All Questions
Tagged with ca.classical-analysis-and-odes fourier-analysis
250 questions
2
votes
0
answers
2k
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 ...
- 121
3
votes
0
answers
204
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=\...
- 4,058
6
votes
0
answers
211
views
Regularity of $|u|^{\alpha}$ when $u$ is Schwartz
Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). ...
2
votes
1
answer
123
views
Inversion of a Fourier-like transformation on $\partial B_1(0)$
I have a function $A(x) = \int_{\partial B_1(0)} e^{ikx} B(k) dk$ in $3D$ and I want to find $B(k)$ for a given $A(x)$. How can I do that?
In my use case it would be enough to invert $R_n^m(x) = \int_{...
- 21
6
votes
0
answers
120
views
Condition on the support of $f$ which ensure that $\widehat{f}$ has a zero-measure nodal region
Suppose that $f\in L^2(\mathbb{R})$ is non-zero and compactly supported. Then its Fourier transform $\widehat{f}\neq 0$ is analytic, and in particular the nodal set $\{\xi\in\mathbb{R}\,s.t.\,\widehat{...
- 282
1
vote
2
answers
140
views
Extending a discrete singular kernel
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
3
votes
1
answer
171
views
Discrete singular integrals
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
2
votes
0
answers
221
views
Turán–Nazarov's lemma for algebraic polynomials?
Nazarov proved a version of Turán's lemma in Complete Version of Turan’s Lemma for Trigonometric Polynomials on the Unit Circumference, which is now known by the name Nazarov–Turán's lemma. A special ...
- 399
3
votes
0
answers
216
views
The Fourier transform of a compactly supported smooth function on Lie groups over $\mathbb{Q}_S$, where $S$ contains finitely many primes and $\infty$
Let $G$ be a semisimple Lie group defined over global Field $\mathbb{Q}$. Let $S$ be a set of finitely many non-Archimedean places including Archimedean places. Let $P_{0}=M_{0}A_{0}N_{0}$ be the ...
- 31
2
votes
0
answers
216
views
Boundedness of solutions to second order ODE
Let $q(x)$ be a probability density function over $[0,1]$. Let $\lambda > 0$ and $f: [0,1] \to \mathbb{R}$ be any solution to the following ODE:
$$
\lambda f''(x) + q(x) f(x) = 0, \text{for all }x \...
- 515
12
votes
1
answer
727
views
A generalization of Rubio de Francia's inequality
Suppose that $\{I_m\}$ is a sequence of pairwise disjoint intervals in $\mathbb{Z}$. The well known Rubio de Francia's inequality says that for any function $f\in L^p(\mathbb{T})$, $2\le p<\infty$, ...
5
votes
3
answers
2k
views
Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 595
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
6
votes
2
answers
437
views
Matrix-valued ordinary differential equation with symmetry
I am considering the following equation
$$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(...
- 192
4
votes
1
answer
191
views
Scaling of double convolution
I am interested in the scaling of
$$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$
In particular, I suspect that
$$F(...
- 192
6
votes
1
answer
365
views
Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?
Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm
$$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$
...
- 873
-1
votes
1
answer
655
views
What does $O(N)$ mean in this article and how does it imply this lemma?
In this article the author proves the following lemma:
LEMMA: $\forall N \in \Bbb N$, there exists $v=v_N$ with compact support so that
$$[M_S(M_S v)^\delta(x)]^{1/\delta} \geq c_\delta NM_S v(x), \...
- 135
1
vote
1
answer
133
views
Integrability of fractional heat kernel
In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$
where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
- 109
1
vote
0
answers
70
views
Function of several variables whose hessian is a Hankel matrix
First of all, let me apologize because I asked this question a few days ago on https://math.stackexchange.com, but I did not get any reply.
I am studying a function $f:\mathbb{R}^n\rightarrow\mathbb{R}...
- 297
3
votes
1
answer
203
views
Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$
Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
user139844
2
votes
0
answers
169
views
Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
- 839
1
vote
1
answer
389
views
When are Fourier cosine coefficients convex?
In the question When are Fourier coefficients monotonic it was determined that, if a function $f$ is (the restriction to $[0,2\pi]$) of a completely monotone function, then its Fourier coefficients, ...
- 595
5
votes
1
answer
314
views
A simple oscillatory integral with a non-smooth phase
Let $\phi\in C_c^\infty(\mathbb{R})$ be an even function such that $\chi_{(-1/2,1/2)}\le\phi\le \chi_{(-1,1)}$, where $\chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $\...
- 421
7
votes
2
answers
455
views
On a monotonicity property of Fourier coefficients of truncated power functions
Is it true that
$$a_{k,n}:=\int_0^{2\pi}x^k\cos(nx)\,dx$$
is nonincreasing in natural $n$ for each $k\in\{0,1,\dots\}$?
This question is related to this previous one.
Twice integrating by parts, one ...
- 128k
22
votes
2
answers
2k
views
When are Fourier coefficients monotonic?
Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by
$$
\hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
- 595
1
vote
1
answer
307
views
Convexity of discrete Fourier transform
Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
- 595
4
votes
2
answers
456
views
Find an element with given periodicity
Sorry for all the confusion. I think what I am actually asking is: Can we find an explicit smooth non-zero function on $\mathbb R^2$ that satisfies
$$f(x_1,x_2) =e^{-i\pi x_2} f(x_1+1,x_2) \text{ and }...
- 192
1
vote
0
answers
690
views
What is the Jacobi-Anger expansion of $\sin^{[k]} (\theta) $?
Cross-post from MSE.
The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \...
- 4,829
3
votes
1
answer
187
views
Improving dispersive estimates for linear KdV
Consider the equation
$$\partial_t u(t,x) = -\partial_{xxx} u(t,x)$$
for $x \in \mathbb R.$
It is well-known that in general we have
$$\Vert u(t) \Vert_{L^{\infty}} \le C t^{-1/3} \Vert u_0 \Vert_{L^1}...
- 173
0
votes
1
answer
88
views
Integration against a certain Fourier transform
I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being ...
- 89
1
vote
0
answers
151
views
Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
5
votes
1
answer
228
views
Does such a function exist?
I am looking for a function with the following property:
Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$
I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$
I am trying ...
- 536
2
votes
1
answer
295
views
Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms
Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded:
$$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
- 179
2
votes
1
answer
495
views
Fourier transform of a function of bounded variation
I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
- 1,879
1
vote
0
answers
103
views
Integrability of Fourier transform of truncated fractional power
Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...
- 1,879
5
votes
0
answers
168
views
Sobolev extension from a discrete set of points
Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...
- 501
20
votes
1
answer
1k
views
Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
- 178
3
votes
1
answer
163
views
Example of a bounded function whose mean-zero mollification diverges at a point
For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\...
- 421
3
votes
1
answer
488
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
- 751
2
votes
0
answers
113
views
Inequality about exponential integrals
I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
- 3,062
4
votes
1
answer
285
views
Idea behind Carleson's theorem modern proof "intitial reductions"
I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.
For any $f \in L^2(\mathbb{R})$, let $\...
- 489
7
votes
1
answer
355
views
Compactly supported probability measure in high dimensions with fast Fourier decay?
For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
- 91
12
votes
1
answer
748
views
Bounding a Fourier coefficient of a non-negative periodic function in terms of its $L^2$-norm
This question is motivated by the earlier MO question: Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$ .
It is a cleaned up ...
- 43.7k
3
votes
0
answers
237
views
Reconstructing a sine wave using square waves and Möbius inversion: L² convergence?
Let $s$ be the (“square wave”) $1$-periodic real function such that $s(x) = 1$ if $0<x<\frac{1}{2}$ and $s(x) = -1$ if $\frac{1}{2}<x<1$ (and maybe $s(0)=s(\frac{1}{2})=0$ for the sake of ...
- 32.5k
27
votes
1
answer
2k
views
Linear combination of sine and cosine
I was explaining to my students the other day why $\cos(2x)$ is not a linear combination of $\sin(x)$ and $\cos(x)$ over $\mathbb{R}$. Besides the canonical method of using special values of sine and ...
- 960
7
votes
2
answers
824
views
Fourier series of smooth functions in infinitely many variables
Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier ...
- 401
3
votes
1
answer
246
views
$f \in L^p(\mathbb{R}^2)$ for all $p \geq 1$, and $f$ has zero integral. What can we say about this function's fourier Transform?
Let $\psi$ be an smooth admissible Shearlet with compact support, cand let $\mathcal{M}$ be a bounded region in $\mathbb{R}^2$ and let $m= \chi_{\mathcal{M}}$ be the characteristic function of $\...
- 237
-1
votes
1
answer
70
views
Is this kind of interpolation correct?
Let $f=\sum f_j$ be a finite sum. Assume that
$$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$
$$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$
Then can we conclude that for $2<p<\infty$
$$\|f\|_p\le C^{1-\...
- 1
3
votes
0
answers
140
views
Decay of Laplace (or Mellin) transform beyond region of convergence?
Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
- 20.2k
0
votes
1
answer
111
views
Averaged Parseval Relation for Sampling a Function on Integers
This was asked a long time ago on math.stackexchange with no answers.
Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is $[1,N],$ and that $\mid f\mid$ is ...
- 10.4k