Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
33 views

Closed form of $\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^2}$

Given vectors $m_{j}\in\mathbb{Z}^{n},M=(m_{1} \ldots m_{n}),\det(M)\not =0$. Is it possible to find a closed form of: $$S=\frac{1}{{\pi}^n}\sum_{k\in \mathbb{Z}^n}\prod_{j=1}^n\frac{1}{1+(m_{j}^{T}k)^...
Quý Nhân's user avatar
5 votes
2 answers
103 views

On the continuity a function given by evaluating compact subsets of smooth functions

Let $M$ be a compact connected smooth manifold. Write $C^{\infty}(M)$ for the Frechet space of the smooth real-valued functions on $M$ equipped with the usual $C^{\infty}$-topology. Given a compact ...
S.Z.'s user avatar
  • 557
12 votes
0 answers
117 views

When could a diligent calculus student compute all Picard iterates algebraically?

As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
James E Hanson's user avatar
0 votes
0 answers
46 views

Fractal dimension using wavelets [closed]

I'm trying to estimate the fractal dimension of a function. I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation $$ \alpha = -...
user38747's user avatar
5 votes
1 answer
726 views

Can the Pythagorean theorem be proved using imaginary numbers?

Can the Pythagorean theorem be proved using imaginary numbers? The proof must avoid circular reasoning, of course. I asked essentially the same question at MSE, but did not receive a definitive answer,...
Dan's user avatar
  • 3,577
-5 votes
0 answers
86 views

Every smooth function contains a bijection [closed]

Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
John Wayne's user avatar
-3 votes
1 answer
104 views

when does $h$ exist?

Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
Roy Burson's user avatar
-4 votes
0 answers
63 views

Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs? [closed]

To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$ $$ \eta(s) = \sum_{n=...
Roy Burson's user avatar
-3 votes
0 answers
76 views

Exercise generalizing (related to) Hölder's inequality

I came across this exercise and feel absolutely stuck: Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
HZA's user avatar
  • 1
2 votes
0 answers
117 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
0 votes
0 answers
67 views

$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable

Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported. I am wondering if we can ...
Jens Fischer's user avatar
2 votes
0 answers
50 views

Convergence of finite-difference method for Cauchy-Riemann equations

Let $I\subseteq \mathbb{R}$ an open interval. Let $f:I\rightarrow \mathbb{C}$ real analytic. Suppose we want to numerically compute an analytic extension of $f$. We will assume the following: we are ...
Plemath's user avatar
  • 312
4 votes
1 answer
286 views

Eigenvalue of a convolution and a restriction?

Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
92 views

About BMO space on smooth open bounded domain

Let $\Omega$ be any open domain in $\Bbb R^d$. Define the $\text{BMO}(\Omega)$ space as $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
3 votes
0 answers
89 views
+50

Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps

Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function $$ m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)}, $$ where $\lambda_{\max}$ denotes the largest eigenvalue....
Ran's user avatar
  • 73
2 votes
0 answers
80 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
53 views

Description of all biholomorphic maps from annulus [duplicate]

Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected? In ...
Jinyang wu's user avatar
2 votes
0 answers
229 views

A deceptively simple regularity problem for functions on the plane

By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer: Consider a twice ...
vmist's user avatar
  • 989
0 votes
0 answers
83 views

Singular behavior of zeros of incomplete zeta function

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
Zhobbyist's user avatar
1 vote
0 answers
39 views

Currents with logarithmic poles compared with those with no poles

I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by $$ '\...
neander's user avatar
  • 161
12 votes
1 answer
398 views

Is $X\times X$ homeomorphic to $X$ for a space of probability measures?

Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if $$ \int_S ...
Christian Remling's user avatar
3 votes
0 answers
67 views

Effective action of unbounded operators on subspaces outside their domains of definition

Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
Qualearn's user avatar
  • 133
3 votes
2 answers
147 views

Vector bundles over a Stein space are projective

It is a "well known" fact that locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules (see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
Tim's user avatar
  • 1,109
0 votes
1 answer
160 views

Weak convergence of $f(x,e^{itx})$

This is the desired result (what I want to prove): $$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$ Given that $f\in C([a,b]\times\{e^{i\...
Quý Nhân's user avatar
9 votes
1 answer
302 views

What are the points of the algebra of polynomial functions on an arbitrary vector space?

Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
Dima Roytenberg's user avatar
0 votes
0 answers
46 views

Amenability of locally convex algebras

Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.). Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
Onur Oktay's user avatar
  • 2,605
1 vote
0 answers
88 views

Density of a subset of Schwartz space in the fractional Sobolev space

It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
Nirjan Biswas's user avatar
1 vote
1 answer
154 views

Is smoothness preserved under an isometric isomorphism?

Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
Tuh's user avatar
  • 113
2 votes
0 answers
68 views

Bessel spaces and Triebel Lizorkin

It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
Guillermo García Sáez's user avatar
1 vote
0 answers
58 views

duality of sobolev spaces. Representation of elements in the dual

I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
Alucard-o Ming's user avatar
0 votes
0 answers
43 views

questions on stochastic kernels and pushforward operator

Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
andy's user avatar
  • 11
4 votes
0 answers
92 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
Akira's user avatar
  • 825
0 votes
0 answers
90 views

How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?

I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
Tuh's user avatar
  • 113
0 votes
1 answer
127 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\...
Ali's user avatar
  • 4,115
1 vote
0 answers
148 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
3 votes
0 answers
78 views

Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$

In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
Józef Zápařka's user avatar
0 votes
1 answer
239 views

Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]

Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
H Tomasz Grzybowski's user avatar
2 votes
1 answer
201 views

Section 3 of Atiyah's "On analytic surfaces with double points" — some questions

I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4. Near the end of section 3, ...
maxo's user avatar
  • 129
4 votes
0 answers
101 views

There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions

Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$. Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
Pierre PC's user avatar
  • 3,669
1 vote
0 answers
49 views

What makes the generalized projection different than metric on a Banach space?

I have came across the notion of generalized projection in Banach spaces, introduced by Ya. Alber and has seen many iterative algorithms being solved by using this projection. It helps in finding the ...
PPB's user avatar
  • 85
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
0 votes
0 answers
77 views

Nice formula for powers of modified Bessel function

Let $K_\nu(z)$ be the modified Bessel function of second kind. I am looking the geometric series $$1+aK_v+(aK_v)^2+(aK_v)^3...$$ I know there are formula for product of two such functions. I would ...
CO2's user avatar
  • 275
3 votes
0 answers
160 views

Gowers' dichotomy for quotients

Gowers' dichotomy establishes that every infinite dimensional Banach space contains a closed infinite dimensional subspace that has an unconditional basis or it is hereditarily indecomposable. A ...
M.González's user avatar
  • 4,461
0 votes
0 answers
57 views

Complexity of evaluation of analytic functions

Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
roignoirewg's user avatar
1 vote
0 answers
53 views

Can one explicitly define a right inverse for a convolution operator on the space of entire functions?

A result of Meise and Taylor in 1988 shows that every non-zero convolution operator on the Frechet space $H(\mathbb{C})$ of all entire functions on $\mathbb{C}$ has a continuous linear right inverse $...
David Walmsley's user avatar
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
52 views

On distributions and kernels

Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
G. Blaickner's user avatar
  • 1,429
0 votes
1 answer
171 views

Is the evolution family self-adjoint?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
Akira's user avatar
  • 825
1 vote
1 answer
61 views

Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
user2103480's user avatar
1 vote
0 answers
84 views

Does sets of positive capacity rule out constant functions?

Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by \begin{align*} \text{Cap}_{p}(K, U) := \inf \left\{ \int_U |\...
Guy Fsone's user avatar
  • 1,101

1
2 3 4 5
256