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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110 views

Schwartz class estimation.

I have a function $f\in \mathcal{S}$ (i.e of Schwartz class), and I want to show there exist constants $C,k>0$ s.t $$\|f\|_p \leq C(\sup_{x\in \mathbb{R}} |f(x)| + \sup_{\mathbb{x\in \mathbb{R}}} ...
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0answers
597 views

What is the fourier transform of $\operatorname{sinc}^4(kt)$?

I have to use Parseval's Theorem. I used it and ended with the integral of $(\operatorname{sinc}^2(kt))^2$. I know the Fourier Transform of $\operatorname{sinc}^2(kt)$ is the triangle function but I ...
5
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0answers
254 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
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2answers
363 views

Cooley-Tukey Algorithm?

Why does the Cooley-Tukey Fast Algorithm take $O(n \log n)$ time? The book derives this from the fact that evaluation takes time: $T(n) = 2T(n/2) + O(n)$ and then uses the Master Theorem to arrive ...
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2answers
280 views

proof that translation of a function converges to function in $L^1$

Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}||f_a -f ||_1=0$$ I know that there exists $g\in C(\mathbb{R})$ s.t $||f-g||_1 \leq ...
6
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3answers
1k views

Known proofs of Wirtinger's Inequality?

I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if $f:[0,\pi]\to \mathbb{C}$ is $C^1$ and $f(0)=f(\pi)=0$ then \begin{equation} \int_0^\pi |f(t)|^2 dt \leq \int_0^\pi ...
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1answer
113 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
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1answer
615 views

How to use joint characteristic function to calculate characteristic function for single variables? [duplicate]

Possible Duplicate: probability question on characteristic function It is a problem in my practice exam. Defined on some common probability space, two random variables $X$, $Y$ have the ...
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2answers
99 views

a Function with several periods

A periodic function is given by $ f(x+nT)=f(x) $, with 'n' an integer and T the period. My question is if we can define a non-constant function with several periods; by that, I mean $ ...
5
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1answer
405 views

Poisson summation formula and Schwartz functions

I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on ...
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1answer
155 views

Can a function with suport on a finite interval have a Fourier transform which is zero on a finite interval?

If $f$ has support on $[-x_0,x_0]$ can its Fourier transform $\tilde{f}$ be zero on $[-p_0,p_0]$? If so, what is the maximum admissible product $x_0p_0$?
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0answers
269 views

Very tricky Fourier transform

I'm trying to evaluate the following integral using complex function theory: \begin{equation} ...
2
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1answer
240 views

Fourier transform of derivative for the bounded function

I was going through the derivation of Wigner distribution properties, and encountered a certain step in the proof I could not justify. Namely, the step requires the following equality to be true: ...
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1answer
1k views

A function and its Fourier transform cannot both be compactly supported

I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the ...
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1answer
273 views

Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
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1answer
719 views

Dirichlet kernel.

I have a function $h\in L^1(\mathbb{T})$, and I want to show that: $$\int_{\pi\geq |t|>\delta>0} h(x+t)D_N(t) dt/2\pi \leq \xi_N(h,\delta)$$ where $\xi_N(h,\delta) \rightarrow 0$ as ...
5
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1answer
278 views

$L^p$ norms of Fourier transform of solutions of hyperbolic Burger's equation at the time of first blow-up

I am struggling to understand the behavior of the Fourier transform (in the $x$ variable) of initially smooth solutions of the hyperbolic Burger's equation in 1-D, $ \partial_t u + u~ \partial_x u ...
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1answer
819 views

Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions)

I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting ...
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2answers
637 views

Signal with finite length in time and frequency

Is it possible for a signal to have finite length in both the time domain and the frequency domain? Or does the finite length of one necessarily imply that the other has infinite length? (By "finite ...
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1answer
253 views

Finding the eigenvalues of the sum of circulant and diagonal matrices - What am I doing wrong?

Saw this question about the eigenvalues of the sum of circulant and diagonal matrices on MO and, since I recall my prof mentioned circulant matrices and Robert Gray's book, I thought I'd give it a ...
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0answers
253 views

Complex numbers and Fourier transform

Here I am stuck in solving Fourier transform and the funny part is that I am stuck in the basics, in the complex part. I hope someone can help me solve this part. $$ 3 + 3 ( \cos \frac{4\pi}3 + j ...
0
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2answers
286 views

$\hat f_\lambda(x) = \hat f(x/\lambda)$

This is a small part of a larger problem I am trying to solve. This is stated as a basic property of the fourier transform. First we define for $f \in L^1(\mathbb{R}^d)$ and $\lambda \neq 0$, $$ \hat ...
13
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1answer
448 views

Laplace transform identity

Is there a function equal to its Laplace transform? I mean $$ \int_{0}^{\infty}dt\exp(-st)f(t)= f(s).$$ Of course I know $f(t)=0 $ satisfy the equation. For the case of the Fourier transform, I ...
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4answers
883 views

Computing the Gaussian integral with Fourier methods?

There are many proofs that $$\int_{-\infty}^\infty e^{-x^2} \, \mathrm dx = \sqrt{\pi}.$$ For example, using a change to polar coordinates, differentiation under the integral sign, and the theory ...
5
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1answer
446 views

Rate of Fourier decay of indicator functions

The Fourier transform of the indicator function of an interval $$\widehat{\chi}_{[a,b]}(\xi)=\int^b_{a} e^{i \xi x}dx=\frac{e^{i\xi b}-e^{i\xi a}}{i\xi}$$ has decay $O(|\xi|^{-1})$ as ...
2
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1answer
174 views

Fourier transform and distibution beloning to S'

I need prove, that a distribution $$\langle F_f,\phi \rangle= p.v. \int\limits_{-1/2}^{1/2}\frac{\phi(t)}{t\cdot \ln{|t|}}\mathrm{d}t$$ belongs to $S^\prime$ (adjoint to Schwartz space) and I need ...
3
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1answer
204 views

Discrete Fourier transform of a particular sequence of real numbers

This may be an elementary question, but I don't know much about the discrete Fourier transforms (DFT). Suppose I have a sequence $\{x_n\}_{n=0}^{N-1}$ of $n$ real numbers such that $x_0\geq |x_n|$ ...
6
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2answers
189 views

Fourier Transform $\displaystyle{F}^{-1}({e^{i\xi^3t}})$

The problem is$$u_t+u_{xxx}=0,u(x,0)=f(x),$$ Use Fourier Transform we get$$\overline{u}=e^{i\xi^3t}\overline{f},$$I want to solve out $u$ . Thus I want to know ...
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2answers
269 views

Understanding Matrix Formula with Scant Knowledge of Linear Algebra

$n$ is a power of $2$. $M =\pmatrix{ 1& x_0 & x_0^2 & \dots &x_0^{n-1}\\\ 1& x_1 & x_1^2 & \dots &x_1^{n-1}\\&& \vdots\\1& x_{n-1} & x_{n-1}^{2} ...
3
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1answer
147 views

Function of $C_0(\mathbb{R})$

I need to prove that $$g(x) = \text{p.v.} \int\limits_{-1/2}^{1/2}\frac{e^{-itx}}{t\cdot \ln{|t|}}dt $$ is function of $C_{0}(\mathbb{R})$. So, I need to prove that $$ ...
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0answers
485 views

How to use Fubini's theorem when integrating over Euclidean Space

I am currently studying the Fourier transform in Euclidean Space $\mathbb{R}^n$, using Knapp's book "Basic Real Analysis". Upon proving the property that the Fourier transform turns Convolution in ...
2
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1answer
172 views

Convolution of function

I need find $x^2\cdot e^{{-x^2}/2} * e^{{-x^2}/2}$. I used statements, that $\widehat{xf}=i \widehat{f}'$ and $\widehat{f*g}=\sqrt{2\pi}\widehat{f}\cdot \widehat{g}$. So, $\widehat{x^2\cdot ...
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1answer
199 views

Example of a function that is not the Fourier transform of any function of L1 (R).

Let $$ g(x) = v.p.\int_{-1/2}^{1/2}{e^{-itx}\over t\ln{|t|}}dt. $$ Please help me prove, that this function is not the Fourier transform of any function of $L_1(\mathbb{R})$.
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0answers
91 views

Fourier transform of $ 1/|x|^{k}$ [duplicate]

is ist possible to find the Fourier transform (direct and inverse ) of $ f(x)= \frac{1}{|x|^{k}} $ for $ k=1,2,3,......$ this function has a severe singularity at $x=0$ so i think this will exists ...
2
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1answer
147 views

Problems on Schwartz Functions

(1) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (2) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ? (3) What ...
5
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1answer
1k views

Fourier transform in $L^p$

Let the $f$ be a function in $L^s$ where $s \in [1,\infty) $. For which $r$ Fourier transform $\hat{f}$ belongs to $L^r$? I'd be grateful for any kind of help including providing a literature or ...
3
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2answers
774 views

Fourier Transform of a frequency linearly modulated signal

I'm working on an oscillating signal whose trend can be modelled as a frequency linearly varying function. An example may be as follows: $$ \Gamma(t)=\sin(2\pi\nu(t)t) $$ with $$ \nu(t)=\nu_0 + at $$ ...
5
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4answers
690 views

What is the relationship between generalized functions and things like the Riesz representation theorem?

I just watched this video of Prof. Osgood's lecture on Fourier Transforms, and it seems to me that there's some connection between his talk of distributions (generalized functions) and the usual ...
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2answers
542 views

Discontinuous Fourier transforms?

What's an example (or even better a large class of examples) of an $L^2$ function whose Fourier transform is discontinuous?
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1answer
191 views

Can you Fourier transform probabilities?

If I have a rect function , and I convolute it with it's self, I get a triangle function. If I convolute with a rect function again, I get a bell-curve. I can continue, so long as I know how to ...
2
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1answer
173 views

Smoothness of Fourier series

In a book from differential equations I found the following theorem, without proof and references: Let functions $f, g: R \rightarrow R$ be continuous and $2\pi$-periodic and let $m\in N$. Assume ...
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2answers
149 views

Is it true that a fourier transform of $f$ never vanishes if the translates of $f$ is $L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ and let $V_f$ be the closed linear subspace of $L^1(\mathbb{R})$ generated by the translates $f(\cdot - y)$ of $f$. If $V_f=L^1(\mathbb{R})$, I want to show that $\hat{f}$ ...
3
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2answers
377 views

Fourier integral/ Fourier transformation of an oscillatory function with FFT

$f(x) = \cos(x^2)$ and $g(k) = \sqrt\pi \cos((\pi k)^2 - \pi/4)$ are a Fourier pair. I want to reproduce $g(k)$ by Fourier integrating $f(x)$ using FFT, i.e. approximating ...
2
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1answer
1k views

Fourier transform over a diagonal matrix

Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is: Under ...
3
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1answer
569 views

Integrability of the Hilbert transform of a Schwartz function

Given a Schwartz function $f\colon\mathbb{R}\to\mathbb{R}$, define its Hilbert transform by $$(Hf)(x)=\frac{1}{\pi}\left(\int_{|t|\leq 1}\frac{f(x-t)-f(x)}{t}\,dt + \int_{|t|\geq ...
3
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0answers
102 views

Truncation in singular integrals

After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ ...
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1answer
294 views

Discrete Fourier transform real signals

The discrete Fourier transform is defined as: $$S(k)=\sum_{n=0}^{N-1}s(n)e^{-j\frac{2\pi}{N}kn}\quad k=0,...,N-1$$ I read that real signals $s(n)$ are: $$S(l)=S(N-l)^*$$ where $S(N-l)^*$ is the ...
6
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1answer
3k views

Three Dimensional Fourier Transform of Radial Function without Bessel and Neumann

I am trying to compute the Fourier transform of $\frac1{|\mathbf{x}|^2+1}$ where $\mathbf{x}\in\mathbb{R}^3$. Just writing out the integral: ...
2
votes
1answer
377 views

Heat equation asymptotic behavior

This was a question on one of my analysis finals, but I was unable to answer it. It seemed very interesting though. Suppose that $f(x, t)$ is a solution of the heat equation $\displaystyle ...
0
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1answer
396 views

Given fourier series,finding functions

4.How to prove that there is a continuous periodic function $f$ (with period $2\pi$), such that $$\hat{f}(n) = \log(n)/(n^{3/2}).$$ $n\neq 0$ and $\hat{f}(0) = -1$. I know only the basics of ...