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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2
votes
2answers
111 views

Bound on the ratio of harmonic and arithmetic mean

Let $a_i>0$ for $i=1,...,n$. It is well-known that $A\ge H$, where $A$ and $H$ are the arithmetic mean and harmonic mean of the vector $(a_i)$, respectively. Is any lower bound on $H/A$ known?
2
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0answers
24 views

Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces

In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
1
vote
0answers
32 views

fractional compact Sobolev embedding on lipschitz domain

Let $\Omega$ be a bounded Lipschitz domain. It is well known that $H^1(\Omega)$ can be compactly embedded into $L^2(\Omega)$. I also found references for the compact embedding $H^\delta(\Omega)\...
1
vote
0answers
31 views

Weighted inner product of independent random unit vectors

Let $u=(u_1,...,u_n)$ and $v=(v_1,...,v_n)$ be independent random unit vectors in $\mathbb{R}^n$. Let $\lambda=(\lambda_1,...,\lambda_n)$ be a fixed unit vector in $\mathbb{R}^n$. What is the ...
0
votes
1answer
83 views

Lifting functions between $L^2$

A map $\pi: X \to Y$, $\mu$ is the measure on $X$, and its push forward is defined by $\nu:=\pi_{*} \mu$. If given $f \in L^2(X, \mu)$, can we find $g \in L^2(Y, \nu)$ such that $g \circ \pi= f$, if $...
1
vote
1answer
40 views

Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
1
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0answers
241 views

A (surprising?) expression for $e$

I apologise if this is off topic. Consider the quantity $$ F(m,n,k)=\frac{(m)_k}{k!n^{k-1} } $$ where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation $$ \sum_{k=1}^{K} ...
-4
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1answer
156 views

Formula for $1^x + 2^x + … + n^x$ when $x$ is complex [on hold]

I would like to know if there is a formula for $$1^x + 2^x + ... + n^x$$ when $x$ is real or complex?
2
votes
1answer
121 views

Weak convergence in $L^p$

My question is probably very basic, sorry about that. Let $\{f_i\},\{g_i\}$ be two sequences converging to 0 weakly in $L^p[0,1]$ for any $p<\infty$. Can one conclude that $\int_0^1f_i(x)g_i(x) dx\...
-1
votes
1answer
129 views

Denominator approximation sequence of a real number

For any positive integer $[n]$, let $[n]=\{1,\ldots,n\}$. Let $r\in\mathbb{R}$. We define for every positive integer $n\in\mathbb{N}$ the minimal difference from a rational with denominator $\leq n$ ...
1
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1answer
150 views

Space derivative of flow of ODE with monotone source

Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
0
votes
0answers
70 views

Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral

The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
4
votes
2answers
186 views

Comparing two limsup's

Let $f\in L^2(0,\infty)$ be a positive, decreasing function. Is it then true that $$ \limsup_{x\to\infty} xf(x) = \limsup_{x\to\infty} \frac{1}{f(x)}\int_x^{\infty} f^2(t)\, dt $$ (and similarly for $\...
12
votes
1answer
625 views

Continuous functions of three variables as superpositions of two variable functions

Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's ...
2
votes
1answer
250 views

Inverse of pseudo differential operator

Let $\operatorname{Op}_h(x,D)(a)$ denote the Weyl-quantisation of a symbol $a$. Is there an explicit way to invert this pseudo-differential operator in an asymptotic series? By this I mean, can we ...
2
votes
1answer
177 views

Limits of a family of integrals

Assume $\lambda_1+\lambda_2=1$ and both $\lambda_1$ and $\lambda_2$ are positive reals. QUESTION. What is the value of this limit? It seems to exist. $$\lim_{n\rightarrow\infty}\int_0^1\frac{(\...
2
votes
1answer
117 views

The radius of an interval's image through a space-filling curve

Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
8
votes
2answers
238 views

Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$. Question Given $\epsilon> 0$, find a "low-degree" ...
1
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0answers
73 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 is ...
2
votes
1answer
181 views

Help in proving, that $\int_{0}^{\infty} \frac{1}{\Gamma(x)} d x=e+\int_{0}^{\infty} \frac{e^{-x}}{\pi^{2}+(\ln x)^{2}} d x$ using real methods only

I hope this does not seem like a too easy question for Overflow. I would like to find an easier method than mine to prove the above statement for the Fransén-Robinson Constant. My first method was to ...
1
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0answers
31 views

Generalization of Lagrange-Burmann to system of self-consistency equations

In my research, I have come across a system of probability generating functions of the following form: $$H_1(x) = x A(H_1(x))B(H_2(x)) \text{,}$$ $$H_2(x) = x C(H_1(x))D(H_2(x)) \text{,}$$ and I am ...
2
votes
0answers
45 views

Extension of a $\delta$-subharmonic function that is subharmonic on a reduced domain

Suppose $B$ is a ball in $\mathbb{R}^{m}$ and $u$ and $s$ are subharmonic on $B$. Suppose there is a closed subset $F$ of the closure of $B$ with no interior such that $v=u-s$ is subharmonic on $B\...
1
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0answers
44 views

Removability of the isolated singularity of real analytic mappings with nondegenerate Jacobian

Let $B^n=\{x\in \mathbf{R}^n: |x|<1\}$$(n>2)$. Consider real analytic mappings $f_1:B^n\setminus \{0\}\to B^n$, $f_2:B^n\setminus \{0\}\to \mathbf{R}^n$ and $f_3: B^n\setminus \{0\}\to S^n$ ...
8
votes
1answer
117 views

Second derivative of integral function

Let $f: \mathbb R^2 \rightarrow \mathbb R$ be a smooth strictly convex function with unique minimum at $0$ such that all level sets $A_x:=\left\{z ; f(z) \le x \right\}$ are compact. Imagine something ...
3
votes
1answer
91 views

What fraction of a charge is induced on a surface via balayage?

Consider a smooth, bounded domain $\Omega \subset \mathbb{R}^3$, and place a charge $q>0$ at a point $z\in\mathbb{R}^3\setminus\overline\Omega$. Via the concept of balayage, there is an induced ...
27
votes
26answers
42k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
3
votes
1answer
68 views

Asymptotic behaviour of function using Fox $H$-function representation

In equation (9) of this paper, it is claimed that the limiting behaviour $$ \int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
4
votes
1answer
97 views

Holder inequality with respect to convex function

Given $a, T>0$, by Holder's inequality we have $$ \int_T^{T+a}f(s)ds\leq \left(\int_0^{T+a}|f(s)|^2ds\right)^{1/2}\cdot\sqrt{a}. $$ Do we have similar result if we replace $|x|^2$ by some convex ...
1
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0answers
55 views

Recurrence involving families of orthogonal polynomials

Let $ \forall n \in N, n\geq 1$ $$ R_n(x)=(-1)^n n! \displaystyle \frac{(x-1)...(x-n)}{(x(x+1)..(x+n))^2}$$ thus by decomposition in simple element it's easy to see that $$ (1): \quad R_n(x)= \...
0
votes
3answers
115 views

Does the Leibniz (product) rule hold for the spectral fractional Laplacian?

Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
0
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0answers
21 views

Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
22
votes
5answers
883 views

Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
3
votes
1answer
83 views

Is a bounded sequence of $H^1(\Omega)$ tight?

Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
3
votes
2answers
180 views

Lipschitz constant of a function of matrix

The function is given by $f(X) = (AX^{-1}A^\top + B)^{-1}$ where $X$, $A$, and $B$ are $n \times n$ positive definite matrices. I'm trying to find the Lipschitz constant such that $\| f(X)-f(Y) \| \...
0
votes
1answer
84 views

Zeros of a Lipschitz function [closed]

So my question is the following: Let f be a real Lipschitz continuous function defined on a an interval of R. Consider the set of points that are zeros of the function and every neighborhood of the ...
1
vote
0answers
31 views

The parameter regularity of power sum

Let $f(x,s)=\sum_{n=0}^\infty a_n(s)x^n$ where $|a_n(s)|\le1$ is a bounded function theory. Suppose for every $|x|<1$, $f(x,s)$ is Holder-$\alpha$ for $s$-variable, i.e. $|f(x,s_1)-f(x,s_2)|\le C|...
10
votes
2answers
701 views

Computing the sum of an infinite series as a variant of a geometric series

I came across the following series when computing the covariance of a transform of a bivariate Gaussian random vector via Hermite polynomials and Mehler's expansion: $$ S = \sum_{n=1}^{\infty} \frac{\...
0
votes
0answers
129 views

On the convergence of $\sum_{n\geq 1} \frac{\sin (2^n)}{n^s}$

What is the radius of convergence of the aforementioned series ? If I recall correctly, I once saw a post here on MO claiming that it converges for $\Re(s) > 1/2$, but I can't seem to find the post ...
1
vote
0answers
63 views

Identifying a determinantal condition

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it? Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \...
2
votes
1answer
181 views

Choosing finite subsets of natural numbers

Let $t>0$ and $\delta\in\big(0,\frac12\big)$ be fixed. For any $k\in\mathbb{N}$ let $I_k,J_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $|I_k|,|J_k|$, ...
0
votes
1answer
54 views

Continuity of $\arg\min$

Let $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a continuous function. Let $A = \{(x,\arg\min_x f(x,y)) | x \in \mathbb{R}^n\}$. Is there necessarily a continuous function $g: \mathbb{R}^...
1
vote
1answer
78 views

Box counting dimension of a set and Lipschitz functions

If $f$ is Lipschitz, then the following holds for the Hausdorff dimension: $$\dim_H f(A) \le \dim_H A.$$ Is the same true for the box counting dimension?
3
votes
0answers
46 views

Solving for a monotone function - contraction operator for functions?

I want to solve a problem for an increasing function $g(x)$, for $x \in [0,1]$ and with $g(0) = 0$ and $g(1) = 1$. The solution will be solution to the following equation $\forall x$, $f_1(x) = f_2(g(...
5
votes
1answer
126 views

Stronger version of Besicovitch covering theorem

I'm wondering if the following strengthening of the Besicovitch covering theorem holds: Suppose $A\subset\mathbb R^n$ is a bounded subset and suppose $x\mapsto r_x$ is a function $A\to(0,\infty)$. Is ...
0
votes
0answers
45 views

What are the sets on which norm-closedness implies weakly closedness?

Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
1
vote
0answers
140 views

Reference Request: Stone-Weierstrass on Other Topologies

Let $X$ be a compact subset of $\mathbb{R}^n$. The classical Stone-Weierstrass theorem describes dense subsets of $C(\mathbb{R}^n,\mathbb{R})$ when it is equipped with the compact-open topology. ...
0
votes
1answer
54 views

Lower semicontinuity of a multi-valued map $F:X\to 2^Y$ in term of net

Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is lower semicontinuous at $x_0\in X$ if for each $y_0\in F(x_0)$ and any neighborhood $U\in \mathcal N(...
5
votes
1answer
230 views

Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?

PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim If the integral $$ \int_0^{2\pi} e^{...
3
votes
1answer
85 views

The optimal asymptotic behavior of the coefficient in the Hardy-Littlewood maximal inequality

It is well-known that for $f \in L^1(\mathbb{R^n})$,$\mu(x \in \mathbb{R^n} | Mf(x) > \lambda) \le \frac{C_n}{\lambda} \int_{\mathbb{R^n}} |f| \mathrm{d\mu}$, where $C_n$ is a constant only depends ...
0
votes
1answer
168 views

Integration by parts formula for the double Riemann-Stieltjes integral

In my research the following integration by parts formula for the double Riemann-Stieltjes integral $$\int\limits_{[a,b]\times[c,d]}f(x,y)\,dg(x,y)=f(b,d)g(b,d)-f(a,d)g(a,d)-f(b,c)g(b,c)+f(a,c)g(a,c)...