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
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2answers
62 views

Fourier Transform of $\exp(-t)$

$$f(t)= \begin{cases} e^{-t} & 0<t<1 \\ 0 & \text{otherwise} \end{cases}$$ How can I solve this function's Fourier transform? I am stuck at here: Daniel R - OP \begin{align} ...
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1answer
17 views

Existence of certain function in Schwartz space

Suppose a polynomial $P(x_1,\cdots,x_n)$ is given. Does there exist a function $\phi\in\mathcal{S}(\mathbb{R}^n)$ such that supp$(\phi)\subset\mathbb{S}^{n-1}$ and ...
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0answers
19 views

One simple question about Fourier transformation of system of PDE's

Let's assume set of equations $$ \tag 1 \frac{\partial \mathbf A}{\partial t} = \Delta \mathbf A + a [\nabla \times \mathbf A] - d\mathbf b_{k} (\mathbf b_{k} \cdot \mathbf A), \quad \mathbf A(0) = 0 ...