# 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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### 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 ...

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votes

**0**answers

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 ...

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**0**answers

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 ...

**-1**

votes

**0**answers

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 ...

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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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**0**answers

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 \...

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vote

**1**answer

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

**0**answers

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 ...

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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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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 ...

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vote

**1**answer

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 \...

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**0**answers

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 ...

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vote

**1**answer

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_{...

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votes

**0**answers

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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votes

**0**answers

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 \...

**-1**

votes

**0**answers

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 ...

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**0**answers

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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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 ...

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**2**answers

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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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 ...

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votes

**1**answer

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
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...

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votes

**1**answer

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 \...

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votes

**1**answer

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 $x>0$,
...

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vote

**1**answer

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{...

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votes

**0**answers

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

**1**answer

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}...

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vote

**1**answer

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\...

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votes

**3**answers

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

**1**answer

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\...

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votes

**1**answer

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 ...

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vote

**1**answer

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/...

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**0**answers

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(...

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votes

**1**answer

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 $\boldsymbol{b}$ ...

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votes

**1**answer

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 ...

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**1**answer

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 ...

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votes

**1**answer

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 \...

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votes

**1**answer

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 \...

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**1**answer

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 ...

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votes

**1**answer

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.
...

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votes

**1**answer

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 ...

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**1**answer

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 $F \in L^p(\mathbb{R}^n)$ ...

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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 ...

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**0**answers

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_{...

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vote

**1**answer

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(\...

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**1**answer

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 ...

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vote

**1**answer

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 $\mu$...

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vote

**2**answers

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^\...

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**0**answers

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

**1**answer

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

**1**answer

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} ...