# 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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votes

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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**

votes

**0**answers

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

**0**answers

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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vote

**0**answers

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

**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 $...

**1**

vote

**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 ...

**1**

vote

**0**answers

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?

**2**

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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votes

**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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vote

**1**answer

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

**0**answers

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

**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 $\...

**12**

votes

**1**answer

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

**1**answer

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

**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{(\...

**2**

votes

**1**answer

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

**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" ...

**1**

vote

**0**answers

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

**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 ...

**1**

vote

**0**answers

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

**0**answers

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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**0**answers

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

**1**answer

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

**1**answer

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

**26**answers

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

**1**answer

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

**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 ...

**1**

vote

**0**answers

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

**3**answers

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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votes

**0**answers

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

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

**3**

votes

**1**answer

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

**2**answers

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

**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 ...

**1**

vote

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

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

**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}^...

**1**

vote

**1**answer

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

**0**answers

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

**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 ...

**0**

votes

**0**answers

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

**0**answers

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

**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(...

**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^{...

**3**

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 ...

**0**

votes

**1**answer

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)...