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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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0answers
41 views

Asymptotic behavior of an exponential of an integral

Let $G(x)$ be the following function $$ G(x) = \sum_{i \in I} \alpha_i (t+x)^{\lambda_i} x^{1-\lambda_i} $$ where we know that the coefficients sum up to one ($\sum_{i \in I} \alpha_i = 1$ ) and that ...
4
votes
0answers
47 views

Notions of $\beta$-Hölder smoothness when $\beta\in (1,2]$: are they equivalent?

I posted the following question on StackExchange a few months ago (https://math.stackexchange.com/questions/2898620/notions-of-beta-h%C3%B6lder-smoothness-when-beta-in-1-2-are-they-equivalent), but ...
2
votes
0answers
23 views

A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: $$ f(t)\int_0^t {a(s)ds} + \int_0^t {a(t - s)} f(s)ds = 0 $$ My question is : under what condition ...
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0answers
27 views

Minimum of linear combination of logarithms [on hold]

I have to minimize the following function $\sum a_i log(1-b_ix)$ on $0\leq x\leq \bar{x}$ with $a_i \in R$ and $0\leq b_i<1$. In my case all the $a_i$ have the same positivity except one, but i don ...
10
votes
2answers
274 views

Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?

Good morning, I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
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0answers
57 views

A property about a MNC

Let $E, \left \|\cdot \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \...
1
vote
1answer
76 views

Multidimensional improper Riemann integrals with oscillatory kernels: Existence

I have asked this question three weeks ago here https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930 but received no relevant answers. Let $n\geq 2$ ...
0
votes
0answers
44 views

A comparison between a function and its convolution

Assume that $f$ is a $L^p$ integrable function for $1\le p\le P_0$, with $P_0$ a positive constant. L is a smooth compactly supported function. Define $L_\epsilon(x) = 1/\epsilon^n L(x/\epsilon)$. Is ...
0
votes
0answers
42 views

Existence of a function satisfying some integral conditions

I need help to prove the existence of a real function $h(x) \in C^1$ with condition that near zero $h(x) \sim \ln(x)$ and near infinity $\lim h(x)_{x \to \infty} = \infty$ such that following ...
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vote
0answers
84 views

How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by $$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$ is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
1
vote
1answer
75 views

A question on a special “metric”

Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
2
votes
0answers
47 views

Proof of a technical fact in the book of Schapire and Freund on boosting

Disclaimer: I asked this question on math.stackexchange.com two weeks ago but it has not been answered yet so I figured that I might as well try to also post it here. I am currently looking at ...
1
vote
1answer
91 views

When do supremum and expectation commute?

This is an alternative form of the question in When do maximum and expectation commute? When we looking for conditions on $G(t,x(t))$ such that $$ \sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...
0
votes
0answers
42 views

Does lattice mod preserve direction?

For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly ...
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0answers
100 views

The notion of ordered Banach space [on hold]

In this paper the authors used the notion of ordered Banach space. Definition: Let $\mathcal E$ be a Banach space with the norm $\left \| . \right \|$, whose positive cone is defined by $K=\{x\in \...
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0answers
39 views

Approximating non differentiable function with smooth function when gradient descent [on hold]

I am trying to use gradient descent to do numerical analysis of certain composition of functions. Problem is that the composed function contains non differentiable function in between differentiable ...
0
votes
0answers
53 views

Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$. Which vector fields $F$ are so that ${\rm div} (F)= f$ ?
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0answers
122 views

$L^1$-continuity estimate for ODE solutions in terms of $L^1$ distance of vector fields (only one of them being Lipschitz)

Consider the following ODE initial value problems \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in ...
16
votes
2answers
562 views

“Insanely increasing” $C^\infty$ function with upper bound

Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...
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0answers
124 views

When a potential is real analytic

Suppose $B$ is the ball of center $a$ and radius $R>0$ in $ \mathbb{R}^{n} $ $n>1$. Suppose also that $u$ is subharmonic and real analytic on a neighborhood of the closure of $B$. We know that ...
2
votes
1answer
89 views

Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction. Imagine one has two evolution equations $$\partial_t u = p(x,\partial_x,f)u$$ $$\...
2
votes
1answer
129 views

Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
2
votes
1answer
163 views

Simple but entangled inequalities

Do there exist functions $F,G$ on $[0,1]$ with $0\le F,G< 1$, such that for all $x, y\in [0,1]$ with $x+y\le 1$, the following hold? 1) $G(x)\le x$, 2) $G(1)<1$, 3) $F(x)>0$ if $x>0$, ...
1
vote
1answer
78 views

About exchanging min and max and correctness of an inequality

Let $v_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \ $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{*},i^{*})$ to the problem \begin{...
2
votes
0answers
61 views

Generalization of regularly varying functions

A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$, $$ \lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a) $$ for some function $g(a)&...
3
votes
1answer
261 views

A moment inequality

Let $\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$, where $x(t)$ and $f(t)$ are real valued continuous functions for $t\in[0,1]$, and $f(t)\geq0$. Is it possible to show that $\left(\chi(0)\chi(2)-\chi(1)^{2}...
1
vote
1answer
122 views

Convolution with Schwartz class function

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution). Consider distribution as follows: $$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...
7
votes
3answers
397 views

For what sets does the Lebesgue Differentiation Theorem hold in one dimension?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
2
votes
1answer
137 views

Does the Lebesgue Differentiation Theorem hold for regular polytopes?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
6
votes
1answer
208 views

Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?

Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$? It is known that for $n = 2$, the function $\displaystyle ...
1
vote
1answer
91 views

A question about real convex functions

This is a follow-up of a popular exercise found in Rudin's Real and Complex analysis. It is known that if a continuous function $f:\left]a,b\right[\to \bf R$ satisfies the inequality $f((x+y)/2)\le 1/...
0
votes
0answers
41 views

What can we say about the Bargmann transform of bounded function?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ Now we define $$ H(t)= H(...
2
votes
1answer
151 views

Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
2
votes
1answer
101 views

On the continuity and injective-ness of Gauss quadrature scheme for numerical integration, with weight function identically $1$

Fix an integer $n\ge 2$. Let $[a,b]$ be an interval and $f: [a,b]\to \mathbb R$ be a continuous function and for $x_1,...,x_n$ being the Gaussian Quadrature nodes in $[a,b]$, and Gaussian Quadrature ...
8
votes
1answer
599 views

Expressions for the inverse function of f(x) = ln(x)e^x

Can the inverse of $ ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function expression ...
2
votes
1answer
152 views

Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$ $$\int_{\mathbb{R}^+} f \in \...
6
votes
1answer
326 views

Interesting behaviour of binomial coefficients

Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...
3
votes
1answer
107 views

How to prove that the following function is monotonically increasing?

$$f(x) = \frac{\sqrt{x}\int_a^1 e^{-xs} s^{b+1}~ \mathrm{d}s}{\int_a^1 e^{-xs} s^{b} ~\mathrm{d}s}:\left]0,+\infty\right[\ \to \mathbb{R} $$ where $0 < a < 1$ and $b > 0.$ By applying ...
4
votes
1answer
99 views

Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...
0
votes
1answer
89 views

$\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk

Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup_D |f|\leq1$. Could ...
1
vote
1answer
108 views

Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results. Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
9
votes
2answers
424 views

Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
0
votes
0answers
143 views

Is it a weaker condition for the symmetry of mixed derivatives?

Asked in MathStackExchange and I think it may be harder than usual problems. Conditions: $f:\mathbb{R^2}\mapsto\mathbb{R}$ has second-order derivatives on some neighborhood of the zero point. $f_{...
1
vote
1answer
40 views

Monotonicity given an implicit function containing a Measure integral

The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers! Knowing that $$f(\...
5
votes
1answer
323 views

Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
1
vote
1answer
59 views

Riesz measure of a smooth subharmonic function on a ball

Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...
1
vote
2answers
54 views

Alternative characterization of epi-convergence

I am struggling with the proof of a property of epi-convergence. We need the following definitions: For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...
1
vote
0answers
115 views

The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
0
votes
1answer
106 views

Regularity in Orlicz spaces for the Poisson equation

I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007) $$\Delta u=f \quad \quad \quad \quad (1)$$ Lemma 2: There is a constant $N_1 >1$ so that ...
3
votes
1answer
347 views

Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...