Questions tagged [real-analysis]

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

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Bound for the laplacian of a strictly convex function from above by the gradient of it

Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$ I am trying to show that $$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
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1 vote
1 answer
67 views

Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?

I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$. I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
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2 votes
1 answer
62 views

Zeros in $[0,1]$ of functions $f \in \mathrm{span} \{ p(x - \lambda_k)e^{\lambda_k x} : k=1,\dots, n \}$

Let $n \in \mathbb N$, let $p:\mathbb R \to \mathbb R$ be a real polynomial, and let $\lambda_1< \lambda_2 <\dots < \lambda_n$. Now let $$ f \in \mathrm{span} \left \{ p(x - \lambda_k)e^{\...
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-1 votes
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Commutativity of convolution and pointwise application of functions

Question: is it possible to have for non-trivial functions $f,g,h$ examples of $f(\lbrace g*h\rbrace(t))\equiv \lbrace g*f\rbrace(h(t))$, i.e. that e.g. applying a function $f(x)$ to a time-series $h(...
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A maximization problem over a subspace of Sobolev space

Let $E\subset \mathbb R^2$ be a "nice" region (e.g. connected, open and bounded). Denote by $H^1_0$ the Sobolev space on $E$, i.e. $$H^1_0:= \left\{f: E\to\mathbb R:\quad f \big|_{\partial E}...
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1 answer
72 views

Integral and inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
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1 answer
158 views

Integral with inequality

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$ ...
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Inequality on matrix trace

Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems : $$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma ...
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4 votes
2 answers
123 views

Lebesgue differentiation theorem at boundary points for Sobolev traces

$\newcommand{\R}{\mathbb R}$ Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$. Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then $$ u(x)...
3 votes
1 answer
82 views

Analyticity of central stable manifolds

Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...
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Measurability of a process defined by an integral

Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$ Let $\mathcal{P} \subset \mathcal{B}(\mathbb{R}_+) \otimes \mathcal{F}$ the $\sigma$-algebra generated by $\{\{0\}\times F_0\}...
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3 votes
2 answers
190 views

Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$

I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function \begin{equation} f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr) \end{equation} attains its maximum inside the ...
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1 vote
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On optimizing a multivariate quadratic function subject to certain conditions

The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
-3 votes
0 answers
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Sobolev embedding [migrated]

I was trying to understand Sobolev embedding, some results about this topic are not clear to me. My question is the following: what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for $W^{...
2 votes
0 answers
75 views

Sonin inversion formula, equivalence of two solutions of an integral equation

Let me first specify the problem I am facing, and then below explain where it arises. Given a function $f(x)$ on the interval $0<x<1$ and a real number $s\in(-1,1)$ I consider the integral ...
3 votes
2 answers
152 views

Does this condition imply absolute continuity?

Let $f: [0, 1] \to \mathbb R$ be a measurable function. Define the (possibly infinite valued) upper and lower Dini derivative $D^+ f, D^- f: [0, 1] \to [-\infty, \infty]$ by $$D^+ f (x) := \limsup_{y \...
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1 vote
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64 views

Functions $f: \mathbb R \to \mathbb R$ such that $\det [f(a_j-b_k)]_{j,k} \neq 0$ for all $a_1,b_1, \dots, a_N,b_N$ and all $N \in \mathbb N$

A function $f: \mathbb R \to \mathbb R$ is called totally positive if for every $N \in \mathbb N$, every $a_1< a_2< \dotsb < a_N \in \mathbb R$ and every $b_1 < b_2 < \dotsb < b_N \...
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1 vote
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105 views

Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
6 votes
2 answers
343 views

Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators $$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$ and $$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
1 vote
1 answer
58 views

Asymptotic properties of weighted random walks / infinite convolutions of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. real-random variables. Let further $0<c<1$. I'm interested in the asymptotic properties of $$ \sum_{k=1}^n c^k X_k. $$ I can prove that this ...
2 votes
1 answer
150 views

Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it

I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such ...
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1 vote
2 answers
74 views

Measurability of Brjuno numbers

A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...
-2 votes
1 answer
151 views

Simple closed form for $\int \lfloor x \rfloor dx$? [closed]

Wolfram Alpha claims there is no closed form in terms of standard funcions for $\int \lfloor x \rfloor dx$ but we believe we found simple closed form agreeing with experimental data. Define $i_1(x)=x -...
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Asymptotic expansion of the laplace transform for $s\to\infty$

Let $F(s)$ denote the Laplace transform of $f(t)$. Expanding for large $s$ one can obtain the following relation: $$F(s)\approx f(0)/s+f'(0)/s^2+2!f''(0)/s^3+3!f'''(0)/s^4+...\quad s\gg1$$ See e.g. a ...
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0 votes
1 answer
93 views

Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$

$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that \begin{equation} \sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq ...
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2 votes
0 answers
60 views

Power series of the modified Bessel function of the second kind

I am looking for a power series representation of $$ \frac{1}{K_{\nu}(x)}, $$ where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer. I know that ...
17 votes
3 answers
780 views

Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds

For real variable $x$, the function \begin{equation} f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n \end{equation} clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
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6 votes
2 answers
367 views

Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?

The question is as in the title: Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
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0 votes
1 answer
131 views

What does "a universal tree" mean?

It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the ...
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3 votes
1 answer
88 views

Inequality: multivariate normal distribition

Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$ Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>...
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0 votes
1 answer
87 views

Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?

Let $X := \mathbb R^n$, $C_b(X)$ the space of all real-valued bounded continuous, $C_c(X)$ the space of all real-valued continuous functions with compact supports, and $C_c^\infty(X)$ the space of ...
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3 votes
1 answer
95 views

Is the smallest root of this quartic always the closest point on the Hyperbola? [closed]

Let $a>b>0$. Suppose we want to minimize $$ f(x)=(x-a)^2+(1/x-b)^2, $$ over $x>0$. Equating $f'(x)=0$ leads to the quartic equation $$ g(x)=x^4-ax^3+bx-1=0. \tag{1} $$ Question: Is the ...
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5 votes
1 answer
182 views

An inequality for polynomials

I have been thinking about the validity of the following inequality: if $P(z)=\sum_{k=0}^na_kz^k, a_n\neq 0$ and $P(z)$ is non-zero in $|z|<1, $ then for $\theta \in [0, 2\pi],$ and $p>0$ \...
5 votes
0 answers
103 views

Regularity of the spherical mean of a compactly-supported function

The problem Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$. Then, consider ...
6 votes
1 answer
242 views

A characterisation of continuous real functions

Let $f: \mathbb R^n \to \mathbb R$ be a measurable function. We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
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2 votes
1 answer
133 views

Asymptotic analysis of an expression involving a Fox's H function

One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...
0 votes
1 answer
65 views

Lower bounds on eigenvalues of sum of two matrices (one of them is symmetric)

Let $L$ be a matrix with eigenvalues($\lambda$ $\geq$ 0). If I add a constant value (say $a$) to all the elements of $L$, what can we say about the minimum eigenvalue of this perturbed matrix? Note: $...
1 vote
1 answer
117 views

Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$

Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE. Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) :=...
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2 votes
2 answers
144 views

Convergence of series related to partial fraction expansion of cotangent function

I am looking at the convergence of the series $$ \cos(t\theta) = \frac{\sin(\pi t)}{\pi} \cdot \Bigg[\frac{1}{t} + 2t \sum_{k=1}^\infty (-1)^k \frac{\cos(k\theta)}{t^2 - k^2}\Bigg].$$ Here $t\in\...
3 votes
1 answer
349 views

Boyd & Chua 1985: Is the proof of Lemma 2 correct?

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$I'm reading this article by Boyd and Chua [1], in which they prove the approximability of arbitrary time-invariant (TI) operators ...
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1 vote
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45 views

Necessary and sufficient conditions so that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
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5 votes
0 answers
119 views

Weaker versions of the Riemann series theorem in constructive mathematics

The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real ...
1 vote
1 answer
63 views

When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...
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2 votes
0 answers
49 views

A division of real analytic functions

Problem statement Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$. Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\...
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2 votes
0 answers
109 views

A technical question concerning convolution product

Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$. Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
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12 votes
2 answers
430 views

Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$

I'm interested in the asymptotics of $$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$ as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...
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1 vote
1 answer
57 views

Best projection on non-convex discrete set with two constraints

I want to compute the projection of a vector $\left( x\right) _{1\leq i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set $$ S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
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1 vote
0 answers
101 views

Uniformly open map on a dense subset

Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion. I think the ...
3 votes
0 answers
127 views

Harmonic polynomial of degree 3

Let $f:\Bbb R^3\to\Bbb R^3$ be a function defined by $$ \begin{split} f(x,y,z) = & \,\Big\{a_1 x y z +a_2\left(-x^3+3 x y^2\right) +a_3\left(3 x^2 y-y^3\right) +a_4\left(3 y^2 z-z^3\right) \\ &...
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3 votes
1 answer
181 views

Dividing a spherical cap into $n$ equal wedges

This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown. Motivation: Optimal ways to cut an orange. In this problem, we have a spherical ...

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