The tag has no usage guidance.

learn more… | top users | synonyms

-4
votes
0answers
51 views

Fourier transform of $|x|^{-\alpha}$ [closed]

Let $k(x)=|x|^{-\alpha}$ ,$x\in\mathbb{R}^d, \,0<\alpha<d $. Why the Fourier transforms of $k$ is $\hat{k}(\xi)=C|\xi|^{d-\alpha}$ with $C=C(d,\alpha)$ ? Thank you .
0
votes
0answers
97 views

Fourier transform [closed]

Let $k(x)=|x|^{-\alpha}$ ,$x\in\mathbb{R}^d, \,0<\alpha<d $. Why the Fourier transforms of $k$ is $\hat{k}(\xi)=C|\xi|^{d-\alpha}$ with $C=C(d,\alpha)$ ? Thank you .
-3
votes
0answers
30 views

Need an example of multidimentional fourier transform [closed]

Please, take the function, for example sin(xy)+cos(xz) from dimention 3 to R, and give me a multidimentional fourier for it. I'll be also thankful for general multidimentional fourier transform ...
0
votes
0answers
70 views

Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
3
votes
0answers
52 views

Conceptual explanation for Poisson summation formula

The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have $$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf ...
28
votes
3answers
3k views

What is the actual meaning of a fractional derivative?

We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called ...
2
votes
1answer
108 views

Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
2
votes
2answers
274 views

Are the zeroes of the Fourier Transform of compactly supported functions isolated?

I have a continuous function $f$ on a locally compact Abelian group $G$ with compact support, and I would like to say that the zeroes of $f$ are sparse in some sense (isolated would be good, uniformly ...
7
votes
2answers
498 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
7
votes
1answer
2k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand $K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed form or some kernel like structure. If I try to take ...
6
votes
1answer
153 views

Is this function Schwartz?

I already asked this question here on MSE, didn't get an answer, and I'm still stuck with it. Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ ...
6
votes
1answer
266 views

G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...
2
votes
0answers
342 views

A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
3
votes
2answers
297 views

Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem. I must let you know from the beginning that it's not an easy one. "Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ , ...
1
vote
0answers
62 views

Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
0
votes
0answers
70 views

Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...
2
votes
1answer
97 views

Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$. Here, $F$ denotes the ...
7
votes
1answer
102 views

A case of nested central limits

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. We are interested in $N$ large and prime. The Fourier transform ...
6
votes
1answer
160 views

Statistical independence and the Fourier transform

Consider the random variable $S=(s_0, \dots ,s_{N-1})$, a sequence of signs uniformly distributed on the hypercube $\{-1,1\}^N$. With the Fourier transform we can define $N$ random walk variables $$ ...
0
votes
1answer
75 views

Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my ...
4
votes
3answers
391 views

Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation: $$\mathbb{E}[(c+e^X)^{-n}]$$ where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
1
vote
2answers
219 views

Proof of Biot-Savart theorem in 3D using distributions and fourier transform

While studying the book Vorticity and Incompressible Flow by A. Majda and A. Bertozzi I came across the following Biot-Savart theorem in 3D: The solution to: $$div \ v=0$$ $$curl(v)=\omega$$ is ...
7
votes
1answer
362 views

Proof of a Fourier pair with Bessel functions?

How can we prove that the Fourier transform of the function $$ f(x) = \begin{cases} (a^2-x^2)^{c/2} BesselJ[c,b\sqrt{a^2-x^2}] & \text{for }x^2 < a^2\\ 0 & \text{otherwise} \end{cases} $$ ...
4
votes
1answer
437 views

Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
0
votes
1answer
117 views

Simplifying an expression using tools from Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help: $f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{ \omega r^{-\gamma}}} ...
3
votes
0answers
126 views

Isbell duality in Joyal and Street's Introduction to Tannaka Duality

In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...
0
votes
0answers
64 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
2
votes
0answers
85 views

Generalization of Pitt's theorem

Pitt's theorem (Pitt 1937), states that the one-dimensional Fourier tranform is well defined and continuous between the weighted spaces $L^p(\mathbb{R},|x|^{bp}dx)$ and $L^q(\mathbb{R},|x|^{\beta ...
0
votes
1answer
152 views

Fourier series and transform related to Epicycles

Let $\gamma:\mathbb{R}\to\mathbb{C}$ be a continuous periodic curve having a bounded variation. 1) Is it true that one can find a sequence of numbers $(r_n)_{n\in\mathbb{N}}\subset (0,\infty)$ and ...
7
votes
2answers
556 views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
0
votes
0answers
71 views

Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...
5
votes
1answer
138 views

How are the infinity norm of Fourier transforms of sign vectors distributed?

This is a follow up to an earlier resolved question. Define the $n$-dimensional discrete Fourier transform via the matrix $$ D_{s,t} := \omega^{st}, $$ where $\omega=\exp(-2\pi i/n)$. Notice that $D$ ...
6
votes
0answers
89 views

Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for ...
0
votes
0answers
99 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
1
vote
0answers
114 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
1
vote
1answer
176 views

Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that $T(x):= \sum_{n \in \mathbb{Z}} ...
0
votes
0answers
124 views

Problem with operator and Fourier transform

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 ...
0
votes
0answers
35 views

Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?

I have been learning Wiener amalgam spaces. In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$ Take $f(x)= ...
3
votes
1answer
144 views

Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
1
vote
0answers
52 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

[I have asked this question on S.E. M; but I have not got any answer; and hope this is o.k. for M.O] Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and ...
4
votes
2answers
248 views

Generalization of Lévy's continuity theorem for nuclear spaces

I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions ...
2
votes
0answers
105 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n ...
3
votes
0answers
98 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = ...
1
vote
1answer
175 views

Is there a Poisson Summation formula for imprimitive Dirichlet characters?

I was wondering if there exists a Poisson Summation formula (like the one existing with primitive character) for imprimitive Dirichlet characters ? For a primitive Dirichlet character $\chi$ we have: ...
3
votes
2answers
530 views

Fourier transform of compactly supported distribution is smooth

My advisor made the comment that if $u\in \mathcal{E}'$ is a compactly supported distribution, then $\hat{u}(\xi)\in C^{\infty}(\mathbb{R}^n)$ is actually a smooth function (not merely a distribution ...
2
votes
0answers
325 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in ...
19
votes
2answers
3k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
5
votes
0answers
130 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
2
votes
1answer
279 views

Connection between the Fourier transform of f and |f|

If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and $$ ...
1
vote
0answers
33 views

Multidimensional Filters

Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then ...