# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

2,703
questions

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### 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 ...

**2**

votes

**2**answers

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?

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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)\...

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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 ...

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votes

**1**answer

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 $...

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**1**answer

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 ...

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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**

votes

**1**answer

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?

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votes

**1**answer

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\...

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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-...

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**1**answer

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 $\leq n$ ...

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votes

**2**answers

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 $\...

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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 ...

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**1**answer

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 ...

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votes

**1**answer

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{(\...

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**2**answers

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" ...

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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\...

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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 $f_3: B^n\setminus \{0\}\to S^n$ ...

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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 ...

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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 ...

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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}...

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votes

**1**answer

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 ...

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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)= \...

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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)?

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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^{-...

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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: ...

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votes

**1**answer

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 ...

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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|...

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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 ...

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votes

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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{\...

**22**

votes

**5**answers

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 $(Tf)(x) = f(x+1)$...

**0**

votes

**1**answer

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}^...

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votes

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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(...

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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 ...

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**1**answer

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 ...

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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.
...

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**1**answer

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 $|I_k|,|J_k|$, ...

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votes

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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) \| \...

**5**

votes

**1**answer

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^{...

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votes

**1**answer

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 ...

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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
$$
\...

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237 views

### A conjecture concerning symmetric convex sets [closed]

Question:
Let's suppose that $S \subset \mathbb{R}^n$ is convex and symmetric so:
\begin{equation}
x \in S \iff -x \in S \tag{1}
\end{equation}
Now, if we define the radius of $S$ as $R$ such that:
...

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112 views

### Limitations on method of Lagrange multipliers [migrated]

My general question is this:
What are the conditions (if any) such that the method of Lagrange multipliers will NOT find all the critical points of a differentiable function?
To give some context to ...

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votes

**1**answer

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(...

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25 views

### Underspecified Riccatti-type ODE

I came across the following Ricatti-type ODE in my reading
$$
\begin{aligned}
\partial_t \psi(t,x) &= \Psi(\psi(t,x)),\\
\psi(0,x)&=x,\\
\Psi(x)&\triangleq \partial_t\psi(t,x)|_{t=0^+}.
\...

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50 views

### Is the formulation of the dual to the Kantorovich problem unique?

Lots of definitions follow, the question is further down.
Let $\Gamma(\mu_1,\mu_2)$ be the set:
$$\Gamma(\mu_1,\mu_2) = \{ \gamma \text{ probability measure on } \mathbb{R}^d \times \mathbb{R}^d ...

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71 views

### Average number of pieces of a random piecewise-linear function

Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...

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**0**answers

161 views

### Orthogonality relation in $L^2$ implying periodicity

Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...

**-1**

votes

**1**answer

107 views

### Is there a name for this family of integral?

This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.

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76 views

### Elementary proof of growth estimate for a polynomial via size from its zero set

The paper Asymptotic properties of polynomials and algebraic functions of several variables by Gorin contains the following.
Lemma 3.1. Let $f\in \mathbb R[x_1,\dots,x_n]$. Suppose $f$ has a root in ...