The fourier-transform tag has no usage guidance.

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380 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 ...

**1**

vote

**2**answers

272 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: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...

**7**

votes

**1**answer

1k views

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

Physical Motivation : Hear to these audio files S_f and P_f. S_f is Fourier partial
sum and P_f is the new reconstruction, both use spectrum only in the
region (0,4KHz) for reconstructing the ...

**10**

votes

**0**answers

357 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 ...

**7**

votes

**0**answers

320 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 ...

**5**

votes

**0**answers

79 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**

votes

**0**answers

121 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

77 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

388 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**

votes

**0**answers

117 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**

votes

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87 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) =
...

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votes

**0**answers

144 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

**0**answers

216 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) &=& ...

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votes

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102 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 ...

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181 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**

votes

**0**answers

109 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**

votes

**0**answers

63 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 ...

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votes

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99 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 ...

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votes

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64 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 ...

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votes

**0**answers

116 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**

votes

**0**answers

143 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 ...

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votes

**0**answers

297 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 $f$ ...

**2**

votes

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195 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 ...

**1**

vote

**0**answers

109 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

47 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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311 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 ...

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vote

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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 ...

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161 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}
...

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84 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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vote

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78 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 ...

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vote

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58 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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119 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 $X$
$$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$
...

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vote

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139 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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vote

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140 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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172 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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296 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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56 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 $i=1...n$ (there are also amplitudes and phase shifts, but let's ignore those for now). I want to solve for $\vec ...

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55 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 ...

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48 views

### Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...

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65 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 ...

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95 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 ...

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119 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 ...

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34 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)= ...

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102 views

### A question on a Fourier transform

(This may sound too simple a question to be asked in MO. But it is a research problem but not an exercise, so I post it here. If it is no appropriate, I will move it to Stackexchange.)
Let $0 < a ...

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102 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 $f(x-x_0)$? ...

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131 views

### Solvable PDEs and their Green's functions

I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...

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90 views

### Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$

Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal ...

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115 views

### Min of a real-valued Fourier transform

Let $P$ be a compact, convex, symmetric, $d$-dimensional body in $\mathbb R^d$, and let $\mu$ be a (necessarily) symmetric probability measure on $P$, so that
$\mu_P(x) = \mu_P(-x)$, for all $x \in ...