Questions tagged [fourier-analysis]

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

Filter by
Sorted by
Tagged with
6 votes
0 answers
193 views

Product of distributions under wavefront set condition is zero

Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\...
Ceka's user avatar
  • 501
14 votes
1 answer
613 views

The first case of the strong Littlewood conjecture

Let $A$ be a set of $n$ integers and consider the quantity: $$\int_{0}^1 \left| \sum_{a \in A} e^{2\pi i a x} \right|dx. $$ The (now solved) Littlewood conjecture is the claim that this quantity is ...
Mark Lewko's user avatar
  • 11.7k
2 votes
0 answers
214 views

Fourier transform of Dirac delta distribution

Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$ $$ V(...
Guido Li's user avatar
5 votes
1 answer
298 views

The discrete Fourier transform's Gaussian-like eigenvector

I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue? $\begin{bmatrix} 2 & 1 & 0 & 0 & \cdots & 0 &...
bobuhito's user avatar
  • 1,537
2 votes
1 answer
138 views

The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$

Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
zoran  Vicovic's user avatar
1 vote
2 answers
626 views

Decay of the Fourier transform of a non-differentiable function

It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
Tony419's user avatar
  • 401
1 vote
1 answer
170 views

Improving the intuition for the 2d fourier transform [closed]

As far as I understand, the 2d fourier transform is calculated as following: ...
dmmpie's user avatar
  • 111
1 vote
0 answers
54 views

Fourier spectrum of two isomorphic functions

Let $p$ be some prime. Let $f \colon \mathbb{Z}_p \to \{0,1\}$ be some function, and define the function $g \colon \mathbb{Z}_p \to \{\pm 1\}$ as $g(x) = (-1)^{f(x)}$. What can be said about Fourier ...
Woka's user avatar
  • 11
-3 votes
1 answer
98 views

Asking for reference about a relation related to Fourier transform [closed]

Sorry for the not-perfect question. I am asking for a reference for the following relation: $$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$ Could ...
Mr. Proof's user avatar
  • 159
6 votes
2 answers
324 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
Ali's user avatar
  • 4,045
6 votes
0 answers
193 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
Tutukeainie's user avatar
0 votes
1 answer
206 views

Constant in Amrein-Berthier uncertainty principle

Let $S,\Sigma$ in $\mathbb{R}^d$ be finite measure set. The Amrein-Berthier uncertainty principle states that there exists $C=C(S,\Sigma)>0$ such that for all $f\in L^2(\mathbb{R}^d)$, $\int_{\...
Chris's user avatar
  • 301
3 votes
1 answer
217 views

A sharp estimate for an oscillatory integral with a simple phase

Let $\alpha>1$ not necessarily an integer, and let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\...
Medo's user avatar
  • 676
4 votes
1 answer
239 views

How to unperiodise a function

We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by $$ \begin{align} F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
spaceman's user avatar
  • 575
2 votes
0 answers
52 views

Approximating an infinite family of holomorphic functions by polynomials in relative error

I think I just proved a theorem I haven't found in the literature, and I think it must generalize. I therefore have two questions. First, if this is in the literature, what is it called? Second, what ...
Sébastien Loisel's user avatar
2 votes
2 answers
267 views

$H^s$ norm of non-integer power of functions

Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $. My ...
Chushamm's user avatar
  • 105
2 votes
1 answer
233 views

An observation of Nazarov on $\Lambda(p)$-systems

I have been reading this note of F. Nazarov "Summability of large powers of logarithm of classic lacunary series and its simplest consequences (1995)", available here https://users.math.msu....
S117's user avatar
  • 21
0 votes
1 answer
388 views

A problem of Fourier transform and Hölder condition

Suppose that $f$ is continuous on $[0,1]$. Thus, $f\in L^1(\mathbb{R})$ and its Fourier transform exists, as $$ \hat{f}(\xi) := \int_\mathbb{R} e^{-2\pi i x \xi} f(x)dx, $$ which can also be written ...
Watheophy's user avatar
  • 419
5 votes
0 answers
86 views

Lower bound for nonconventional ergodic averages in finite fields

Let $p$ be a sufficiently large prime number and $f\colon\mathbb{F}_{p}\to\mathbb{R}_{\geq 0}$ be a function bounded by 1 such that the average of $f$ over the finite field $\mathbb{F}_{p}$ is at ...
Wenbo Sun's user avatar
0 votes
0 answers
69 views

Reverse Hölder for possibly singular matrix weights

I'm working on some nonlinear partial differential equations and I have been led to the following puzzle. Let $W(x)$ be a symmetric positive semidefinite-valued function of $x \in \mathbb{R}^d$ (a &...
Sébastien Loisel's user avatar
2 votes
0 answers
192 views

Fourier transform of unbounded linear operator

I am trying to construct Fourier transform of a family of unbounded linear operators. Here is the construction. Fix $H$ a Hilbert space. Let $D\subset H$ be a fixed dense subset. Denote by $L(H)$ some ...
Ken.Wong's user avatar
  • 493
0 votes
0 answers
122 views

An exercise about sum-product estimate

I am struggling with 1.11 exercise from the George Shakan "Discrete Fourier Transform". Let $A \subset \mathbb{Z}/q\mathbb{Z}$ be any set not containing zero with $|A|>\sqrt2q^{5/8}$. ...
Sei's user avatar
  • 11
4 votes
0 answers
68 views

Local energy estimate in a semiclassical regime

Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
J.Mayol's user avatar
  • 489
1 vote
0 answers
82 views

Interpolation between projective and injective spaces

Suppose $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $X$ is a Banach space and $L_p(\Omega;X)$ be the corresponding Bochner space for $0<p\leq\infty.$ Is it true that the complex ...
A beginner mathmatician's user avatar
2 votes
0 answers
155 views

Product of Heavisides: calculus vs Fourier transform vs wavefront set

I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
Evangeline A. K. McDowell's user avatar
0 votes
1 answer
217 views

Fourier transform of a Radon measure [closed]

Let $\mu$ be a Radon measure on $\mathbb R^d$ with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its ...
Bazin's user avatar
  • 15k
0 votes
0 answers
70 views

Alternative to the Sampling Theorem / Invertible transform with sampling criteria

I seek a transform $T$ that operates on real-valued $x(t)$, that Is perfectly invertible Has discrete counterpart with continuous reconstructor Provides conditional reconstruction guarantees ...
OverLordGoldDragon's user avatar
2 votes
1 answer
188 views

Non-Fourier complete orthogonal basis?

The Fourier Transform (FT) Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero Is invertible: info-preserving, has inverse function Is energy-...
OverLordGoldDragon's user avatar
1 vote
1 answer
190 views

Explanation of a step in a work by C. E. Kenig and A.D. Ionescu

I am studying the work Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
Mr. Proof's user avatar
  • 159
1 vote
2 answers
151 views

Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?

Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral. $$ I = \int_{\mathbb{R}} \int_{\mathbb{...
Johnny T.'s user avatar
  • 3,537
3 votes
0 answers
198 views

Finding minimum of function using its Fourier transform

Say we have a function $f:\mathbb{R}^n \to \mathbb{R}$ such that $$ f(x) = \int_{\mathbb{R}^n} \psi(k) e^{ik \cdot x} dk $$ where $\psi$ is the Fourier dual of $f$. Say we further know that $\psi >...
spaceman's user avatar
  • 575
1 vote
1 answer
309 views

A particular commutator of the discrete Fourier matrix

For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
ABB's user avatar
  • 3,898
7 votes
1 answer
587 views

A lecture by Rudin

Is it available a written version of this lecture by Rudin on the relation between Fourier analysis and the birth of set theory? https://youtu.be/hBcWRZMP6xs If not Rudin himself, maybe someone else ...
Alessandro Della Corte's user avatar
2 votes
0 answers
149 views

An oscillatory integral

Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates? \begin{...
Dapao Zhang's user avatar
2 votes
1 answer
462 views

Shift-invariant spaces

We can define a shift-invariant space as $$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$ where convergence of the series is taken to be in $L^2(\...
AKG's user avatar
  • 49
7 votes
0 answers
141 views

Inequality of product of discrete cosines

Let $k,a,b,c$ be odd positive integers. Consider the following inequality: $$ \sum_{x,y \in [k]} \cos^a\bigg(\frac{2\pi}{k}\cdot x\bigg) \cdot \cos^b\bigg(\frac{2\pi}{k}\cdot y\bigg) \cdot \cos^c\bigg(...
Tamir Dror's user avatar
4 votes
0 answers
79 views

Does this sequence of functions converge in a distributional sense?

Let $f\in W^{1,12/5}(\mathbb{R}^3)$ (time-independent), let $K^{\epsilon}$ be a uniformly in $\epsilon$ bounded sequence in $L^{1}\cap L^{7/5}(\mathbb{R}^3)$ and let $$\tilde{K}^{\epsilon} := K^{\...
Jakob Möller's user avatar
3 votes
2 answers
331 views

A Sobolev embedding theorem for functions on spheres

$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: $$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
Dapao Zhang's user avatar
1 vote
1 answer
424 views

Fourier transform of the fractional Poisson kernel

Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows: $$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
Student's user avatar
  • 601
1 vote
0 answers
150 views

Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
Zhan's user avatar
  • 63
12 votes
0 answers
236 views

Pointwise convergence of trigonometric series

$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, ...
xXF's user avatar
  • 221
5 votes
0 answers
202 views

Majorizing $|\{\alpha\}-1/2|$ by trigonometric polynomials

Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$...
H A Helfgott's user avatar
  • 19.3k
3 votes
1 answer
654 views

Equivalent action of convolution of directional derivative

I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented. Let $k*f(x)...
Mirar's user avatar
  • 350
8 votes
1 answer
977 views

Who introduced the discrete Fourier transform?

I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...
AChem's user avatar
  • 803
0 votes
1 answer
603 views

Fast decaying Fourier coefficients for indicator function

Let $0 \leq a < b \leq 1$. I wanted to compute the Fourier series expansion of the indicator function $f = \chi_{[a, b]}$ of the interval $[a, b]$, as $$ f(x) = \sum_{k\geq 0}a_k e(kx). $$ My ...
Melanka's user avatar
  • 577
2 votes
0 answers
1k views

Stein's book on harmonic analysis

My background : I am a Math PhD student. I will most probably work in harmonic analysis on Euclidean spaces. I am a fan of Folland's Real analysis and I have thoroughly studied first 8 chapters of ...
risefrominfinite's user avatar
0 votes
0 answers
113 views

A weaker condition on Fourier coefficients for boundedness of the function

Let $f : \mathbb{S}_1 \rightarrow \mathbb{C}$ be a square-integrable fnction and let $(\widehat{f}_k)_{k \in \mathbb{Z}}$ be its Fourier-coefficients. It is very well known, that the condition $\sum\...
Tardis's user avatar
  • 1,077
1 vote
0 answers
73 views

A problem arising from Wiener-Levy theorem on the real line

Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
ABB's user avatar
  • 3,898
0 votes
0 answers
65 views

Extracting the point mass measure of some type of positive measures

Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals. Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
ABB's user avatar
  • 3,898
3 votes
0 answers
159 views

The inversion formula for the square root of a positive function

Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
ABB's user avatar
  • 3,898

1
3 4
5
6 7
30