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

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0
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1answer
39 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
3
votes
1answer
98 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
5
votes
2answers
118 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
3
votes
1answer
224 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
2
votes
0answers
61 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written ...
4
votes
3answers
173 views

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$. ...
1
vote
1answer
93 views

A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
0
votes
0answers
28 views

Minimum size set of partition-of-unity smooth functions with bounded support [closed]

Let D be the unit disk in $R^2$, and take fixed $\epsilon <1$. I am looking for smallest possible set ${\left\{f_i\ \right\vert{}\ {0\!\leq\!f}_i\!<\!1\}}_{i=1}^n\,\,$ with the following ...
3
votes
1answer
72 views

Is the variation of two BV functions the same in the set in which they coincide?

Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set $A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$ is it true that ...
2
votes
0answers
57 views

Asymptotic behavior of a function [migrated]

Let $n\geq 2$, and define $$\phi(t) = \int_{S^{n-1}} \cos(t \omega_1) d\omega,$$ where $S^{n-1}$ is the unit sphere in $n-$dimensional euclidean space, and $d\omega$ denote the surface area on ...
1
vote
2answers
109 views

Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set ...
-1
votes
0answers
27 views

Total measure and Riesz theorem [migrated]

As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial. Riesz representation theorem states the following: Given $X$ a locally compact ...
-4
votes
1answer
143 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
0
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0answers
48 views

Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
-4
votes
0answers
29 views

Help with math Real Analyis [closed]

Let {An} be a sequence in R and let f be continuous real valued function on all R.If {An} converges to L,prove that {f(An)} converges to F(L).?
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0answers
20 views

Is scaling mapping uniformly continuous for any metric in $R$? [migrated]

For example, is $f(x)=2x$ as a function of $R\rightarrow R$ uniformly continuous for a general metric $d(x,y)$? Or is the following true? $\forall \epsilon > 0, \exists \delta $, such that ...
1
vote
1answer
84 views

Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
-1
votes
0answers
11 views

How can i get real analog of complex function? [migrated]

I have a function: sin(wt-jT) (1.1), where j - complex number I transform it to function with real arguments: ...
0
votes
0answers
20 views

Correct definition of submodularity

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
1
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0answers
50 views

Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty ...
2
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0answers
150 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral ...
0
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0answers
47 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
2
votes
1answer
233 views

Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
3
votes
0answers
123 views

Extreme derivatives in Baire class 1

In the 1994 volume of "Differentiation of Real Functions" A. Bruckner poses the following problem (p.41): "Find necessary and sufficient conditions on a continuous function $F$ that its Dini ...
-2
votes
0answers
105 views

Conjecture about tail events

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \tau_{A} := \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ ...
1
vote
1answer
73 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
1
vote
0answers
63 views

The real method of interpolation and operator ideals,

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding) ...
-2
votes
1answer
89 views

Is this intergral inequality valid? [closed]

Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
-1
votes
0answers
70 views

inverse Fouier transform from partial Fourier transform

It is known that a function $u(\mathbf{x})\in C_0^{\infty}(\Omega)$ can be reconstructed from its Fourier transform using inverse Fourier transform. $$u(\mathbf{x}) = \mathcal{F}^{-1} (\widehat{u}) = ...
2
votes
1answer
117 views

Approximation of the cumulative normal distribution

As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...
4
votes
1answer
612 views

When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...
14
votes
3answers
449 views

Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$

Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$. I have the following theorem: For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which ...
3
votes
1answer
50 views

Injectivity of vector functions: Numerical Verification

Problem Setup Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact. The function $f(\cdot)$ can only be evaluated numerically. ...
1
vote
1answer
136 views

Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that $$\int_0^T\Vert ...
2
votes
1answer
81 views

Scaling of distributions

Suppose we have a sequence of $L^1(\mathbb{R})$ functions $p_\epsilon$ with $\|p_\epsilon\|_{L^1} \leq 1$ for all $n$. Suppose we know that $p_\epsilon \to 0$ in distributions. Is it obvious that ...
2
votes
1answer
102 views

Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$. If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$, $$ ...
1
vote
1answer
105 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ ...
4
votes
1answer
104 views

Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be ...
4
votes
1answer
114 views

Compact, not local uniform convergence of sequences of functions on the rationals

I stumbled upon the following elementary problem while trying to come up with a certain counterexample in category theory. (Basically, I am interested in the constant sheaf of $\mathbb F_2$-vector ...
3
votes
1answer
177 views

Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?

I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of ...
0
votes
1answer
69 views

A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...
3
votes
0answers
92 views

Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
1
vote
0answers
75 views

monotonicity of a function

I want to know if the function below is monotonically decreasing for all $a,b >0, a\neq b $ \begin{equation} x\rightarrow \frac{\sinh^2((a-b)x)}{\sinh(2ax)\sinh(2bx)} \text{, $x >0. $} ...
1
vote
0answers
52 views

Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum: $ \inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt, $ where $ ...
-1
votes
1answer
51 views

Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
3
votes
1answer
148 views

Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e. $$\rho(\mu,\nu)<\varepsilon,$$ then there exist two random ...
6
votes
1answer
134 views

Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
1
vote
0answers
89 views

Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...
2
votes
1answer
70 views

Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha ...
1
vote
0answers
29 views

Constant periodic Sobolev embedding

Dear mathoverflowers, I would like to have a reference regarding the optimal constant in the Sobolev embedding $$ \|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s}, $$ ($H^s$ denotes the standard L^2 ...