All Questions
389 questions
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 1,247
0
votes
0
answers
56
views
Existence of a couple of functions solution of a differential equation (with additional constraint)
I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...
- 1,199
0
votes
0
answers
227
views
Negative Sobolev norm of non-zero mean non-periodic function on bounded space
The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows:
$\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, ...
- 318
3
votes
2
answers
1k
views
Fourier transform inversion theorem for a function not in L1 or L2
For $\frac{1}{4}<a<1$ consider the following function:
$$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$
If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...
- 1,199
3
votes
0
answers
168
views
Zak transform and VMO
The Zak transform of a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ is defined as follows:
$$
Zf(x,\omega) := \sum_{k\in\mathbb Z}f(x+k)e^{-2\pi i k\omega},\quad (x,\omega)\in Q_0 :=(0,1)^2.
$$
...
11
votes
1
answer
668
views
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
- 7,789
1
vote
1
answer
536
views
Incoherence of Fubini therorem with integral on Fourier series
I ask this question because of the apparent incoherence of the value of following integral:
$$I=\int_{0}^{1} \int_{0}^{\infty} \left|\sum_{n=1}^{\infty} f(nx) e^{2 i \pi n y} \right|^2 dx dy$$
...
- 1,199
5
votes
0
answers
120
views
Geometric characterization of Silva distributions
There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.
A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...
- 1,017
1
vote
0
answers
233
views
Fubini: can we interchange integration order on this double integral (with Fourier series product)
Can we interchange the order of integration of following double integral ?
$$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$
Where $F(x,y)= \...
- 1,199
2
votes
1
answer
181
views
On a paper by Adams and Frazier
I am reading a paper by Adams and Frazier (namely Adams, Frazier, Composition operators on potential spaces. Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165, available here), whose main purpose is ...
- 3,146
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
1
vote
0
answers
100
views
Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion
Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$:
$$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
- 141
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
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
1
vote
0
answers
124
views
Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
10
votes
1
answer
433
views
Shift invariant subspaces of $l^1$
There is a simple characterization of shift-invariant closed subspaces of $l^2$: for any measurable subset $S$ of $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$, the set of elements of $l^2$ whose Fourier ...
- 42.8k
3
votes
0
answers
126
views
An identity of operator norms and de Leeuw's theorem
Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$
$$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
- 187
5
votes
1
answer
595
views
Explicit Paley-Wiener function
By a Paley-Wiener function I mean a function $f(z)$ that is the Fourier image of a test function. Equivalently, by Paley-Wiener theorem, $f(z)$ is an entire function that is of rapid decay on the real ...
- 1,959
7
votes
0
answers
1k
views
Books on von Neumann algebras
I am interested in non-commutative $L^p$ spaces. I have a very basic background on von Neumann algebras. But all the papers appearing now a days really requires very deep knowledge of von Neumann ...
- 455
4
votes
1
answer
277
views
Does the Fourier transform preserve the separation property?
The space of Schwartz functions on the plane is denoted by $\mathcal{S}$.
The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.
...
- 356
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
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 187
1
vote
0
answers
202
views
Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 259
5
votes
1
answer
2k
views
Injectivity of the Fourier transform on $L^1$ without inversion
Is there a proof of the injectivity of the Fourier transform on $L^1({\bf R})$ that does not rely on an inversion formula?
The proofs I have seen in the literature ultimately rely either on the ...
- 18.7k
8
votes
2
answers
613
views
Pairs of elementary Fourier transforms in $L^2$
It is customary to teach Fourier transform on the real line by starting with functions from $L^1$, $L^2$ or the Schwartz space. It is not so easy to illustrate the theory by computing explicit pairs ...
- 18.7k
5
votes
1
answer
249
views
If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
For any $f\in L^1(\mathbb{R}^n)$ and $1\le j\le n$, recall that the Riesz transform $\mathcal{R}_jf\in L^{1,\infty}(\mathbb{R}^n)$ is given by
$$ \mathcal{R}_jf:=c_n\lim_{\epsilon\to 0}\left(\frac{x_j}...
- 3,146
3
votes
1
answer
195
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
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
2
votes
1
answer
250
views
Density in the Space of absolutely convergent Fourier series
It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/...
- 125
27
votes
5
answers
3k
views
Nice applications for Schwartz distributions
I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and ...
2
votes
1
answer
699
views
Schwartz kernel theorem
I would like to understand how the Schwartz kernel theorem works for some more difficult cases and therefore would like to discuss an example from scratch:
Let the Dirichlet Laplacian on the half-...
- 23
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
2
votes
1
answer
183
views
is this weighted-maximal function unbounded?
The Hardy-Littlewood maximal operator
$$Mf(x)=\sup_{x\in B}\frac1{\vert B\vert}\int_B\vert f(y)\vert dy$$
where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$.
It is ...
- 43.2k
5
votes
0
answers
166
views
Fourier basis for sub-Gaussian spaces?
Let $(\mathcal{X}, \pi)$ be a probability space such that $\pi$ has full support. Consider $L^2(\mathcal{X},\pi)$ to be the inner product space of function $f: \mathcal{X}^n \to \mathbb{R}$, with ...
- 519
2
votes
1
answer
336
views
Separability of $L^1$ in $L^2$ topology
In the space $L^1(0,1)$ take the topology generated by the $L^2$-balls
$$B^2_r(f)=\{g\in L^1(0,1):\; \|f-g\|_2<r\}.$$
Is $L^1(0,1)$ separable in this topology?
- 23
1
vote
0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
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
$$T_\lambda f(t)=\int \frac{\...
- 171
2
votes
0
answers
125
views
Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle
I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
- 141
1
vote
0
answers
127
views
What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
- 41
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
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
3
votes
3
answers
580
views
Approximate identities and pointwise convergence
I'm studying Fourier analysis and have a question about approximate identities.
Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $...
- 41
15
votes
2
answers
681
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,649
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
$$
\|f\|^2_{H^{-1}(\mathbb{T})}\lesssim \int_\mathbb{T} |\sin(x)f(x)|^2 \; ...
- 165
0
votes
1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
7
votes
1
answer
1k
views
Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?
I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...
- 85
1
vote
1
answer
378
views
Easy Garding Inequality
Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
- 4,145
3
votes
1
answer
256
views
On construction of a $\mathbb{Q}$ periodic function with Fourier series
Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series:
$$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$
Using ...
- 1,199