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.
1,600 questions
3 votes
1 answer
137 views
$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions. Mainly ...
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4 votes
0 answers
206 views
Traces mixing tensor products of Fourier coefficients on finite symmetric groups
Let $G$ be a finite (symmetric) group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$. Let $f:G\to \{0,1\}$ be a Boolean function on $G$ ...
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0 votes
1 answer
207 views
Pointwise estimate for a multilinear operator
Let $0<\alpha<1$, and let $$f=\frac{\chi_{[0,1]}}{x^{a}},\quad 0<a<1,$$ $$g=\chi_{[0,1]},$$ $$h(x)=\frac{\chi_{[0,1]}}{|x-1|^{b}},\qquad 0<b<1.$$ I have a multlinear operator on $L^{...
- 790
0 votes
0 answers
105 views
Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
- 75
4 votes
1 answer
383 views
Bounding the largest Fourier coefficient of $f$ minus a class function on symmetric group $S_n$
Let $G=S_n$ be a symmetric group. A class function on $G$ is any $g:G\to\mathbb C$ that is constant on conjugacy classes, i.e. $g(xy)=g(yx)$ for any $x,y\in G$. Let $\mathcal C$ denotes the set of ...
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1 vote
0 answers
77 views
Space of interpolating functions with constraints on interpolation
Disclaimer: I am a first year mathematics student who is trying to write an applied math paper, so my question might seem trivial. Definitions: Let $N \in 2 \mathbb{N}$ and $u \in \mathbb{R}^N $ be a ...
1 vote
0 answers
165 views
English translation of van der Corput's 1939 proof for three-term progressions in primes
I've seen van der Corput's paper "Über Summen von Primzahlen und Primzahlquadraten" [Mathematische Annalen 116 (1939), 1–50] referenced here and there. It proves that there are infinitely ...
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1 vote
0 answers
64 views
Asymptotics of the projection on the kernel of the coboundary operator on infinite cubical lattice
Consider the infinite cubical lattice $\mathbb{Z}^d \subset \mathbb{R}^d$ as a polyhedral cell complex and write $C^k_{(2)}(\mathbb{Z}^d)$ for the Hilbert space of real oriented $\ell_2$ $k$-cochains ...
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1 vote
0 answers
93 views
Can one approximate every continuous function on an abelian group simultaneously in $L^{1}$ and $L^{2}$ by functions with compact Fourier transform?
Let $G$ be a locally compact abelian group and let $\widehat{G}$ denote its dual group, and let $L^{1}(G)$ and $L^{2}(G)$ be defined via the Haar measure. Consider the space $$ K^{1}(G):=\{ h\in L^{1}(...
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3 votes
0 answers
248 views
About bilinear arguments of the Fourier restriction
A bilinear argument of the Fourier restriction (not bilinear Fourier restriction) can be described follows: This excerpt is from the book of C. Demeter, Fourier restriction, decoupling, applications. ...
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1 vote
0 answers
170 views
Reference needed for some theta function upper bound estimates
Let $(V=\mathbb{R}^n,\langle,\rangle)$ be a positive definite real quadratic space. Let $x=(x_1,\ldots,x_n)\in V$ be the usual local chart and let $P(x)=f_1(x_1)f_2(x_2)\cdots f_n(x_n)$ be a "...
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0 votes
0 answers
106 views
$\ell^2$ decoupling parabolic rescaling
Suppose $0<\delta\leq\sigma<1$. For simplicity, assume $\sigma,\delta$ are dyadic and their ratio is clean. Let $T_u(x,y)=(\sqrt{\sigma}(x+2uy),\sigma y)$, a linear map, where $[u,u+\sqrt{\sigma}...
- 111
2 votes
0 answers
223 views
Gauss Sums over Conjugacy classes of $M_n(F_q)$
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\Tr{Tr}$Let $V := M_n(F_q)$ be $n \times n$ matrices over a finite field $F_q$. Let $X$ be a conjugacy class in $V$ whose characteristic ...
- 621
2 votes
0 answers
70 views
Orthogonal feature functions for model fitting on unbounded intervals (like Chebyshev polynomials for bounded intervals)
This question is rummaging around in my head for quite some time. I will start with exposition on "model fitting" and then explain how Chebyshev polynomials are perfect on bounded intervals ...
- 343
6 votes
1 answer
448 views
Functions with compactly supported Fourier transform
The following is a repeat from the following question on MathStack Exchange. I am just hoping to have more success here. This is a follow-up from that question. The question is this: I want to ...
1 vote
0 answers
173 views
Diffusion approximation to the Schrödinger equation with low frequency initial data
I'm trying to figure out if the solutions to the Schrödinger equation and the diffusion equation are close for low-frequency/smooth initial data. I.e., I want to bound $$ \|u(t,x) - v(t,x)\|_{L^\infty}...
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0 votes
0 answers
324 views
Boundedness of an explicit function in mixed $L^p$ norm
For $j=1,2,3$, let $\alpha_{j},\beta_{j}\in (0,\frac{1}{2})$, $1<p_{j}<2$, and let $\theta_{j}=\beta_{j}-\alpha_{j}$. Assume also that $\theta_{j}p_{j}<1$, $j=1,2,3$ and consider the region $$...
- 790
1 vote
0 answers
180 views
An inequality for the sum of integrals in Bourgain's paper
I'm now (quickly) reading Bourgain's paper: J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205-224 (https://www.ams.org/journals/jams/2017-30-...
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3 votes
0 answers
293 views
Singularities of exponential series $\sum a_n \exp(i \sqrt{n} z)$
More generally I'm interested in series of the form $f(z):=\sum a_n \exp(i n^{1/d} z)$ that have analytic continuation, and finding restrictions on where their singularities can be. The motivation is ...
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1 vote
0 answers
217 views
Coordinate-free proof for exterior derivative in Fourier space
In this comment, Terry Tao points out that the exterior derivative of a $k$-form $\omega$ on $\mathbb{R}^n$ may be motivated from a Fourier analytic perspective, as satisfying $$ d \omega = (i \xi \...
19 votes
0 answers
607 views
When is the series $\sum_{n=1}^\infty \frac1{a n^2 + b n + c}$ rational?
The question was originally posted on Math Stack Exchange, but no answers were received. Even starting a bounty didn’t get any responses Here Let $a,b,c$ be integers such that $a\neq 0$ and $$ a n^2 +...
- 199
5 votes
1 answer
355 views
Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following statements of the Wiener-Tauberian theorem: Version 1: Every proper closed ideal in $L^1(\mathbb R)$ is contained in a maximal ...
- 322
8 votes
1 answer
741 views
Bourgain and Demeter's Proof of $\ell^2$ decoupling Lemma 3.3
I posted in stackexchange, but then was told that overflow may be a more appropriate place to ask. I'm currently reading Bourgain and Demeter's "The Proof of the $\ell^2$ Decoupling Conjecture”, ...
- 111
8 votes
1 answer
420 views
Discrete Fourier transform converges to torus and real Fourier transforms in what sense?
On the group $\mathbb Z/N\mathbb Z$, we have characters $\chi_j(k) = e^{2\pi i j k/N} \in \text{Hom}_{\mathbf{Grp}}(\mathbb Z/N \mathbb Z, \mathbb T)$, and so equipping with counting measure $\#$, we ...
- 1,235
5 votes
2 answers
366 views
Sharp $L^{\infty}$ Bernstein inequality for bandlimited functions
I'm interested in finding a proof in the literature for the following result: Let $f(x)$ let be a smooth function $\mathbb{R} \rightarrow \mathbb{R}$ such that $\hat{f}$ exists and is supported on $[-...
1 vote
0 answers
72 views
Approximate spectral projections $\chi(\sqrt{-\Delta} - \lambda)$ of Laplace-Beltrami eigenfunctions on Riemannian manifolds
I have been working my way up to reading Restrictions of Laplace-Beltrami Eigenfunctions to Submanifolds by Berq, Gerard, and Tzvetkov for the past two weeks by reading Chapters 0, 1, 3, and 4 from C. ...
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0 votes
0 answers
36 views
Does replacing "finite relative extrema" with "finite strict extrema" in Dirichlet's convergence theorem maintain Fourier series convergence?
In Stein and Shakarchi's book: Fourier Analysis: An Introduction, page 128, the author said that ``Dirichlet's theorem states that the Fourier series of a real continuous periodic function $f$ which ...
- 329
2 votes
1 answer
246 views
Is a continuous function with finitely many strict extrema on a closed interval necessarily a bounded variation function? convergence of Fourier
We know that if $f$ is a continuous function on $[a,b]$ and has finite number of relative maxima and minima on $[a,b]$, then $f\in \mathrm{BV}([a,b])$. In fact, under this condition, we can prove that ...
- 329
1 vote
0 answers
78 views
Reference request: $C_0$ version of Wiener's Tauberian theorem
Norbert Wiener has written 2 version of his Tauberian theorem for translations of functions: one for $L^1$ and one for $L^2$. I am wondering whether a similar statement exists for uniformly continuous ...
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2 votes
0 answers
80 views
Inhomogeneous version of $L^\infty_x L^2_t$ local smoothing estimate for general dispersive symbols?
Consider the group $e^{t (i \partial_x)^{2j+1}}$. We have a well-known homogeneous local smoothing estimate $$ \| \partial_x^j e^{t (i \partial_x)^{2j+1}} u_0 \|_{L^\infty_x L^2_t} \leq C \|u_0\|_{L^2}...
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1 vote
1 answer
225 views
Let $f\in C_c(\mathbb R)$. Then $\exists$ $\{f_n\}$ in $C_c^3(\mathbb R)$ s.t. $f''_n\to f$ pointwise and $\{f''_n\}$ is uniformly bounded
Let $f\in C_c(\mathbb R)$. Then show that there exists a sequence $\{f_n\}$ in $C_c^3(\mathbb R)$ s.t. $f''_n\to f$ pointwise and $\{f''_n\}$ is uniformly bounded.
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0 votes
1 answer
205 views
Broad estimate in the proof of Bourgain's Decoupling inequality
As I read Guth's decoupling lecture 9, Decoupling Lecture 9, I didn't understand the proof of Broad estimation as described below. How to derive the first inequality in the "Broad Estimate"? ...
- 819
2 votes
1 answer
206 views
Question about B. Connes proof of decay of $l^2$ shells in the spherical summation problem for multiple trigonometric series
The spherical uniqueness theorem for multiple trigonometric series states that if: $$ \sum_{|n|< R} a_n e^{i \langle n, x \rangle } \to 0 $$ for every $x \in \mathbb{T}^d$ then each $a_n$ is zero. ...
- 21
4 votes
1 answer
343 views
What does a square-function estimate tell us in harmonic analysis?
I’ve noticed in harmonic analysis that many researchers are interested in the so-called square-function estimate (see arXiv:1906.05877 or arXiv:1909.10693). However, what I’m not clear about is what ...
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2 votes
1 answer
265 views
Absolute integrability and Fourier transform
In engineering we are mainly interested in linear time-invariant (LTI) systems which are bounded input, bounded output (BIBO). It's easy to prove that BIBO condition is equivalent to $$\int_{-\infty}^{...
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1 vote
0 answers
213 views
The Fourier transform of a $C_c^1$ function is in $L^1$: reference?
Let $f$ be a compactly supported $C^1$ function. Then $\widehat{f}$ is in $L^1$. I know two proofs of this result: (a) it is a weaker version of a result by Bernstein, which you can find, together ...
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10 votes
0 answers
321 views
The discrete uncertainty principle and entropy
If $G$ is a finite abelian group, $f : G\to {\bf C}$ is a function, and $\hat f$ is its Fourier transform, then $$|{\rm supp}(f)| \cdot |{\rm supp}(\hat f)| \ge |G|.\tag 1$$ This is the discrete ...
- 1,251
1 vote
1 answer
242 views
When is a function of exponential type $\sigma$ a Fourier transform?
If $\phi:\mathbb{R}\to \mathbb{C}$ is an integrable function supported on $[-1,1]$ (say), then its Fourier transform is an entire function $F$ of exponential type $2\pi$. However, the converse is not ...
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4 votes
0 answers
164 views
Example of Fourier transform of $L^1$ functions with certain regularity and growth condition
It is well-known that, vaguely speaking, for nice functions, the momentum operator corresponds to differentiation via the Fourier transform. However, I need some help in resolving the following ...
- 1,110
1 vote
0 answers
92 views
What will happen if the exponent index in Gagliardo-Nirenberg inequality is chosen less than $ 1 $?
The well-known Gagliardo-Nirenberg inequality in bounded domains are given as the following theorem. Let $\Omega \subset \mathbb{R}^n$ be a measurable, bounded, open and connected domain satisfying ...
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0 votes
0 answers
119 views
Decay of the discrete Fourier transform
Let $G$ be a finite abelian group of order $n$. Let $\hat{G}$ be the dual group of $G$, i.e., the group of characters on $G$. For $f:G\to \mathbb{C}$, we define its Fourier transform $\hat{f}:\hat{G}\...
- 448
2 votes
1 answer
352 views
Fourier transform of functions which is compactly supported
Can one give me an explicit proof that if the Fourier transform of $f$ is compactly supported in a region, then $f$ is essentially constant on the dual region,i.e., $f \sim 1 $ on the dual region,i.e. ...
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2 votes
1 answer
234 views
On existence of a suitable density for a sequence
Suppose that $\lambda_1<\lambda_2<\ldots$ is a sequence of positive real numbers such that $$|\{n\in \mathbb N\,:\, \lambda_n \leq \lambda\}| \leq \sqrt{\lambda} \quad \forall\, \lambda>>1....
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1 vote
0 answers
192 views
Choosing phases to minimize the $L^\infty$ norm of a trigonometric polynomial
Let $a,\varphi\in \mathbb{R}^N$. Consider the trigonometric polynomial $$f(\phi;t):=\sum_{n=1}^Na_ne^{i \phi_n} e^{i nt}.$$ My question is: what can be said about the quantity $$\omega(a)=\inf_{\phi\...
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1 vote
1 answer
228 views
Continuous large sieve inequality
In Tao's blog we can find this exercise: Let $[M,M+N]$ be an interval for some $M \in {\bf R}$ and $N > 0$, and let $\xi_1,\dots,\xi_J \in {\bf R}$ be $\delta$-separated. For any complex numbers $...
- 13
0 votes
0 answers
114 views
Can Parseval's theorem be extended to the "nested" Fourier series representation of $f(x)$?
This is a cross-post of this question I posted on Math StackExchange a couple of months ago that has not yet received any answers or even comments (other than a single comment of my own). Assuming ...
- 1,221
4 votes
2 answers
1k views
Does this function exist?
Is it possible to construct a compactly supported smooth function $\phi\geq 0$ such that $\operatorname{Supp} \phi\subseteq[1/2,2]$ and $\phi(t)+\phi(t/2)=1$ for all $t\in [1,2]$?
- 1,970
3 votes
0 answers
106 views
Paley-Littlewoord/paraproduct theory: Frequency of multiplication with respect to the japanese bracket
I'd like to prove (or disprove) the following bound for high frequencies $j$ of the dyadic decomposition of $\langle z \rangle^\alpha f$, for some $\alpha > 0$: $$ | \triangle_j (\langle z \rangle^\...
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4 votes
1 answer
339 views
On optimum constant for upper density of zeros of an entire function
Let $f\in L^2(\mathbb R)$ be a not identically vanishing function that is compactly supported in the interval $(-\sigma,\sigma)$ for some fixed $\sigma>0$. For each $z\in \mathbb C$ let us define ...
- 4,189
0 votes
0 answers
99 views
What does the condition $\int_0^{d_n} f(t)e^{-2\pi i t n} \, dt = 0$ for all integers $n$ and a dense sequence $(d_n)$ imply?
According to the uniqueness theorem in Fourier analysis, for every $f \in L^1[0,1]$ that satisfies $$ \int_0^1 f(t)e^{-2\pi i t n} \, dt = 0 $$ for every $n$, one has $f=0$. Now suppose that $(d_n)_{n ...
