**8**

votes

**2**answers

299 views

### Is the Fourier transform of $e^{-|x|^n}$ positive?

Let
$$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$
Is $\Phi$ positive everywhere in $\mathbf{R}^n$?
Could someone helps me answer this question or gives a reference for it? Thanks.

**2**

votes

**1**answer

210 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 ...

**8**

votes

**2**answers

518 views

### Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n ...

**0**

votes

**0**answers

47 views

### Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...

**1**

vote

**1**answer

113 views

### harmonic balance method for a 2-mass 3-spring system [closed]

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...

**1**

vote

**1**answer

58 views

### Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by
$$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...

**2**

votes

**0**answers

95 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
...

**0**

votes

**0**answers

36 views

### Supremum distribution of band-limited functions with random spectrum

Consider the properties of band-limited functions $f_N:[-\pi,\pi]\to\mathbb{R}$ defined through their Fourier series $f_N(x)=\sum_{n=-N}^N c_n e^{inx}$ where $c_n=a_n+i b_n$ and both ...

**5**

votes

**0**answers

116 views

### Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm.
A nonclassical polynomial of degree $d$ is a ...

**10**

votes

**0**answers

189 views

### Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...

**9**

votes

**2**answers

549 views

### Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...

**-1**

votes

**1**answer

139 views

### separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...

**1**

vote

**2**answers

237 views

### Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...

**1**

vote

**0**answers

96 views

### Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: ...

**2**

votes

**1**answer

59 views

### What are the known conditions for the log of the Fourier transform of a 2D real discrete signal to have no branch cuts?

Suppose we sample a real 2D signal, f(x,y), at N by N evenly spaced points in x and y. Then we compute the Fourier transform of the sampled signal, F(u,v), and then take the log of F. There will be a ...

**13**

votes

**3**answers

428 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

**1**

vote

**1**answer

116 views

### A metric on the set of BV functions, is it mentioned/studied in literature?

I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$.
Given any $x,y \in ...

**3**

votes

**0**answers

135 views

### Proof without distributions

I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...

**2**

votes

**1**answer

100 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any ...

**2**

votes

**0**answers

69 views

### Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind
$$
I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...

**8**

votes

**1**answer

286 views

### Maximal $L_1$ norm of Fourier Transform of a Subset

Let $A_n$ be the following $n\times n$ matrix: $(A_n)_{i,j}= \frac{1}{\sqrt{n}}\omega_n^{i\cdot j}$ for all $0 \le i,j <n$, where $\omega_n=e^{\frac{2\pi i}{n}}$.
I want to understand how $A_n$ ...

**2**

votes

**0**answers

92 views

### Error term for a Fourier integral

There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable,
$$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$
So it should be that
...

**10**

votes

**2**answers

279 views

### Bounding exponential sum with square roots

It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...

**2**

votes

**1**answer

111 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 ...

**8**

votes

**2**answers

525 views

### Can exponential sums be small on a whole interval?

This is almost certainly routine to an analyst, so forgive me in advance.
Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i ...

**3**

votes

**2**answers

169 views

### Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; ...

**3**

votes

**1**answer

80 views

### Spaces $C^\infty(\mathbb T^n\times \mathbb R^n)$, $C^\infty_0(\mathbb T^n\times \mathbb R^n)$ and $\mathscr{S}(\mathbb T^n\times \mathbb R^n)$? [closed]

Is there any characterization of the space $C^\infty(\mathbb T^n\times \mathbb R^n)$ that I can take as a definition of it?
I assume it would be something like this:
$$C^\infty(\mathbb T^n\times ...

**1**

vote

**0**answers

100 views

### Looking for some “nontrivial” examples of pseudodifferential operators/symbols

I'm reading up on $\Psi DO$'s and trying to find some examples of symbols that are not quite so trivial.
Obviously, the first example of a symbol that most people talk about is just a polynomial in ...

**0**

votes

**1**answer

80 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 ...

**2**

votes

**1**answer

66 views

### Locality of homogeneous pseudo-differential operator

Let $P$ be a polynomial in several variables, and let $P(D)$ be the corresponding differential operator. Obviously, $P(D)$ is a local operator, in the sense that I need only to know the function $u$ ...

**0**

votes

**0**answers

34 views

### Wiener amalgam space $W(\mathcal{F}L^{2}, L^{1}) \subset L^{1}$?

(I have asked this question on SE but could not get any answer and hope this is o.k for MO)
Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= ...

**2**

votes

**0**answers

90 views

### Swapping sums and integration for a kernel in Fourier space (the non absolutely convergent case)

Under what conditions on $c_{r}^{m}$
does
$$\int_0^{2\pi} k(p,q)\exp(-inq)dq=\sum_{r=0}^{\infty}c_{r}^{m}\exp\left(-imp\right) \text{ in } L^2_{per}$$
hold for
...

**0**

votes

**0**answers

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 ...

**7**

votes

**1**answer

379 views

### Proof of a Fourier pair with Bessel functions?

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

**2**

votes

**1**answer

77 views

### Elaboration of a certain section of a paper by Thanigasalam

In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...

**1**

vote

**2**answers

278 views

### Derivative of Band-limited functions [closed]

I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy
$$\int_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means ...

**0**

votes

**1**answer

120 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}}} ...

**4**

votes

**0**answers

266 views

### Localization arguments in the paper 'the proof of $l^2$ decoupling conjecture'

I am currently reading Jean Bourgain and Ciprian Demeter's 2015 paper The proof of the $l^2$ decoupling conjecture and would appreciate some help in understanding localization argument used in that ...

**4**

votes

**1**answer

301 views

### What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?

Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...

**2**

votes

**2**answers

163 views

### Composition operators on fractional-order (periodic) Sobolev spaces

(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...

**3**

votes

**1**answer

173 views

### Fourier Transform of Eisenstein Series - Sum of Divisors or Ramanujan Sums?

I am stuck on this computation of the Fourier coefficients of Eisenstein series. For $\Gamma = SL(2, \mathbb{Z})$ and $\Gamma_\infty = \left\{ \left( \begin{array}{cc} 1 & m \\ 0 & 1 ...

**1**

vote

**1**answer

71 views

### A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ...

**1**

vote

**1**answer

251 views

### The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator

Consider a general bilinear multiplier operator:
$$
T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, ...

**7**

votes

**2**answers

362 views

### Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?
$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$
Here, $|x|$ denotes the pointwise absolute ...

**2**

votes

**0**answers

61 views

### Is this similarity to the Fourier transform of the von Mangoldt function real? [duplicate]

This question proposed in SEM but no answer and it's interesting to know connection
between Fourier analysis and number theory .
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = ...

**3**

votes

**0**answers

66 views

### Fourier coefficients of positive polynomials

Let $p(x) \geq 0$ be a positive polynomial on the hypersphere ($x \in S^{n-1}$) satisfying $\int_{S^{n-1}} p(x) = 1$. Writing $p(x) = \sum_{j=0}^s p_j(x)$ where $p_j(x) = \sum_m p_{jm} s_{jm}(x)$ with ...

**0**

votes

**1**answer

182 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 ...

**8**

votes

**2**answers

776 views

### Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?

**4**

votes

**0**answers

429 views

### Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...

**0**

votes

**0**answers

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 ...