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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\...
Ali's user avatar
  • 4,135
1 vote
0 answers
144 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
zoran  Vicovic's user avatar
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 ...
António Borges Santos's user avatar
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
Ryo Ken's user avatar
  • 113
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)}\,...
Ali's user avatar
  • 4,135
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
Ali's user avatar
  • 4,135
4 votes
2 answers
360 views

Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
TaQ's user avatar
  • 3,584
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. ...
Ali's user avatar
  • 4,135
0 votes
1 answer
414 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
  • 61
11 votes
3 answers
1k views

"Simple" integral equation

Let $H(z)$ be a continuous solution of the problem $$ H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1. $$ Is it true that $H(0)=1-\ln2$? The question ...
AAK's user avatar
  • 283
1 vote
1 answer
90 views

The number of roots of pseudo-exponential polynomials

Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
ABB's user avatar
  • 4,058
3 votes
0 answers
244 views

Norm on the space of real analytic functions

The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
Wreck it Ralph's user avatar
1 vote
0 answers
113 views

Computing a limit for the Weierstrass function

Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
Ali's user avatar
  • 4,135
4 votes
1 answer
305 views

Holomorphic extension of the Fourier transform of a measure

If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
user509119's user avatar
3 votes
1 answer
166 views

A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?

So I am wondering if there exists a general procedure for the following problem: given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
Sidharth Ghoshal's user avatar
7 votes
1 answer
368 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} \...
abbyJeffers's user avatar
8 votes
3 answers
429 views

A density claim

Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true: If $f\...
Ali's user avatar
  • 4,135
2 votes
2 answers
281 views

Most general reverse Hölder inequality for polynomials

Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$, $$\|p\|_{L^\infty(a,b)} \...
Sébastien Loisel's user avatar
5 votes
1 answer
353 views

Family of functions with prescribed derivatives

Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
Ali's user avatar
  • 4,135
1 vote
1 answer
210 views

On a property of complex exponentials

Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that $$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
Ali's user avatar
  • 4,135
0 votes
0 answers
251 views

How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
zoran  Vicovic's user avatar
0 votes
0 answers
120 views

How to prove an equality involving Laguerre polynomials

Assume $\mu<0$. Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$ and $f\in L^2(\Bbb C)$. How to prove that $$\sum^\infty_{k=0}\int_{\Bbb C} f(z)\frac{1}{2(2k+1)-\mu}L^0_k(\lvert z\rvert^2)...
zoran  Vicovic's user avatar
2 votes
0 answers
72 views

Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
Ali's user avatar
  • 4,135
3 votes
1 answer
424 views

Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$. What i want to ask is, if $S_C$ is nonempty for some $...
W.J.'s user avatar
  • 379
1 vote
2 answers
113 views

$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute. I'm ...
Fetchinson0234's user avatar
2 votes
1 answer
145 views

Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral $$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$ Where $\psi(2^{-k} \xi)$ is a smooth ...
Mr. Proof's user avatar
  • 159
10 votes
1 answer
349 views

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 ...
Ali's user avatar
  • 4,135
0 votes
0 answers
112 views

How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?

Let $ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $. Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
xiuhua's user avatar
  • 101
3 votes
1 answer
137 views

Estimate the homogeneous components of a polynomial against its maximum

Let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed. (I.e., the above sum ranges over ...
fsp-b's user avatar
  • 463
1 vote
2 answers
588 views

Inequality between coefficients of a polynomial and its supremum

For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
fsp-b's user avatar
  • 463
4 votes
1 answer
317 views

Taylor coefficients of Hadamard product

I imagine this to be a very classical question in complex analysis: Consider the Hadamard product $$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$ where $E_1(z):=(1-z)e^z$ is the first elementary ...
Guido Li's user avatar
11 votes
1 answer
582 views

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 ...
Ali's user avatar
  • 4,135
4 votes
1 answer
636 views

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>...
Ali's user avatar
  • 4,135
2 votes
0 answers
65 views

On a question relating integral equation:

I don't know if the following question qualifies as research level. If it isn't, sorry. Set the following terminology: $ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$ $\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
GSA_1's user avatar
  • 41
1 vote
2 answers
106 views

'Partial boundedness' of continuously parametrised power series

Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space. Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by $...
fsp-b's user avatar
  • 463
6 votes
1 answer
182 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
Ali's user avatar
  • 4,135
4 votes
1 answer
390 views

Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
Kung Yao's user avatar
  • 192
2 votes
1 answer
163 views

Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)

In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven: Lemma: Let $p, q\in \mathbb{R}[x]$ be stable monic ...
Strickland's user avatar
5 votes
2 answers
338 views

Approximation of analytic function by a fixed number of monomials

This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials $ \sum_{n=0}^K \frac1{n!} x^n $ ...
Catologist_who_flies_on_Monday's user avatar
2 votes
0 answers
162 views

Hilbert transform on weighted Sobolev spaces

Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
Ali's user avatar
  • 4,135
-1 votes
2 answers
129 views

Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]

Under which condition can it form a Hilbert space? Or what space can it form? You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
Nan Zhang's user avatar
9 votes
1 answer
607 views

Interpolation theory and $C^k$-spaces

Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
Jan Bohr's user avatar
  • 779
11 votes
1 answer
411 views

A density question for the Hilbert transform

Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions $$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
Ali's user avatar
  • 4,135
1 vote
0 answers
84 views

A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$. Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
Ali's user avatar
  • 4,135
0 votes
1 answer
322 views

Injectivity of analytic functions

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions: Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
Rahul Sarkar's user avatar
1 vote
0 answers
203 views

Construction of weight function to satisfy condition on given functional

Consider the following function : $$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$ Here, $\omega(z)$ is a weight we are going to consider The following two conditions should meet for $\...
bambi's user avatar
  • 375
3 votes
2 answers
498 views

if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?

This question is related to solving $f(f(x))=g(x)$. Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user avatar
2 votes
1 answer
103 views

A density question

Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that $$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
Ali's user avatar
  • 4,135
3 votes
1 answer
237 views

Isoperimetric inequality for analytic functions on an annulus

Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that $$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
MathLearner's user avatar
1 vote
0 answers
196 views

Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
bambi's user avatar
  • 375