Questions tagged [analytic-functions]
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169 questions
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How do we know that global analytic functions exist?
Let $(M,m_0)$ be a real analytic manifold. Let $\pi$ be a cotangent at $m_0$. How can one prove that there exists a global analytic function $\theta:M\rightarrow \mathbb{R}$ such that $d\theta(m_0)=\...
- 1,329
4
votes
1
answer
199
views
Entire function of order 1 and type $\infty$ with prescribed indicator function $h_f(\theta)$
I would like to know if there exists an entire function $f$ of order 1 such that the indicator
$$
h_f(\theta) := \limsup_{r\to\infty} \frac{\log |f(re^{i\theta})|}{r}
$$
satisfy
$$
h_f(\theta) = \...
- 245
9
votes
1
answer
665
views
What actually is the "right way" to view the analytic continuation of the Bell numbers?
Dobinski's Formula gives a curve which interpolates the Bell Numbers
$$ B(x) = \frac{1}{e}\sum_{k=0}^{\infty} \frac{k^x}{k!} $$
This formula is rather elegant except an awful $\frac{0^x}{0!}$ in the ...
- 3,169
3
votes
1
answer
149
views
Show directly that the analytic besov spaces are subset of the bloch space
Let $\operatorname{Hol} (\mathbb D)$ denote the space of all analytic functions, defined on the unit disc of the complex plane. For $0<p<\infty$ and $n\in \mathbb N$, such that $np>1$, we ...
1
vote
0
answers
100
views
Upper bound of first moment of GL(3)/GL(n) L function
I have a question.
Let $f$ be a Maass form for $\mathop{SL}_3(\mathbb{Z})$ which is an eingenfunction of all the Hecke operators. Let $(A(n,m))_{n,m}$ be its Fourier coefficient and
\begin{equation*}
...
- 19
4
votes
1
answer
357
views
On optimum constant for upper density of zeros of an entire function
Let $f\in L^2(\mathbb R)$ be a not identically vanishing function that is compactly supported in the interval $(-\sigma,\sigma)$ for some fixed $\sigma>0$. For each $z\in \mathbb C$ let us define ...
- 4,189
4
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0
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153
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Is there a theory for integration of non-analytic function being analytic
Let $f:\mathbb{R}\times \mathbb{R}_{>0} \rightarrow \mathbb{R}$ be a function. Suppose $f(\cdot,y):\mathbb{R}\rightarrow \mathbb{\mathbb{R}}$ is integrable for every fixed $y$ and $f(x,y)$ has no ...
1
vote
0
answers
80
views
Commutants of analytic Toeplitz operators on the harmonic Bergman space
It is known that, in the analytic Bergman space $L_a^2(\mathbb{D},dA)$, if the Toeplitz operator $T_{f}$ whose symbol $f$ is an analytic nonconstant function on the unit disk, commutes with $T_{u}$ ...
- 97
6
votes
1
answer
338
views
Can we prove that Schwarz-Christoffel transform works for all polygons, without using the Riemann mapping theorem?
The Schwarz-Christoffel transform is
$$S(z)=\int_0^z \frac{1}{(w-A_1)^{\beta_1}\dots (w-A_n)^{\beta_n}}dw$$
where $A_1<A_2<\dots<A_n$ are points on the real axis, and also $\beta_k$ are real ...
- 464
0
votes
1
answer
185
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,189
3
votes
0
answers
205
views
Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
- 2,679
2
votes
1
answer
315
views
Can solution of heat equation become constant in finite time
Consider parabolic equation
$$
u_t = u_{xx},
$$
where $x \in (0, 1)$, $t \in (0, T)$ with nonhomogenious Dirichlet boundary conditions.
$$
u(0, t) = \psi_0(t) \in C^0,
$$
$$
u(1, t) = \psi_1(t) \in C^...
7
votes
1
answer
494
views
Rational points on an analytic curve
Let $\Gamma$ be a transcendental analytic curve in $\mathbb{R}^2$. I am interested in the topology of its rational points $\Gamma(\mathbb{Q}):=\Gamma\cap\mathbb{Q}^2$.
We know by Pila-Wilkie that if $\...
- 73
2
votes
1
answer
141
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,726
3
votes
1
answer
310
views
Prescribe the type of an entire function which inverse zeros are summable
According to Lindelöf's theorem, given points $z_i\in \mathbb C\setminus \{0\}$ ordered by increasing modulus with possible repetitions, we can define a function
$$
f(z)=\prod_{n=1}^\infty (1-z/z_n)e^{...
- 1,362
6
votes
1
answer
525
views
Analyticity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\setminus\{0\}$ but non-analytic at $x=0$. Is the convolution (...
- 403
1
vote
0
answers
88
views
Simple smooth functions on Bolza surface
Consider the Bolza surface, a compact Riemann surface of genus 2.
It is an octagon in the Poincaire disk model with opposite sides identified.
I would like to write down some analytic expressions for ...
- 231
1
vote
1
answer
256
views
Generalisation of Paley–Wiener type results for unbounded sets
Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
- 1,362
3
votes
1
answer
147
views
When entire or meromorphic map of finite type restricts to a Galois covering?
Suppose that $f \colon \mathbb{C} \to \mathbb{C}$ is an entire map of finite type, i.e., with finitely many singular values. Then we can consider the restriction $f| \mathbb{C} \setminus f^{-1}(S_f) \...
- 51
3
votes
1
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272
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Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere
I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove
$$f=0\textit{ on } M_1=\{(z_1,\...
- 431
3
votes
1
answer
188
views
Growth of preimages of singular values of finite type entire map
Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire map having precisely two distinct singular values $w^1$ and $w^2$. If $w^i$ has infinitely many preimages under $f$, we write $(z_n^i)_{n \in \...
- 51
0
votes
1
answer
97
views
Integration algorithm and analytic property
This question is the continuation of the previous one.
In the article about the integration of analytical polynomial - time computable function $f(x)$ with the Taylor series $$f(x) = \sum_{n=0}^{\...
- 81
1
vote
1
answer
161
views
Characterizing the unimodular functions from the closed disk $\overline{\mathbb{D}}$ to $\mathbb{C}$ with constraints
Let $\mathbb{D}$ be the open disc. It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a ...
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3
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426
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A strictly increasing, analytic function that goes through key points of the iterated logarithm?
Is it possible to create a function $f(x)$ that:
is strictly increasing (at least for $x>0$)
is real analytic
goes through all the points where the iterated logarithm would increment value?
i.e. [...
- 71
3
votes
0
answers
151
views
Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
- 759
3
votes
1
answer
385
views
Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?
It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting.
It is known that if $L$ is a uniformly elliptic operator, with ...
- 1,795
7
votes
1
answer
589
views
Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
- 191
17
votes
3
answers
1k
views
A kernel 'more analytic' than $\exp(-x^2)$
I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \...
- 1,389
8
votes
1
answer
218
views
Separate holomorphicity implies holomorphicity on analytic varieties
Suppose that $M$ and $N$ are two complex analytic varieties and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
- 845
2
votes
0
answers
78
views
power of analytic function is still analytic in Krasner sense
In page 54 of his book, "Analytic elements in $p$-adic analysis" Escassut claims that if $f$ is analytic in Krasner sense in a set $D$ of a ultrametric field, so is $f^n$ for any positive ...
- 4,306
3
votes
1
answer
259
views
Uniformly closed ideals of smooth/real analytic functions
Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
- 845
4
votes
1
answer
313
views
Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...
- 203
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0
answers
162
views
Geometric interpretations of $A_k$ singularities on plane curves
Definition: A real smooth analytic function $f$ defined on some neighborhood of $t_0$ is said to have an $A_k$ singularity at $t_0$ if and only if $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...
2
votes
0
answers
145
views
Real analytic periodic function whose critical points are fully denegerated
I have asked this question on MathStackExchange. My question: is there any non-constant real analytic function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that, $$\nabla f(x_0)=0 \Rightarrow \nabla^2 f(...
- 121
9
votes
1
answer
362
views
Is there a largest o-minimal structure all of whose definable functions are analytic?
In the paper "Quasianalytic Denjoy-Carleman classes and o-minimality" by J.-P. Rolin, P. Speissegger and A. J. Wilkie, it is proven that there is no largest o-minimal structure on the real ...
- 333
3
votes
1
answer
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Does Noetherianity imply division theorem?
I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum.
Let $\mathcal{O}_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-...
- 1,036
2
votes
1
answer
269
views
Existence of the special entire Hardy space function with infinitely many zeros in the strip
Question. Does there exist an entire function $h$ satisfying three following assertions:
$h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane;
$zh - 1$ belongs to $H^2(\mathbb{C}...
- 607
4
votes
2
answers
295
views
Existence of nonzero entire function with restrictions of growth
Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
$F(z)\neq 0$ for all $z\in \mathbb{C}$;
$1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...
- 607
5
votes
1
answer
444
views
Real-analytic analogue of Schwartz functions
Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
- 53
0
votes
1
answer
301
views
A holomorphic function in the open unit disk satisfying certain properties
Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a_n|$ is divergent, where $a_n$'s coefficients in the Taylor series expansion ...
- 165
3
votes
0
answers
82
views
lower bound for zero multiplicity of function formed from determinant of functions
I have a family of single-variable analytic functions, $D(z)$, formed as follows.
Let $L$, $n$ and $T_{0}$ be positive integers, $\varphi_{1},\ldots, \varphi_{L}$ be analytic functions in ${\mathbb C}$...
- 31
2
votes
0
answers
134
views
A division of real analytic functions
Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
- 1,036
5
votes
2
answers
645
views
How to compute $\sin(\frac{d}{dx})f(x)$?
Assuming $f(x)=e^{-x^2}$ for $x$ in $[-10,10]$, I have tried the following:
Fourier transform $\mathcal{F}$: $\frac{d}{dx}$ can be diagonalized as $\mathcal{F}^{-1} i\omega \mathcal{F}$. Therefore, $\...
- 370
2
votes
1
answer
352
views
On a lemma of Łojasiewicz in complex analysis of one variable
Context. The question arises from my former question on the remainder of a power series. Precisely, I was trying to understand if the boundary behavior of power series considered by Ricci in his paper ...
- 6,830
4
votes
1
answer
216
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
- 324
3
votes
3
answers
502
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
- 324
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votes
0
answers
221
views
If $\lim_{t\to +\infty} \int_{0}^{\pi} f(x)\exp(e^{xt}) \, dx=0$ then $f=0$ a.e?
Question, Let $f \in L^1(\alpha, \beta)$ , $\beta>0$ and
$$
F(x)= \int_{\alpha}^{\beta} f(t)\exp(e^{xt}) \, dt
$$ such that $\displaystyle \lim_{x\to +\infty}F(x)=0$. Does this imply that $f$ is ...
- 1,513
2
votes
0
answers
155
views
Existence of analytic function in disk algebra [closed]
Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
- 149
0
votes
0
answers
145
views
Predual of $H^{\infty}(\mathbb{D})$
Is the predual of $H^{\infty}(\mathbb{D})$ contained in the maximal ideal space of $H^{\infty}(\mathbb{D})$?
- 149
1
vote
0
answers
195
views
Analyticity of solutions to Schrödinger's equation
Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...
- 439