All Questions
389 questions
2
votes
0
answers
136
views
Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following three statements of the Wiener-Tauberian theorem:
version 1: If $I$ is a closed ideal in $L^1(\mathbb R)$, such that the set $...
- 650
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
2
votes
0
answers
186
views
Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
- 171
2
votes
0
answers
125
views
Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle
I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
- 141
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
2
votes
0
answers
183
views
Fourier series and regular distribution
Assume you have a distribution $K$ on $\mathbb{T}$, the torus, such that $\sum_{n=-\infty}^{\infty} |K(e_n)|^2$ is finite, where $e_n := e^{in\cdot}$ are the Fourier basis. Does this imply that the ...
- 95
2
votes
0
answers
106
views
Hilbert Transform and multiplier in $\mathbb{C}(X)$
I found myself trying to solve an equation of that kind :
$$
H f= R f,
$$
where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...
- 3,425
2
votes
0
answers
93
views
Multiplier operators on anisotropic weighted $L^2$ spaces
Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...
- 21
2
votes
0
answers
224
views
On uniform or simple convergence of Poisson Summation formula
Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$):
$$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n =1}^{\...
- 1,199
2
votes
0
answers
276
views
pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?
Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$
Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in \ell^...
- 1,051
1
vote
3
answers
849
views
How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?
$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times g(...
- 175
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
1
vote
1
answer
4k
views
how to prove the range of a closed linear operator is closed ?
The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is ...
- 1,644
1
vote
1
answer
487
views
Fourier Transform of an even function
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$, where $f$ is an even continuous function.
Denote
$$
F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\...
- 343
1
vote
1
answer
536
views
Incoherence of Fubini therorem with integral on Fourier series
I ask this question because of the apparent incoherence of the value of following integral:
$$I=\int_{0}^{1} \int_{0}^{\infty} \left|\sum_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$
...
- 1,199
1
vote
1
answer
378
views
Easy Garding Inequality
Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
- 4,143
1
vote
2
answers
854
views
Derivative of Band-limited functions [closed]
I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy
$$\int_{-\infty}^\infty f(x)^2dx=c.$$
(For pure mathematicians: "bandlimited" means ...
- 21
1
vote
1
answer
289
views
Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$
Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is,
$$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$
with the ...
- 1,051
1
vote
1
answer
298
views
Is this set a Riesz Basis of $L^2(0,\pi)$
Let $A=\{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\} _{n=1}^\infty$, where $a$ is a positive real number. Is $A$ a Riesz Basis of $L^2(0,1)$?
- 11
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 407
1
vote
1
answer
925
views
Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
- 5,405
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
1
vote
1
answer
121
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,143
1
vote
1
answer
113
views
The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 3,477
1
vote
1
answer
433
views
Why complex conjugate in definition of the Fourier transform?
Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
...
- 21.3k
1
vote
1
answer
230
views
Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 521
1
vote
1
answer
322
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 $F$ ...
- 4,058
1
vote
1
answer
484
views
When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?
Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows:
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$
It is ...
- 1,051
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T],
$$
where $(f_n(t))_n$ is the unknown ...
- 617
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
1
vote
1
answer
203
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. ...
- 159
1
vote
1
answer
128
views
Computation on the Hardy space
Why
$$
\Pi_+ \left(\frac{\overline{z}}{1-\overline{qz}}f\right)= \frac{f(z)-f(\bar{q})}{z-\overline{q}}, \quad f\in H^2(\mathbb D),$$
where
$q\in \mathbb C$,
$\Pi_{+}$ is the Szegö projector:
$$\Pi_{+...
- 97
1
vote
1
answer
439
views
Well-known conditions for the Fourier inversion formula
Let $f\in L^1(\mathbb{R})$.
One may easily check that
$$(*)~~~f', f''\in L^1(\mathbb{R})\Rightarrow \int_\mathbb{R}|\hat{f}| ~\text{is finite} \Rightarrow \int_\mathbb{R}\hat{f}(s)e^{2\pi is x}ds ~\...
- 4,058
1
vote
1
answer
228
views
Discrete harmonic analysis with infinite/unbounded number of variables
Is there any study of harmonic analysis for Boolean functions of the form $f:\{0,1\}^*\to \{0,1\}$, or $f:\{0,1\}^\omega\to \{0,1\}$?
That is, similar notions to standard harmonic analysis of $\{0,1\}^...
- 203
1
vote
1
answer
194
views
$\|f\|^2_{H^{-1}(\mathbb{T})}\lesssim \int_\mathbb{T} |\sin(x)f(x)|^2 \; dx$?
I have been stuck in this question for a while, and I would appreciate any new ideas. I have been considering the inequality
$$
\|f\|^2_{H^{-1}(\mathbb{T})}\lesssim \int_\mathbb{T} |\sin(x)f(x)|^2 \; ...
- 165
1
vote
1
answer
672
views
Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation
Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...
- 225
1
vote
1
answer
319
views
The $L^2\times L^2\to L^2$ norm of the bilinear multiplier operator
Consider a general bilinear multiplier operator:
$$
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,
$$
where $\Pi$ is the torus, $n\in\mathbb{Z}$, $m$...
- 463
1
vote
1
answer
111
views
How to show such result for generalized $ O(|x|^{-1/2}) $ function?
Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
- 1,219
1
vote
1
answer
346
views
Range of discrete Fourier transform for binary strings
$\DeclareMathOperator\C{\mathbb{C}}$Let X be the set of all $n$-bit binary strings, $x=(x_1,\ldots x_n)$ where where $x_i\in\{-1,1\}$. Now consider the discrete fourier transform $F$, which maps ...
- 680
1
vote
1
answer
134
views
A unique continuation problem
Let $f\in L^{2}(0,1).$ Consider the following unique continuation problem:
$$
\left\{
\begin{array}{ccc}
af(x-r)+bf(x)=0, & \mathrm{if} & x\in (r,1) \\
& & \\
cf(x+1-r)+df(x)=0 &...
- 47
1
vote
1
answer
117
views
Density and Fourier approximation
Let $\mathbb{T}$ denote the 1-d torus and $H^s(\mathbb{T})$ the Sobolev space of order $s\geq0$ of complex-valued functions on $\mathbb{T}$ with the identification $H^0 (\mathbb{T}) = L^2 (\mathbb{T})$...
- 313
1
vote
1
answer
134
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
1
answer
137
views
Uniqueness of solution to system of integral equations
Given the following system of integral equations for an integrable function $f(x)$:
For all integers $k \ge 1$ holds
$\int_{0}^{2\pi} [f(x)]^k e^{(ikx)} dx = 0$.
If $f(x)$ is real-valued and non-...
- 33
1
vote
1
answer
334
views
Orthonormal basis and decay
Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
- 501
1
vote
1
answer
124
views
On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 1,035
1
vote
1
answer
480
views
Is there an asymptotic bound for this oscillatory integral?
I have an oscillatory integral:
$$ \int u(x,y) e^{i\lambda f(x,y)} dx $$
with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying:
$$ \text{Im} f \geq ...
- 93
1
vote
1
answer
211
views
Let $f \in M^{1,1} (\mathbb R)$ (Feichtinger's algebra /Modulation Space). Can we say $Fof\in M^{1,1}(\mathbb R)$; $F$ is an entire function?
The Modulation space ( Feichtinger's algebra),
$$S_{0} (\mathbb R) = M^{1, 1}(\mathbb R): = \{ f\in L^{2}(\mathbb R) : V_{g}(f) \in L^{1}(\mathbb R^{2}) \};$$
where $V_{g}f (x, w)$ is the short- ...
- 1,051
1
vote
1
answer
977
views
Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
- 278
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