# Questions tagged [real-analysis]

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

4,682
questions

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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\...

1
vote

1
answer

67
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### 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^...

2
votes

1
answer

62
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### 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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### 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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29
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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}...

0
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1
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72
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### Integral and inequality

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

0
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1
answer

158
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### 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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0
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94
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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 ...

4
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2
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123
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### 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
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### 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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40
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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\}...

3
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2
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190
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### 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 ...

1
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0
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48
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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) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...

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0
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53
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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
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### 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
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### 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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0
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64
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### 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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### 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
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2
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343
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### 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
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1
answer

58
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### 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
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### 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 ...

1
vote

2
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74
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### 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 ...

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votes

1
answer

151
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### 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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votes

0
answers

24
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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 ...

0
votes

1
answer

93
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### 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 ...

2
votes

0
answers

60
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### 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
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### 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 $\...

6
votes

2
answers

367
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### 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 ...

0
votes

1
answer

131
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### 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 ...

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>...

0
votes

1
answer

87
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### 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 ...

3
votes

1
answer

95
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### 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 ...

5
votes

1
answer

182
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### 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
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### 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
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### 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)|&...

2
votes

1
answer

133
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### 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
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### 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
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### 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) :=...

2
votes

2
answers

144
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### 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
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### 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 ...

1
vote

0
answers

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### 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(...

5
votes

0
answers

119
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### 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
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### 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(...

2
votes

0
answers

49
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### 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^\...

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 ...

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'...

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;\...

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) \\
&...

3
votes

1
answer

181
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### 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 ...