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2
votes
1answer
72 views

Definite positiveness of sinc function in three dimension

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
votes
0answers
48 views

Bounds of the difference of a bounded band-limit: difference version of Bernstein inequality [on hold]

for continues signal (function) we have Bernstein inequality : $$ |{df(t)}/dt| \le 2AB\pi $$ where $A=\sup|f(t)|$ and B is Bandwidth f(t), i.e. if $$ F(\omega ) = \int\limits_{ - \infty }^\infty {...
1
vote
1answer
66 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
votes
1answer
306 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
votes
0answers
70 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 ...
0
votes
0answers
53 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 $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
1
vote
2answers
171 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
votes
1answer
73 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
votes
1answer
112 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
votes
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 \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]...
0
votes
0answers
39 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
votes
1answer
189 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
vote
1answer
70 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
vote
1answer
315 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 ...
0
votes
0answers
42 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
vote
0answers
92 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....
1
vote
0answers
40 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(...
2
votes
0answers
46 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 \...
0
votes
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(...
0
votes
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 ...
0
votes
0answers
69 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-...
4
votes
0answers
95 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, ...
1
vote
0answers
50 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 $f$ ...
1
vote
2answers
209 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 \...
10
votes
1answer
262 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 ...
1
vote
2answers
211 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)]=\...
11
votes
0answers
307 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 ...
3
votes
0answers
44 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 ...
0
votes
0answers
223 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{...
1
vote
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 ...
11
votes
0answers
262 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
votes
1answer
253 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 $$ \...
0
votes
2answers
89 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 $c_n:=\int^{\pi}_{-\pi}\exp\left(\sum_{k=0}^{\infty}a_k\cos(k\theta+\phi_k)\right)\cos(n\theta+\psi_n)d\theta$. ($...
2
votes
2answers
276 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
votes
0answers
147 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 ...
2
votes
0answers
91 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
votes
1answer
109 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)...
0
votes
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 $\...
1
vote
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
vote
1answer
201 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
votes
2answers
506 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 ...
1
vote
0answers
76 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 ...
4
votes
2answers
350 views

Where to find a table of fair Fourier transforms?

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
119 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
172 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. ...
8
votes
1answer
509 views

Fourier transform that is almost a brick wall - but why?

Let $$g(x) := \sqrt{1+x^2},$$ and $$h(x) := g^{-3/2}(x) \exp(-i2\pi g(x)).$$ I can observe that the Fourier transform $|H(f)|$ is almost flat if $|f|<1$, and $H(f)\approx 0, \; |f|>1$. This ...
1
vote
1answer
129 views

Annihilator property dual

Let $G$ be a locally compact group and $\phi$ be in $ L^{\infty}(G)$ that annihilates $I$, where $I$ is a closed ideal of $ L^1(G)$, so by duality we have: $$\int_G f(y)\phi(y)dy=0$$ for all $f\in I$....
2
votes
1answer
240 views

Decay of the Fourier transform of a compactly supported continuous BV function

If $f$ is a compactly supported bounded variation function on the real line $\mathbb R$, its Fourier transform $\widehat f$ can be estimated as $|\widehat f(\xi)| = O(|\xi|^{-1})$, $|\xi|\to\infty$. ...
3
votes
0answers
132 views

Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions

Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set \begin{equation*} {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1) \end{...
6
votes
0answers
176 views

Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$: $$ f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2}, $$ where $\frac{1}{2}<\alpha<1$, and ...