All Questions
Tagged with ca.classical-analysis-and-odes fourier-analysis
250 questions
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
8
votes
0
answers
525
views
Phase perturbations in oscillatory integrals
I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in ...
- 852
1
vote
2
answers
561
views
Convergence of squares of the moduli of partial sums of Fourier series
Let $\mu$ be a complex measure on the unit circle. The Wiener theorem says that the sequence of the Cesaro means of $|\hat\mu_n|$ has a limit. Define $p_n(z)=\sum_{k=0}^n \hat\mu_k z^k$. Then the Abel ...
- 11
6
votes
1
answer
679
views
On the Existence of Certain Fourier Series
Is there an $f\in L^{1}(T)$ whose partial sums of Fourier series $S_{n}(f)$ satisfies $\|S_{n}(f)\|_{L^{1}(T)} \rightarrow \|f\|_{L^{1}(T)}$ but $S_{n}(f)$ fails to converge to $f$ in $L^1$-norm ?
- 643
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
3
votes
1
answer
682
views
Is there any result about the uniform convergence rate of multi-dimensional Fourier series
For example in the 1-dimensional case, it is known that if f satisfies the α-Hölder condition, then
$|f(x)-(S_Nf)(x)|\le K \frac{\ln N}{N^\alpha}$
where $S_N f$ is the n-term partial sum of the ...
- 31
1
vote
2
answers
3k
views
Convergence of Fourier series in L^{\infty}-norm [closed]
As we know, for $1<p<\infty$, the Fourier series of $f\in L^{p}(T)$ converges to $f$ in $L^{p}$-norm.
But is there any results concerning the convergence of Fourier series in $L^{\infty}$-norm?
...
- 643
2
votes
0
answers
231
views
characterization of Hörmander multipliers
Denote $S$ as the space of Schwartz functions, for $v\in S'$, the space of tempered distributions, define an operator $T_v:f\in S \to f*v$. Then space of Hormander Multipliers $M^{p,q}$ can be defined ...
- 331
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 9,756
0
votes
1
answer
779
views
Inversion of Fourier Transformation
As we know, the inversion formula of Fourier transformation holds pointwise for Schwartz class.
We also have a general result concerning the inversion of Fourier transformation on locally compact ...
- 643
0
votes
2
answers
200
views
Good probability measues on $S^1$ reprented by a kernel
I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
- 1,471
0
votes
1
answer
537
views
Proving uniform bound
Hello
I want to prove that
$\lim_{h\rightarrow\infty}\left(\int_{0}^{\infty}\left(\cos ht-1\right)\underset{t}{\triangle}\left[\frac{\phi(t)\exp\left(-itx\right)}{it}\right]dt\right)=-\int_{0}^{\...
- 579
8
votes
1
answer
2k
views
Recent progress on Bochner-Riesz conjecture
Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by
$
\widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi).
$
($(1-|\xi|^2)_+^\delta$ are known as ...
- 505
7
votes
1
answer
822
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 643
3
votes
7
answers
1k
views
Continuous or analytic functions with this property of sinc function
This question is motivated by my previous post in SE (math.stackexchange.com).
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function,...
- 744
2
votes
1
answer
2k
views
Series of squared Fourier coefficients
Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is
$$
g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t}
$$
does the series
$$
\sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}?
$$
...
- 167
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k
15
votes
2
answers
3k
views
Evil Fourier Coefficients
Let $f:[0,1]\to[0,1]$ be the classical devil's staircase.
Has anybody ever computed (or studied) the fourier coefficient of $f(x)$?
Related question: is the fourier series of $f(x)-x$ normally ...
- 373
25
votes
5
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
- 923
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
4
votes
1
answer
1k
views
Decay of the Fourier transform
Suppose $f(z)$ is a function analytic in the strip $|Re(z)|\leq a$. Is the fourier transform $\hat{f}(w)=o(e^{-a|w|})$?
It seems plausible but I can't seem to prove it either.
There is similar ...
- 3,217
14
votes
1
answer
2k
views
Is the Fourier transform of 1/(1-log(1-x^2)) (supported in [-1,1]) integrable?
This question was suggested when trying to find an explicit example of a continuous function with compact support in $\mathbb{R}$ whose Fourier transform is not integrable. The existence of such a ...
- 1,981
10
votes
6
answers
6k
views
Fourier transform of (real) exponential
Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
- 101
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
3
votes
1
answer
2k
views
fourier transform of radon measure
hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
- 979
7
votes
2
answers
286
views
accelerated convergence to the mean using quadratic weights
If the sequence $x_1,x_2,\dots$ is periodic, the unweighted averages $(\sum_{i=1}^n x_i)/n$ converge to the asymptotic average of the $x_n$'s with error $O(1/n)$, but the weighted averages $(\sum_{i=1}...
- 19.7k
8
votes
2
answers
3k
views
Discontinuous convolutions
Is the following true?
The convolution of two infinitely differentiable as well as integrable real functions can be nowhere continuous.
A reference/proof idea would be very helpful.
- 9,631
6
votes
6
answers
3k
views
The maximum of a real trigonometric polynomial
Given the coefficients $a_0,\ldots,a_N$, $b_1,\ldots,b_N$ of a real trigonometric polynomial:
$ f(x) = a_0 + \sum_{n=1}^N a_n \cos(nx) + \sum_{n=1}^N b_n \sin(nx) $
is there any efficient way to ...
- 531
1
vote
1
answer
950
views
The difference between Lebesgue and Hardy spaces
Are there any known inequalities of the following type for $f$ satisfying some conditions:
$$
\|f\|_{H_p(\mathbb{R})} \le C\|f\|_{L_p(\mathbb{R})},
$$
where $H_p$ denotes the real Hardy space and $...
- 979
1
vote
1
answer
706
views
Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
3
votes
2
answers
216
views
one-side estimates for quasi-trigonometric polynomial
Let $f(x)=\Re(\sum_{k=1}^n a_k e^{i\lambda_k x})$ for $0 < \lambda_1 < \lambda_2 < \dots < \lambda_n$ and some complex $a_1$, $a_2$, $\dots$, $a_n$. What is the best (in some sense) ...
- 109k
8
votes
2
answers
2k
views
The dual group of $\mathbb Q$
What is the dual group of the additive group of rational numbers equipped with the standard topology inherited from $\mathbb R$? As a group, this dual group is isomorphic to $\mathbb R$ (see the ...
- 162
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
15
votes
1
answer
13k
views
Fourier transforms of compactly supported functions
One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies ...
- 2,243
1
vote
2
answers
2k
views
Fourier series of B-spline
The Fourier series of a function (B-spline) is given by:
$$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$
But the B-spline has only finite support. How ...
- 267
16
votes
4
answers
11k
views
Fourier transform of Analytic Functions
Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.
I'm trying to construct a function according to some conditions in the frequency domain of ...
- 317
2
votes
1
answer
846
views
Integral involving exponential of fractional power
Can anything be said about the Fourier integral
$\int_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$
where $a > 0$ and $\gamma > 0$?
Can it be related to some special ...
- 185
6
votes
2
answers
5k
views
Can I relate the L1 norm of a function to its Fourier expansion?
I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any ...
- 185
10
votes
2
answers
5k
views
Approximate a probability distribution by moment matching
Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
- 1,839
1
vote
2
answers
3k
views
What does Gibbs phenomenon shows the nature of Fourier Series
As the title shows,we know that there is some points the series not approaching to the function.
Now,take the convergence theorem into consideration.As there is some the first break-points，the series ...
- 57
12
votes
1
answer
3k
views
Lower bounds on (truncated) Fourier transform of functions of constant modulus and bounded derivative
Let $f(x)=e^{i\phi(x)}$ define a function from $[0,1]$ to the complex unit circle through the real smooth function $\phi(x)$. Also, this function is periodic: $\phi(0)=\phi(1)=0\text{ mod }2\pi$ and ...
- 731
41
votes
6
answers
87k
views
Fourier vs Laplace transforms
In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
- 411
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839
11
votes
3
answers
3k
views
A two-variable Fourier series and a strange integral
I have recently had occasion to investigate the Fourier series of the function $f(x,y)=\log({2+\cos 2\pi x} +\cos{2\pi y})$. Accordingly, define
$I(m,n)=\int_{0,0}^{1,1}f(x,y)\cos{2\pi mx}\cos{2\pi ...
- 13.1k
11
votes
2
answers
2k
views
Hypoellipticity of square root of laplacian
It is a well known result (sometimes called the Weyl lemma) that the laplacian in $\mathbb{R}^n$ is hypoelliptic, i.e. if $f$ is a distribution s.t. $\triangle(f)$ is smooth in an open set, than $f$ ...
- 2,479
12
votes
3
answers
2k
views
level sets of multivariate polynomials
Let $p:\mathbb R^n \rightarrow \mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^n\subset\mathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$...
- 621
13
votes
2
answers
862
views
Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
- 261
3
votes
1
answer
270
views
L^p Idempotent multipliers in 2 dimensions
This question is suggested by that about L^p multipliers (and the answer by Michael Lacey in particular).
Let E be a measurable set in the plane and XiE its characteristic function. We say E is an Lp ...
- 2,479
10
votes
4
answers
2k
views
Reading for finite Fourier analysis
Can anyone recommend some good reading for Fourier analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...
- 7,013