Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
4,682
questions
0
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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
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
- 43
1
vote
1
answer
67
views
Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?
I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
- 1,269
2
votes
1
answer
62
views
Zeros in $[0,1]$ of functions $f \in \mathrm{span} \{ p(x - \lambda_k)e^{\lambda_k x} : k=1,\dots, n \}$
Let $n \in \mathbb N$, let $p:\mathbb R \to \mathbb R$ be a real polynomial, and let $\lambda_1< \lambda_2 <\dots < \lambda_n$. Now let
$$
f \in \mathrm{span} \left \{ p(x - \lambda_k)e^{\...
- 180
-1
votes
0
answers
52
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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(...
- 11.7k
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votes
0
answers
29
views
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.
$$H^1_0:= \left\{f: E\to\mathbb R:\quad f \big|_{\partial E}...
- 55
0
votes
1
answer
72
views
Integral and inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 221
0
votes
1
answer
158
views
Integral with inequality
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 221
1
vote
0
answers
94
views
Inequality on matrix trace
Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems :
$$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma ...
- 55
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)...
- 4,144
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 ...
- 1,332
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0
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40
views
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\}...
- 221
3
votes
2
answers
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 ...
- 1,269
1
vote
0
answers
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)$ ...
- 61
-3
votes
0
answers
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^{...
- 35
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 ...
- 158k
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 \...
- 1,928
1
vote
0
answers
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 \...
- 339
1
vote
0
answers
105
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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 ...
- 41.4k
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.
$$
I can prove that this ...
2
votes
1
answer
150
views
Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it
I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such ...
- 1,269
1
vote
2
answers
74
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Measurability of Brjuno numbers
A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...
- 41.4k
-2
votes
1
answer
151
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Simple closed form for $\int \lfloor x \rfloor dx$? [closed]
Wolfram Alpha claims there is no closed form in terms of standard funcions
for $\int \lfloor x \rfloor dx$ but we believe we found
simple closed form agreeing with experimental data.
Define $i_1(x)=x -...
- 23.7k
0
votes
0
answers
24
views
Asymptotic expansion of the laplace transform for $s\to\infty$
Let $F(s)$ denote the Laplace transform of $f(t)$.
Expanding for large $s$ one can obtain the following relation:
$$F(s)\approx f(0)/s+f'(0)/s^2+2!f''(0)/s^3+3!f'''(0)/s^4+...\quad s\gg1$$
See e.g. a ...
- 97
0
votes
1
answer
93
views
Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$
$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that
\begin{equation}
\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq ...
- 1,269
2
votes
0
answers
60
views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...
- 83
17
votes
3
answers
780
views
Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds
For real variable $x$, the function
\begin{equation}
f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n
\end{equation}
clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
- 1,269
6
votes
2
answers
367
views
Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
The question is as in the title:
Is there a nonpolynomial $C^\infty$ function $f$ on $\mathbb{R}$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every natural ...
- 1,269
0
votes
1
answer
131
views
What does "a universal tree" mean?
It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the ...
- 13
3
votes
1
answer
88
views
Inequality: multivariate normal distribition
Let $p(u,x)=\frac{1}{(4\pi u)^{q/2}}e^{-|x|^2/(4u)},u>0,x \in \mathbb{R}^q.$
Prove that for $r\geq 0,c>1$ there exists $C>0$ (depending on $r,c$) such that $$\forall x \in \mathbb{R}^q,u>...
- 221
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 ...
- 375
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 ...
- 6,489
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 $p>0$
\...
- 437
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 ...
- 575
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)|&...
- 1,928
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) :=...
- 6,216
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\...
- 2,690
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 ...
- 33
1
vote
0
answers
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(...
- 528
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 ...
- 1,686
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(...
- 528
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^\...
- 559
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 ...
- 793
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'...
- 528
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
$$
S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
- 45
1
vote
0
answers
101
views
Uniformly open map on a dense subset
Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion.
I think the ...
- 11
3
votes
0
answers
127
views
Harmonic polynomial of degree 3
Let $f:\Bbb R^3\to\Bbb R^3$ be a function defined by
$$
\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) \\
&...
- 61
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 ...
- 500