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**11**

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416 views

### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...

**9**

votes

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148 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

**7**

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340 views

### Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...

**6**

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**0**answers

92 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 ...

**5**

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135 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 ...

**4**

votes

**0**answers

121 views

### Decay to stationarity in a random walk on the hypercube

Let $\mu$ be a probability distribution on $\mathbb F_2^n$. Consider the random walk $X_0,X_1,\ldots$ defined by
$$\begin{aligned} X_0 &= 0\\X_{i+1}&=X_i + Z,\end{aligned}$$
where $Z\sim \mu$ ...

**4**

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3k 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 ...

**4**

votes

**0**answers

96 views

### Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you.
This post is related to zeros of partial sum of Taylor series of $e^x-1$.
Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...

**4**

votes

**0**answers

481 views

### Inverse Fourier Transform involving a Bessel Function, Exponential, and Power

I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...

**3**

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57 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 ...

**3**

votes

**0**answers

131 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 ...

**3**

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**0**answers

101 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) =
...

**3**

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171 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

**3**

votes

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250 views

### How is the deconvolution of a fat gaussian from a polynomial derived?

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let:
$\begin{eqnarray}
p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\
G(x,y) &=& ...

**3**

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105 views

### Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...

**3**

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206 views

### Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?

Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...

**3**

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401 views

### Computing a 2D Fourier transform on a disk

I am looking for a reliable and fast way of evaluating an integral like
$$
F(r, \phi)= \int_0^1 \int_0^{2\pi} f(\rho, \theta) e^{2\pi i \rho r \cos(\theta - \phi)}\rho\, d\theta\,d\rho,
$$
where $f$ ...

**3**

votes

**0**answers

120 views

### Meaning of fractional Fourier transform with imaginary iteration count

(I'm reposting this from math.stackexchange as I didn't get an answer there and thought it might be "advanced" enough for this site.)
As one may know, the Fourier Transform
$$F[f](\nu) = ...

**2**

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**0**answers

170 views

### Can we do better than zero padding of FFT?

My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...

**2**

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134 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 ...

**2**

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111 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 ...

**2**

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367 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 ...

**2**

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70 views

### request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.
Hardy, G.H.
I am not ...

**2**

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**0**answers

130 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**2**

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206 views

### Positive Fourier coefficients for a function $f:\{+1,-1\}^n \to \mathbb R$

This is from my research in computer science where the Fourier transform over $GF(2)^n$ is a tool to study functions on the Boolean hypercube.
For example, the majority function on 3 variables is ...

**2**

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**0**answers

262 views

### Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...

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368 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 ...

**1**

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102 views

### If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4.
However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...

**1**

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78 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 ...

**1**

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68 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) ...

**1**

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115 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

**0**answers

55 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 ...

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34 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 ...

**1**

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170 views

### Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...

**1**

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**0**answers

104 views

### Inversion of Fourier transform of a multivariate gamma distribution in polar form?

Let $\mathbb{S}^{N-1}$ be the unit sphere in $\mathbb{R}^N$ under the Euclidean norm $||\cdot||$. Let $\mu$ be an infinitely divisible Borel measure. If there exists a finite measure $\alpha$ on ...

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90 views

### Cross-correlation of two functions which are not fixed

I am trying to cross-correlate two functions, but one of which is changing each 'step' of the cross-correlation.
I want to cross-correlate T(f) and ...

**1**

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117 views

### Mellin transform of time-shifted function

The Mellin transform of a function $f(x)$ can be written as
$$
\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx
$$
Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...

**1**

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63 views

### Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...

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131 views

### Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the
Fourier transform of a random variable $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

**1**

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264 views

### Relationship between Fourier series & DFT

Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.
However, isn't there a flaw? When I integrate over the continuous ...

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152 views

### How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put
$$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$
and
...

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189 views

### Convergence of the inverse image of a sequence of Fourier transforms

The Fourier Transform is in general terms a continuous mapping, as well as it's inverse when it exists. Is there any abstract result that translates convergence of the transforms of a sequence of ...

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320 views

### Theta functions and Fourier transforms

Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...

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62 views

### Fourier transform in Lorentz spaces

Denote by $E$ the class of nonnegative, even functions on $\mathbb{R}$, monotone
decreasing to 0 on $\mathbb{R}_+$.
In here, Y. Sagher stated the following proposition about Fourier transform of ...

**0**

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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 ...

**0**

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66 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 ...

**0**

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73 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 ...

**0**

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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 ...

**0**

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132 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**

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37 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)= ...