All Questions
297 questions
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534
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Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane
Setting:
Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$.
Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
- 398
1
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1
answer
185
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About the upper bound on the roots of the matching polynomial
Heilman and Lieb had proven that if a graph had $d$ as its maximum vertex degree then the roots of the matching polynomial are bounded from above by $2\sqrt{d-1}$.
Is there a modern exposition of ...
- 617
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1
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3k
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In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
1
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1
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121
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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
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 4,143
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2
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588
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Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
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2
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106
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'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 463
1
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1
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652
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Significance and motivation for outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
- 175
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1
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65
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Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
1
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1
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152
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SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 151
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1
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132
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Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
1
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1
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119
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weakly separated sequences in RKHS are separated by Gleason metric
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, the authors ask to observe that weakly separated in a Reproducing kernel hilbert space implies separated ...
- 151
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1
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90
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The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
- 4,058
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1
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86
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Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 190
1
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1
answer
128
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Computation on the Hardy space
Why
$$
\Pi_+ \left(\frac{\overline{z}}{1-\overline{qz}}f\right)= \frac{f(z)-f(\bar{q})}{z-\overline{q}}, \quad f\in H^2(\mathbb D),$$
where
$q\in \mathbb C$,
$\Pi_{+}$ is the Szegö projector:
$$\Pi_{+...
- 97
1
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1
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597
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A question about the proof of Riesz-Thorin interpolation theorem
I was reading the proof of Riesz-Thorin interpolation theorem in http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf
and get stuck at the last step. We construct the complex ...
- 323
1
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1
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136
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Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
1
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1
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191
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Sequence of smooth maps converging to the identity [closed]
Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
- 27
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1
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215
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About principal values and Wirtinger derivative
Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$.
I'm interested in ...
- 377
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2
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263
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Books on real and/or complex analytic functions on Banach spaces taking values in Banach spaces
I'm looking for good textbooks on the subjects.
If you know one(s), please let me know.
- 1,169
1
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1
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1k
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Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
- 1,644
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1
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780
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An asymptotic series for the digamma function
As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number.
$$
\psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}}
$$
$B_n$ is the first Bernoulli numbers.
How ...
- 1,551
1
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2
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113
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$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$
Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm ...
- 1,150
1
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1
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220
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How to prove that weighted Bergman space is separable.
Let $D$ be a bounded domain in $\mathbb{C}^n$ and $\varphi$ be a non-positive plurisubharmonic function on $D$. The weighted Bergman space $A^2(D,e^{-\varphi})$ is the space of holomorphic functions ...
- 11
1
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1
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168
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Sampling set: relatively dense and uniformly discrete
The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...
- 319
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1
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250
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Square integrability of quotient of square integrable holomorphic functions
Let f be an anti-symmetric function in the Bergman space of the bi-disc. That is, f is square integrable, holomorphic on the bi-disc and $f(z_1,z_2)=-f(z_2,z_1).$ So, $f(z_1,z_2)/(z_1-z_2)$ extends as ...
- 11
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1
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130
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A problem on completeness of a specific space of complex functions
Let $C(D) \cap H(\bar D)$ denote the inner product space of functions these are analytic in unit disk $D$ and continuous in $\bar D$, equipped with the inner product $(f,g)= \frac{1}{2 \pi} \int_{0}^{...
- 13
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1
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137
views
When is a homogeneous polynomial an inner function on the unit torus?
What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
- 11
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1
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195
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System of quadratic complex equations
I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| x_{k}-...
- 13
1
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1
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323
views
Infinite product's question
Given a pair of strictly increasing functions $f,g:\mathbb{N}\to \mathbb{N}$
define:
$P_N(f,g)\doteq \left(z\in \mathbb{C}\mapsto \prod_{i=1}^{f(N)}\left(1+\frac{z}{v_i(N)}\right)\in \mathbb{C}\...
- 1,805
1
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1
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312
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Invertibility of frame/sampling operator on Bargmann-Fock spaces
Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
1
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0
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146
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 ...
- 521
1
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0
answers
71
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Integral formula of quantum dilogarithm
In the paper"Level N Teichmüller TQFT and Complex Chern-Simons Theory" arXiv:1612.06986, the authors study the quantum dilogarithm function:
\begin{equation}
\mathrm{D}_{\rm b}(x,n)=\prod_{...
- 109
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0
answers
38
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About Carleson measures on the Hardy space on the bidisc
I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
...
1
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0
answers
63
views
Extension of meromorphic distribution
Let $W$ be a topological vector space (e.g. Frechet) with a dense subspace $V$. Let $D_s$ be a distribution on $V$ that is meromorphic in $s\in\mathbb C$ and extends continuously to $W$ with respect ...
- 3,799
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0
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115
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Looking for examples of kernels with scalar Pick property but not the complete Pick property
I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy.
A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...
- 151
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0
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125
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Interpolating sequences are strongly separated
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
- 151
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1
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88
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Mean values of polynomial and holomorphic matrices
Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...
- 981
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0
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113
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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 ...
- 4,143
1
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0
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111
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Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
- 11
1
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0
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140
views
Does a Borel transform uniquely determine a Borel measure?
It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question.
I came across the concept of a Borel transform of a Borel ...
- 3,515
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0
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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
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0
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94
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Less strict holomorphy
Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....
- 117
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0
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271
views
Starlike sets in $\mathbb{C}^n$
Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
- 61
1
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0
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74
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Is $A^{-\infty}$ a $Q$-algebra?
Let $\mathbb{D}=\{z\in \mathbb{C}: |z|<1\}$ and
$$
A^{-\infty}=\Big\{f:\mathbb{D}\rightarrow \mathbb{C}\;|\; \exists n\in \mathbb{N} \textrm{ such that }\|f\|_{-n}= \sup\limits_{z\in \mathbb{D}}\;\!...
- 21
1
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0
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139
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Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
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 ...
- 4,143
1
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1
answer
294
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Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
- 31
1
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0
answers
145
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Using Paley-Wiener Theorem to prove the decay of $G(x-y)$
This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
- 1,305
1
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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 $\...
- 375
1
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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)])...
- 375