# Questions tagged [analytic-functions]

The analytic-functions tag has no usage guidance.

146
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### 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 ...

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### 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 ...

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### 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 ...

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### 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)! < \...

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### Separate holomorphicity implies holomorphicity on analytic varieties

Suppose that $M$ and $N$ are two complex analytic varities 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)\...

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### 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 ...

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### 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 ...

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### 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 ...

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### Approximation error of Chebyshev expansion of the second kind

Weierstrass' well known theorem states that every continuous function on $[-1,1]$ can be uniformly approximated to arbitrary precision by a polynomial function. Among these approximations it is known ...

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### 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 $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ ...

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### 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(...

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### 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 ...

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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-...

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149
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### 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}...

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188
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ${\mathbb C}$...

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### 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^\...

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### 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, $\...

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238
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### Predual of $H^{\infty}(\mathbb{D})$

Is the predual of $H^{\infty}(\mathbb{D})$ contained in the maximal ideal space of $H^{\infty}(\mathbb{D})$?

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### 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 ...

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### Existence of an eigenpair for d-bar operator in the unit disck

Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...

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### Mellin transform of the Bessel function $Y_n$ of order $n \geq 2$

The Mellin transform of the function $h$, locally integrable on $(0,\infty)$, is defined by
$$M[h,z] = \int_0^\infty t^{z-1} h(t) dt \tag{1}$$
For some functions $h$ the above integral is not ...

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### Is the space of analytic sections of a vector bundle a Fréchet space?

Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of ...

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### When does an analytic submanifold descend to the quotient?

Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. ...

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462
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### Behavior of $|f'(z)|/(1+|f(z)|^2)$ as $|z| \rightarrow \infty$?

Let $f(z)$ be an entire holomorphic function in $\mathbb{C}$, and consider the real-valued function
$$g_f(z)=\frac{|f'(z)|}{1+|f(z)|^2}.$$
If $f(z)$ is a polynomial, then it is easy to prove that $\...

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### Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...

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### Prescribing variations that preserve the area

Let $(M^3,g)$ be a Riemannian manifold and let $\varphi : \Sigma \to M$ be a two-sided embedding of a closed surface into $M$, with a unit normal $N$. Suppose that $\varphi$ is a regular point of the ...

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### Measure of preimage of Jordan disk under entire map

Let $f\colon\mathbb{C} \to \mathbb{C}$ be an entire map. For simplicity assume that $f$ is of finite type, i.e., it has finite set $S(f)$ of singular values. $S(f) \subset \mathbb{C}$ is a minimal (...

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### Bounded real analytic function with bounded derivative and its higher order derivatives

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a bounded analytic function such that its derivative is also bounded. What kind of bound can we get on the higher order derivatives of $f$? Does it ...

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### On a variant of Carlson’s theorem

My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...

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### Exponentials and other functions of sums of anti-commuting operators

I know that if $A$ and $B$ are commuting operators, then $\exp(A+B) = \exp(A) \exp(B)$. Is there a similar formula if $A$ and $B$ are anti-commuting (that is, $AB+BA = 0$)?
I have developed a formula ...

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### Literature on non-Archimedean analogues of basic complex analysis results

It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...

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### When do volumes depend real-analytically on the parameters defining the regions?

Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$.
For $r \in \mathbb{R}$, let $S_r$ be the sub-level set in $B$ defined by the ...

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### An extension of the Carlson's theorem in complex analysis

For the statement of Carlson's theorem please see,
https://en.wikipedia.org/wiki/Carlson%27s_theorem.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish ...

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598
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### Existence of a smooth compactly supported function

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...

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1
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### Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via
$$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...

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### In search for a counterexample related to the Abel-Stolz theorem

Disclaimer: I posted this question seven days ago here on the Math.SE, with slightly different (however in an inessential way) comments. The question has been upvoted but no answer has been given, so ...

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### A problem related to analytic function

Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...

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### Real analyticity of continuous function via restriction to analytic curves

Suppose $X\subset \mathbb R^n$ is an irreducible real analytic sub-variety (i.e. the set of solutions of a system $f_1=\ldots=f_k=0$ with $f_i$ analytic)
Let $x\in X$ be a point and let $F: X\to \...

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### An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series

Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...

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### Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...

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### Global theory of holomorphic functions [closed]

I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...

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### Is a mixture of real analytic functions again analytic?

Let $$h : \mathbb{R}^2 \to \mathbb{R}^+.$$
Suppose that for each $x$, $h(x, y)$ is a real analytic function of $y$.
Let $\mu(dx)$ be a finite measure on $\mathbb{R}$, and for each $y$, suppose that
$$...