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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4
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0answers
118 views

Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$ \Phi_i(2t)\approx \Phi_i(t), \quad i=1,2 $$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
3
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1answer
111 views

Stone-Weierstrass theorem for modules of non-self-adjoint subalgebras

In "Weierstrass-Stone, the Theorem" by Joao Prolla, there is a Stone-Weierstrass theorem for modules, stated as the following: Let $\mathcal{A}$ be a subalegebra of $C(X, \mathbb{R})$ and $...
7
votes
3answers
245 views

Shrinking subset and product

Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
14
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1answer
800 views

Was Cantor aware of Lebesgue theory of integration?

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
2
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1answer
133 views

A question about series involving a Sobolev functions

Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
3
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0answers
58 views

Elliptic equations in semi-infinite strips

Let $\Omega$ be a bounded domain in $\mathbb R^n$ with smooth boundary. Let $g=(g_{jk})_{j,k=0}^n$ be a Riemannian metric on $\mathbb R^+\times \Omega$ with smooth bounded components. Is there a good ...
2
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1answer
89 views

Integral inequality for Schwartz function

Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that: $$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$ ...
3
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0answers
102 views

Green operator of elliptic differential operator and radius of convergence

Let $E \to X$ be a hermitian vector bundle over a compact Kähler manifold and let $L$ be a self-adjoint elliptic linear differential operator on $E$. Suppose that $E \to X$ and $L$ are real-analytic. ...
2
votes
2answers
56 views

Looking for references on higher order homogenous differential equations and a particular equation I am trying to solve

Most of the theory I know (and found, after some significant amount of searching) on homogenous higher order differential equations (third order onwards) assume constant coefficients: that is, it is ...
2
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0answers
69 views

Distorted elementary functions

Let $f(x)$ be an elementary function defined on $X\subseteq\mathbb{R}$ and $\xi(x), \eta(y)$ strictly monotone for $x\in X,\, y\in f(x)$. Questions: is there an established name for functions of the ...
2
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1answer
130 views

An inequality involving fractional Laplacian

I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function): $$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$ ...
1
vote
1answer
92 views

Shrinking subset with disjoint unions

Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be disjoint finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that ...
1
vote
0answers
100 views

Fourier transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
1
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0answers
80 views

A bilinear estimates involving critical Sobolev norms

Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
5
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2answers
204 views

Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?

This has received no full solution at StackExchange. As per https://dlmf.nist.gov/8.10#E13 we have $$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma% \left(n,n-1\...
11
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1answer
337 views

A density question for the Hilbert transform

Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions $$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
4
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0answers
74 views

Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms

Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
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0answers
45 views

Asymptotic inverses and de Bruijn conjugates (etc.) for complex-valued functions

I recently got my hands on a copy of Regular Variation by Bingham, Goldie, and Teugels ("BGT"), and it's been an absolute revelation for my research. The thing is, my current work centers ...
1
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1answer
269 views

Some fun with special infinite nested radicals

Let us define the following functions: $$f_n(x)=\sqrt{x^{n}-\sqrt{x^{n+1}- \sqrt{x^{n+2}-\cdots}}} $$ $$g_n(x)=\sqrt{x^{n}+\sqrt{x^{n+1}+ \sqrt{x^{n+2}+\cdots}}} $$ with $f(x)=f_1(x)$ and $g(x)=g_1(x)$...
2
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0answers
35 views

Binary law on pairs of finite unions of segments

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
5
votes
1answer
200 views

Does such a function exist?

I am looking for a function with the following property: Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$ I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$ I am trying ...
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0answers
59 views

Well-posedness for a wave equation with degenerate coefficients

Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem: \begin{equation}\label{pf0} \begin{aligned} \begin{cases} t\partial_t(t\partial_t u)-\...
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0answers
63 views

$1$-parameter analytic functions are almost everywhere Morse

Let $I = [t_{0}, t_{1}]$ be a closed interval with $t_{0} < t_{1}$ and let $M$ be a compact real analytic $n$-dimensional manifold without boundary. Furthermore, let $f:I \times M \rightarrow \...
6
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3answers
218 views

Uniformly approximating a function and its derivative using polynomials

I'm struggling either proving or disproving the following statement: Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
7
votes
3answers
278 views

Invertibility of specific function

This is my first post. I'm not a mathematician, just an electronics engineer who loves mathematics. In one of my projects, I arrived at the following function: $$V\left(\varphi\right)=\frac{A\sqrt{\pi-...
0
votes
1answer
79 views

Entropy of a refinement of a partition

We consider a probability space $(X, B, \mu)$. Let $\alpha$ and $\beta$ be countable partitions of X. We suppose $\beta$ is a refinement of $\alpha$, ie that every set in $\alpha$ is a union of sets ...
3
votes
1answer
97 views

To show a set is a set of positive Lebesgue measure in $ \mathbb{R}$

Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure? Notation: $t+E=\{t+e|e\in E\}$
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2answers
184 views

Linear combination of convex functions is constant

Let $\Phi : \mathbb{R}_{++}\to \mathbb{R}$ be a convex function (not necessarily differentiable). Fix an $\alpha \in (0,1)$ and define $$g(t) = \alpha t .\Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)...
-2
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1answer
166 views

Is there a function $f$ that is a finite sum of functions with finite products of the inputs of $f$ as inputs with this property?

Note: This question aims to be a generalization of Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? and Is it possible to create a polynomial $p(x)$ with this ...
4
votes
1answer
486 views

Analytic functions where all derivatives vanish at infinity and which are bounded

Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$. I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
4
votes
1answer
456 views

Higher-order derivatives of $(e^x + e^{-x})^{-1}$

I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$ It is fairly straightforward to obtain $$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...
-1
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1answer
91 views

Limiting points of elementary set

I consider the following set $$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$ Is it possible to identify the closure of $A$ in the reals?
0
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1answer
75 views

Does convergence of a sequence of subharmonic functions imply the vague convergence of their Riesz measures?

Suppose $D$ is a bounded domain of $\mathbb{R}^m$ for $m>1$ and $\{u_n\}_{n\geq1}$ is a sequence of subharmonic functions on $D$. Assume $u_n\to u_0$ pointwise on $D$ and $u_0$ is subharmonic on $D$...
4
votes
1answer
116 views

Inequality involving sigmoid function

Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > ...
1
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0answers
59 views

About the computation of the inverse Laplace transform [closed]

I have several questions about the inverse Laplace transform: If $F(s)$ is a smooth real-valued function vanishing on a large subset of $\mathbb{R}$ (e.g. $F(s)$ is supported on a bounded interval), ...
12
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1answer
237 views

Regarding a positive Lebesgue measure set in $\mathbb{R}^2$

Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure. ...
0
votes
0answers
44 views

Central manifold and fixed point theorems

Let us consider a real dynamical system $s′=g(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: Let $d=d(y,z)$ be a function verifying $u<d<w$ ...
0
votes
1answer
59 views

Convergence in weak dual topology $\sigma(L^\infty, L^1)$

Let $f\in L^\infty(\mathbb{R})\cap C(\mathbb{R})$, that is $f$ is continuous and bounded on $\mathbb{R}$. Let $S_r$ denote the shift by $r\in \mathbb{R}$: $S_r f=f(\cdot-r)$. Suppose $S_{r} f $ ...
2
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0answers
57 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
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0answers
43 views

Non-square multiplication operator matrix

Let $A(x), x\in (0,1)$ an $2\times n$ matrix, with $n\geq 2$. Consider the multiplication operator $K$ on $[L^2(0,1)]^n$ defined as $$K: f(x) \mapsto Kf(x)=A(x).f(x).$$ Intuitively, $$K: [L^2(0,1)]^n ...
2
votes
1answer
170 views

How to give a Borel set whose projection is not Borel?

I wonder that how Suslin got the idea "analytic set" and the advanced knowledge in this field. Can anyone explain this simply or recommend some reference? All discussions and comments are ...
3
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0answers
119 views

How to compute $\lim_{n\to \infty}a_1b_n+a_2b_{n-1}+\cdots+a_nb_{1}$ [closed]

As is well known, Stolz–Cesàro theorem is the following: Let $\displaystyle {(a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$be two sequences of real numbers. Assume that ${\displaystyle (...
-1
votes
1answer
104 views

Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]

Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds? $$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
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0answers
44 views

Characterizing a set of functions

Let $\mathscr C$ be the space of measurable functions $f$ on the interval $[0,1]$ that can be written in the form $$ f(t)=\sum_{k=0}^\infty c_k e^{-kt},$$ for some square summable series $\{c_k\}_{k=1}...
1
vote
0answers
39 views

Non constant delay differential equations

Given $\varphi:[0,1] \to [0,1]$ a continuous function, let $(E)$ be the delay differential equation (I am not sure about the terminology, as the delay is non constant): $y'(t) = y(\varphi(t))$. It is ...
1
vote
1answer
142 views

Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$

Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions). Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$. Link to the problem (paper "...
4
votes
1answer
151 views

Is a specific product function orthogonal to all harmonic functions

Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
9
votes
0answers
175 views

Is every Baire metric space a complete metric space in disguise?

I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here: Let's say that a metric space $X$ is Baire if every countable ...
5
votes
1answer
159 views

First order PDE in complex variables?

Consider the equation $$f'(x)+ g(x)f(x)=0$$ This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$ Similarly, we can look at complex variables and consider the equation and ...
5
votes
0answers
85 views

Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?

While solving a problem in calculus of variations, I came to the following question: Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...

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