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Questions tagged [fourier-transform]

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38 views

From equation to Laplace to z-tranform [on hold]

I have a doubt. Having this relation: p(w)=((rho*c)/2*A)exp-(iomegat)(((1+RO*exp(i*2*kd2)(1+RL*exp(i*2*kd3))/(1-(RORL*exp-(i*2*k*d4))))*U(\omega) Now for the Laplace, I take ts=1/fs, \tau=dn/c and \...
-1
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0answers
36 views

Help with explanation to the notations in a paper about notations and use of Fourier and LaPlace Transformations [on hold]

I'm looking at a paper 'Recent applications of fractional calculus to science and engineering' (https://www.hindawi.com/journals/ijmms/2003/753601/abs/) but some of the notations in it baffled me and ...
1
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1answer
51 views

encryption with wavelet transform

I need to know about an article, book or other reference that deals with encryption using Fourier transform and the Wavelet. (I plan to use matlab or other recommended software.)
5
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0answers
55 views

Are Stochastic Process Characterized by Their conditional Moments

Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that $$ \mathbb{E}\left[\int_s^T\...
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0answers
75 views

Transformation of Fourier Transform

Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also. Is there an expression ...
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0answers
16 views

Solution using Fourier transform to IVPs of elastic wave equations

I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media. \begin{equation} \begin{cases} \rho u_{...
5
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1answer
151 views

Positive-definiteness of radial sinc function in three dimensions

In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
1
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1answer
68 views

Maximum Magnitude Deviation between DFT and DTFT

This is a cross-post from signal processing forum as it was not conclusive. Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then $$ X(\omega) = \sum_{n = 0}^...
9
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1answer
343 views

Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
4
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0answers
82 views

On smoothness of a function and decay of its Fourier transform

I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here. I am interested in a quantitative version of the principle that smoothness of ...
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0answers
79 views

Initial value theorem for Fourier transform

Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
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2answers
173 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
3
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1answer
74 views

Level sums, displacements: how to determine them efficiently?

Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$, $\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$. Define $$S(\delta,m) = ...
3
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1answer
312 views

Expected value of the maximum of the periodogram

Let us suppose that $X_1,\ldots,X_n$ with $n\ge1$ are iid random variables such that $\operatorname EX_1=0$ and $\operatorname E|X_1|^s<\infty$ with some $s>2$ and define the DFT of $X_1,\ldots,...
3
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1answer
147 views

Recovering information for $\sum_{n \leq x}a(n)$ from $\sum_{n \geq 1}a(n)e^{-nx}$

I am wondering if I could deduce the bound for the partial sums \[ \sum_{n \leq x}a(n) \ll x^{A}, \quad x \to \infty \] from the relation \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]...
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0answers
43 views

Composing functions whose derivatives have $L^1$ Fourier transforms

Let $f, g \in C^n(\mathbb{R})$. Given bounds $\|f^{(i)}\|_\infty \le a_i, \|g^{(i)}\|_\infty \le b_i \forall i \in \{1, ..., n\}$, we can derive bounds for $\|(g \circ f)^{(i)}\|_\infty$ using the ...
2
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1answer
226 views

About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
1
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1answer
76 views

Maximum of the periodogram of a truncated sequence

Suppose that $Z,Z_1,Z_2,\ldots$ are iid random variables such that $\operatorname EZ=0$, $\operatorname EZ^2=1$ and $\operatorname E|Z|^s<\infty$ with some $s>2$. Let $\tilde Z_t=Z_tI_{\{|Z_t|\...
1
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1answer
320 views

Which Fourier transform is the correct one?

Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it: $$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$ while from Sokhotski–Plemelj theorem it follows ...
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0answers
46 views

Positivity of an integral with product of functions and their Fourier transforms

What condition two functions $f$ and $g$ defined on $\mathbb{R^+}$ must fulfilled to have: $$\int\limits_{0}^\infty \ln(x) (fg+ \hat{f} \hat{g} ) dx >0$$ Where we note $\hat{f}(x)= 2\int\...
1
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0answers
93 views

Functional equation with Fourier transform

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$: $$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$ Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
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0answers
41 views

Comparison of (square) of a function and its Fourier transform in an integral

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
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0answers
47 views

Existence of unbounded $M \subset \Bbb{R}$ of finite measure s.t. $1_M$ is $L^p$-Fourier multiplier

I would like to know if there is a measurable set $M \subset \Bbb{R}$ such that $M$ has finite Lebesgue measure $0 < \lambda(M) < \infty$, $M$ is unbounded in the sense that $\lambda(M \...
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0answers
89 views

Can we deduce the sign of this integral which includes cosine transform?

Let's $v(t)$ be a negative real function and consider the following integral with $f(t)$ complex (rapidly decreasing at infinity): $$A= \int\limits_{0}^\infty \frac{1}{x} \int\limits_{x}^\infty \Big(...
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0answers
53 views

Solution of a functional equation with cosine transform

What are the functions verifying: $$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$ With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
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0answers
74 views

Convolution of Fourier Transform and Walsh (or Hadamard) Transform

Is it possible to calculate the convolution of the Fourier transform and the Walsh or the Hadamard transform? Since the Fourier transform is sinusoidal while the Walsh or Hadamard transform is non-...
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0answers
97 views

Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?

As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
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0answers
53 views

Support of functions on Minkowski space and their Fourier transform

Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$. Are then $f$ ...
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2answers
236 views

Composition of Riesz potentials

For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by $$ (I_\alpha f)(x) = \frac{1}{\gamma(\alpha)} \int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, , \quad \text{where} \quad \...
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1answer
267 views

Reference request: Fourier transform on the multiplicative group of real numbers

Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
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2answers
213 views

What gives a “Parseval like” equation mixing cosine and sine Fourier transforms ?

Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by: $$\mathcal{F}^c [f (x)]=\int_0^{\infty} f(t) \cos(xt) dt $$ $$\mathcal{F}^s [f (x)]=\...
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0answers
308 views

“Small” zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$

If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
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0answers
46 views

Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions

I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it ...
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0answers
243 views

Eigenvalues of a specific Hankel matrix

I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by \begin{equation} G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2}, \end{...
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0answers
51 views

Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
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0answers
269 views

Fourier Transforms of Convolutions

Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have $$\...
4
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1answer
264 views

Is there a measure on the sphere with positive Fourier transform?

Is it possible to have an even probability measure $\mu$ (that is $\mu(A)=\mu(-A)$ for any set $A\subset \mathbb{R}^d$) supported on the unit sphere $S^{d-1}$ such that its Fourier Transform $$ \...
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2answers
93 views

Fourier series of $\exp(\sum_k a_k\cos(k\theta+\phi_k))$

I need to know the Fourier series of exponential of general function, represented as $c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\psi_n)d\theta$. ($...
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2answers
285 views

Fourier transform inversion theorem for a function not in L1 or L2

For $\frac{1}{4}<a<1$ consider the following function: $$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$ If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...
4
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0answers
153 views

Existence of $A\subset\Bbb{R}^n$ of finite measure and $\hat{1_A}\in\bigcap_{q>1}L^q$, but s.t. for some $1<p<\infty$, $1_A$ is no $L^p$-Fourier mult

I am interested in the following somewhat obscure question: Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its ...
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0answers
94 views

Need to show bounded behavior of a particular Fourier transform

First let me be briefly state the relevant information to my problem: $\beta(s) \in C_0^{\infty}([-1,1])$, and $\beta \equiv 1$ around $s=0$. The $\beta$ I'm using is an even function, but it doesn't ...
2
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1answer
125 views

CTRW: solve a renewal equation

Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f_J(x)$. Suppose the initial distribution $\rho_{X(0)...
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0answers
87 views

On boundedness of a certain term involving Fourier transform

Let $f$ be a function over the reals. Then I have following two questions: Question 1: Do there exist any $f\in L^1(\mathbb{R})$ such that the following expression is bounded for all frequency $\...
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0answers
71 views

When is $|\int_a^b \exp(-izx)f(x) \, \mathrm{d}x| \leq |\int_a^b f(x) \, \mathrm{d}x|$ for a general $f$? [closed]

Obviously, if $f(x) \geq 0 $ on $(a,b)$, the above claim is trivial. We also see that if $a = -\infty, b = \infty$, above is claiming that $|\hat{f}(z)| \leq |\hat{f}(0)|$ for all real $z$, where $\...
1
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1answer
227 views

Fourier transform of delta function restricted to sphere [duplicate]

I want to compute $\mathcal{F}^{-1}\{\delta(|\cdot|-1)\}(x)$, which exactly means the following computation: $$f(x) = (2\pi)^{-n/2} \int_{|\xi|=1}e^{ix\cdot\xi}\mathrm{d}\xi, \mbox{ where }~ \xi \in \...
9
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2answers
529 views

Function and Fourier transform vanish on an interval

I'm no expert on these things (and this may not be cutting edge research level; it's really motivated by this MSE question), but it seems that there are non-zero measures (and also functions (?), I ...
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0answers
77 views

Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)

How can I prove the following inequality about the Fourier transform? $$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
3
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2answers
362 views

Where to find a table of fair Fourier transforms? [closed]

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me. For ...
5
votes
1answer
126 views

Harmonic analysis on nilpotent Lie groups and the Campbell-Hausdorff formula

I am trying to understand the non-commutative analysis for nilpotent Lie groups, so I've been reading Corwin's and Greenleaf's book on the representation theory of nilpotent groups and going through ...
4
votes
1answer
176 views

Does the Fourier transform preserve the separation property?

The space of Schwartz functions on the plane is denoted by $\mathcal{S}$. The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively. ...