All Questions
Tagged with cv.complex-variables harmonic-analysis
73 questions
0
votes
1
answer
118
views
Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
0
votes
0
answers
57
views
Double-periodic functions with (possible) poles
Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
2
votes
1
answer
168
views
Special function in the Hardy space
Let $H^2(\mathbb{D})$ denote the complex Hardy space, this is: analytic functions defined unit disc $\mathbb{D}$ whose coefficients form a sequence in $\ell^2$. Functions in $H^2(\mathbb{D})$ have a ...
- 319
5
votes
2
answers
791
views
How to calculate an integral over the complex unit sphere
We want to calculate the following integral over the complex unit sphere $S^{2n-1}$:
$$\int_{S^{2n-1}} \frac{1 }{|1 - \langle z, \zeta \rangle|^2} \, d\sigma(\zeta),$$
where $ z $ is a fixed point in ...
- 113
1
vote
0
answers
338
views
Recognizing when a $2\pi$-periodic function is a shifted sine
Consider a real analytic function $f(x)$ with a period of $2\pi.$ Let $t\in \operatorname{Range}(f(x))$, $$I(t)=\int_{0}^{2\pi}\log |t-f(x)|dx.$$ Prove that $I(t)$ is equal to a constant if and only ...
- 21
2
votes
1
answer
112
views
On compactly supported functions with prescribed sparse coordinates
Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
- 4,115
0
votes
0
answers
94
views
The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
- 852
2
votes
1
answer
106
views
A modified Paley–Wiener theorem with weaker condition
Let's consider the following argument: let $f$ be a function in $L^2(\mathbb R)$ such that $\hat f$ extends to an entire function on $\mathbb C.$ Assume that for each $t>0$ and $x \in \mathbb R$
$$
...
- 1,755
7
votes
1
answer
367
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 71
1
vote
1
answer
85
views
Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 190
2
votes
1
answer
197
views
Linear elliptic equation
Let $\Delta:=\partial_z\,\partial_{\overline {z}} $ be the Laplacian operator. I look for a particular non-trivial solution $u$ of $$\Delta u=\frac{a}{1-|z|^2}u$$ where $u\in C^2(\mathbb{D})$ and $a\...
- 43
8
votes
1
answer
461
views
On critical points of harmonic functions
Let $u \in C^{\infty}(\mathbb R^3)$ be harmonic. Suppose that $u$ has no critical points outside the unit ball but that it has at least one critical point inside the unit ball.
Does it follow that $u$ ...
- 4,115
1
vote
0
answers
51
views
Application of the $\operatorname{BMO}$, $H^1$ duality
Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
- 111
1
vote
0
answers
47
views
Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
- 981
2
votes
1
answer
128
views
Regarding basis of holomorphic Hardy space
Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain and let $H^2(\partial\Omega)$ denotes a holomorphic Hardy space which is a $L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)(=\...
- 63
2
votes
1
answer
238
views
Extension of a Szegő Kernel to the boundary
Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.
Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
- 63
4
votes
1
answer
191
views
Integral of $\ln(1/|f|)$ for $f$ bandlimited
I came across the following assertion: if $f\in PW_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,...
- 319
7
votes
0
answers
168
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
- 1,170
4
votes
0
answers
173
views
On the best constant for Carleson's embedding theorem
In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $...
- 81
1
vote
1
answer
206
views
Is $\log|h|$ BMO when $h$ is analytic?
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. A function $f$ is called a bounded mean oscillation (BMO for short) function on $\mathbb{D}$ if
$$\sup_{disc\,\subset\, \mathbb{D}}\dfrac{1}{|...
- 11
2
votes
0
answers
120
views
Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?
Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary:
$$
|f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg).
$$
Does it follow that the (...
- 43.2k
11
votes
1
answer
413
views
Estimating the growth of the Taylor coefficients given the growth of the function at the boundary
Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies
$$
|f(z)|\le \frac{1}{(1-|z|)^{k}}
$$
for some fixed $k>0$.
Question: What can I deduce about the growth of the ...
- 43.2k
4
votes
0
answers
104
views
Walsh-Lebesgue type theorem in $\Bbb R^{2m}$ for $m>1$
Is someone aware of any analogue of the Walsh-Lebesgue theorem in $\mathbb{R}^{2m}$ for $m>1$ and dealing with polyharmonic polynomials?
In this post, $\phi$ is said to be polyharmonic in $\mathbb{...
- 41
7
votes
1
answer
204
views
Are $\log(\sigma(A(z))$ subharmonic functions?
Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$.
Is it ...
- 124
2
votes
1
answer
125
views
Twisted winding number
Consider the contour integral
$\frac{1}{2\pi i}\oint_\gamma\chi(z)\frac{dz}{z}\,,$
where $\gamma$ is a (not necessarily simple) closed curve lying in $\mathbb{C}\setminus{0}$ and $\chi\colon\mathbb{...
- 1,453
8
votes
0
answers
339
views
The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures
Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
- 1,284
1
vote
1
answer
261
views
Beltrami equation with harmonic coefficient
I need to find solutions to the Beltrami equation
$$
\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}
$$
for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
- 134
0
votes
0
answers
173
views
Function Spaces on the Open Unit Disk defined by Hardy Space norms
I've been reading up on Hardy spaces and (sub)harmonic functions over the open unit disk $\mathbb{D}\subset\mathbb{C}$, and I've found myself working with atypical objects in mostly-typical situations....
- 1,284
3
votes
0
answers
167
views
Does the maximum principle hold in this pluriharmonic setting?
Let $U \subseteq \mathbb{C}^m$ be open, and let $F: U \to \mathbb{C}$ be a holomorphic function, with real part $u$. We are given a subset $S \subseteq U$ given by finitely many real equalities and ...
- 5,215
5
votes
1
answer
270
views
A domination property for the Hardy space $H^1$
In the theory of Hardy spaces of the unit disc, a fact that is implicitely used quite often is that if $f\in H^p, 1<p<\infty$, then there exists a function $F\in H^p$ such that $|f(z)| \leq |F(z)...
0
votes
0
answers
47
views
Oscillatory integral independent of a parameter
Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral
$$Q(t) :=...
- 222
4
votes
2
answers
629
views
The formula for (and computation of) the inverse p-adic mellin transform
So, after scouring the entirety of the internet, I managed to find one (and, so far, only one) source that actually explains how to invert the $p$-adic mellin transform:
$$\mathscr{M}_{p}\left\{ f\...
- 1,284
5
votes
1
answer
236
views
Which plane curves can be harmonically parametrized?
In this question, a “(closed oriented plane) curve” $\Gamma$ will mean a continuous map $f \colon \mathbb{U} \to \mathbb{C}$ where $\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$ is the unit circle, modulo ...
- 32.4k
6
votes
2
answers
839
views
A harmonic function
Consider the vertical strip of angle $\alpha=\frac{\pi}{2}$
In this case, the harmonic function which is $0$ on the left line and $1$ on the right line is given by $$f(a+ib)=\frac{a}{T}.$$
Now, when ...
user135093
1
vote
2
answers
144
views
Measurability of the angular limit function
Let $\mathbb T$ be the unit circle and suppose that $f\in L^1(\mathbb T)$ is real-valued. Then its Poisson integral $F=P[f]$
is real-valued, too. Let
$$Osz[f](e^{i\theta}):=\limsup_{z\to e^{i\...
- 687
5
votes
1
answer
850
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
- 51
6
votes
0
answers
219
views
Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
- 13.8k
3
votes
2
answers
212
views
Decay of bandlimited function
Consider a function $f\in L^2(\mathbb{R}^d)$ with $\|f\|_{2}=1$ and such that $\hat{f}$ is supported on the ball $B(0,1)$. I am wondering which is the best decay that $f$ can have.
I read on "G. ...
- 319
4
votes
0
answers
210
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
1
vote
1
answer
132
views
Plancharel-Pólya inequality for functions of exponential type
If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$
then it can be proven that $$\int_{-\infty}^{+\...
- 319
1
vote
1
answer
167
views
Sampling set: relatively dense and uniformly discrete
The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...
- 319
3
votes
0
answers
250
views
Dimension of roots of irreducible Schur polynomial on unit circle
Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...
- 31
2
votes
0
answers
88
views
Link between subharmonic and subanalytic functions
Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
- 1,017
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
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
3
votes
2
answers
457
views
Integrality of complex infinite series
Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying
$$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$
$$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...
- 43.1k
4
votes
1
answer
1k
views
upper bound on derivatives of a function defined on an arc
This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...
- 398
3
votes
2
answers
236
views
Sampling Theorem for non-bandlimited Functions
The classical Shannon sampling theorem states that
a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $...
- 41
12
votes
2
answers
1k
views
Fourier transform of the critical line of zeta?
This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
- 4,897
0
votes
1
answer
128
views
On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$
There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...
- 175