Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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-4
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0answers
10 views

Good estimates on the truncation of the exponential series. [on hold]

What are the good (analytic) upper bounds we have on the series $\sum_{k=D}^\infty \frac{a^k}{k!}$ in terms of $a$ and $D$?
2
votes
3answers
184 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...
1
vote
0answers
21 views

The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
2
votes
2answers
170 views

Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange. In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ ...
-2
votes
0answers
98 views

Non-classical real generalization of Stirling formula [on hold]

Let $x$ from $\mathbb{R}_0^+$. (Optionally $x$ can be complex , but will not be discussed.) Be $$F_n(x):=\frac{f(n+1)^x}{\prod\limits_{k=1}^n (1+\frac{x}{f(k)-f(0)})^{g(k)-g(k-1)}}$$ $$S_n(x):=K_f ...
2
votes
1answer
60 views

A continuity/bootstrap argument

I am trying to understand how one can prove the following assertion using a continuity argument: Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to ...
1
vote
0answers
28 views

Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
3
votes
1answer
164 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
0
votes
0answers
42 views

Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, ...
1
vote
1answer
51 views

linear recurrence inequality of positive terms

This is a follow up on my previous linear recurrence inequality question. I have some matrices which satisfy a linear recurrence formula of the form $$ A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad ...
2
votes
1answer
62 views

linear recurrence inequality

Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does not imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the ...
9
votes
4answers
929 views

What's the use of Malgrange preparation theorem?

The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
21
votes
3answers
514 views

Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
13
votes
2answers
337 views

Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...
2
votes
1answer
64 views

On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
-8
votes
0answers
70 views

Masters of Pure Mathematics [closed]

Let f:[0, infinity)-->R be a function such that for any a€R, the seq f(an) converges to 0. Does lim f(x) exist as x--> infinity.?
0
votes
0answers
56 views

Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on ...
4
votes
1answer
256 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} ...
34
votes
1answer
659 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
1
vote
0answers
16 views

Concavity of maxima [closed]

Suppose we have the following optimization problem : $\min\limits_x kf(x) + g(x)$ where $f$ is a decreasing convex function in $x$ and $g$ is an increasing convex function. Can we say that $x^*$ is ...
-1
votes
0answers
33 views

Compact subsets, how to check? [closed]

My question is: How can I see if (in $\mathcal H=\mathcal l^2 (\mathbb{N} )$ $B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$ ,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$ are compacts ...
-1
votes
1answer
167 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
3
votes
0answers
109 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
9
votes
2answers
165 views

Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?

I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
1
vote
1answer
107 views

Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
1
vote
0answers
46 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
0
votes
1answer
51 views

Long term behavior of a certain discrete time dynamical system on graphs

Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$. Assume that each vertex has an ...
2
votes
0answers
39 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
8
votes
1answer
568 views

Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...
2
votes
1answer
145 views

Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture: Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$ Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto ...
3
votes
1answer
285 views

About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
0
votes
0answers
84 views

Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers. Does $$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...
8
votes
1answer
361 views

Why should the map $-\Delta^{-1}$ be continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
3
votes
1answer
167 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
4
votes
0answers
114 views

Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...
5
votes
1answer
600 views

Is the following integral nonzero?

Recently I met an integral as follow: $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq ...
1
vote
1answer
60 views

Partition of Real Number [closed]

Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
2
votes
1answer
926 views

For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that $$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
48
votes
1answer
2k views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
5
votes
2answers
287 views

Brownian motion, quadratic variation, existence of partitions?

Let $B_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t_{k, n} : k = 0, 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = ...
1
vote
0answers
74 views

Convolution Integral involving an unknown function

I've got the following problem I'm working on which is related to some of my research. I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 ...
4
votes
0answers
88 views

Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
2
votes
1answer
101 views

Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
9
votes
1answer
232 views

Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
0
votes
0answers
32 views

On the continuity of Riemann-Liouville integral

For a function $f:(0,1)\to\mathbb{R}$, the Riemann-Liouville integral of $f$ is defined by $$ (I^{1-\nu}f)(t):=\frac{1}{\Gamma(1-\nu)}\int_{0}^{t}\frac{f(s)}{(t-s)^{\nu}}ds, $$ where $\nu\in(0,1)$ is ...
5
votes
1answer
131 views

Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq ...
1
vote
1answer
88 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...
4
votes
2answers
2k views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
1
vote
1answer
113 views

Integration against Borel measures on compact Hausdorff spaces

I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof. Suppose that $X$ ...
2
votes
1answer
84 views

A singular integral of several functions

While playing with some PDE I came across a singular integral that looks something like ...