# Questions tagged [real-analysis]

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

2,703
questions

**3**

votes

**1**answer

143 views

### Differentiability of the blow-up of a function

Let $u \in C^0([-1,1])$ such that $u(0)=0$. Suppose that $u$ satisfies the following property:
For every $\{t_k\}\subset \mathbb{R}$ such that $t_k \to 0$, there exist a real number $\alpha$ (...

**1**

vote

**1**answer

107 views

### Pointless characterization relating between a fractal and its code space

Given an hyperbolic IFS $(X,\{f_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma_N = \{1,\ldots,N\}^{\mathbb{N}}$ and the generated fractal set by $\mathcal{A}$.
There is a continuous ...

**1**

vote

**0**answers

37 views

### Poincaré lemma for gradient times its transpose

Poincaré lemma states that a vector $v_i(x)$ defined on a ball in $R^n$ is the gradient of a function if and only if
\begin{equation}
\partial_i v_j = \partial_j v_i
\end{equation}
or equivalently ...

**1**

vote

**1**answer

67 views

### On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$.
In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...

**1**

vote

**0**answers

57 views

### Real-analytic function with given set of values [closed]

We say that a strictly increasing sequence $x_n$ of reals converges fast
to $x$, if for each $k\in\mathbb{N}$ the sequence $n^k\cdot(x_n − x)$ is
bounded. It is known that there exists a $C^\infty$-...

**22**

votes

**1**answer

2k views

### Does the average primeness of natural numbers tend to zero?

This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...

**2**

votes

**2**answers

170 views

### If the Hausforff dimension of the graph of a function $u$ is $N$ and $\tilde u = u$ a.e. then $\dim_H \mathrm{graph} \, \tilde u = N$ too

Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$.
Let $\tilde u = u$ a.e. Is it true ...

**11**

votes

**2**answers

381 views

### Smoothness of finite-dimensional functional calculus

Assume that $f:\mathbb R\to\mathbb R$ is continuous.
Given a real symmetric matrix $A\in\text{Sym}(n)$, we can define $f(A)$ by applying $f$ to its spectrum. More explicitly,
$$ f(A):=\sum f(\lambda)...

**4**

votes

**1**answer

136 views

### Real analytic function of one variable with given set of values

Given two strictly increasing bounded sequences of reals $x_n$ and $y_n$. What is known about existence of real analytic function $f$ with property $f(x_{n_k})=y_{n_k}$ for some subsequence $x_{n_k}$ ?...

**1**

vote

**0**answers

118 views

### Questions on Riemann's explicit formula

If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...

**4**

votes

**1**answer

296 views

### Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold

How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?

**1**

vote

**1**answer

108 views

### Comparison of two integrals

Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)...

**2**

votes

**1**answer

174 views

### Hausdorff dimension of the graph of a BV function

Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...

**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)...

**1**

vote

**1**answer

99 views

### Is this operator invertible?

Let $T(t)$ be a strongly continuous semi-group on a Banach space $X$, and let $A(\cdot)\in C(0,\tau; \mathcal{L}(X))$ for some $\tau>0$. The operator $G:C(0,\tau;X)\to C(0,\tau;X)$ maps every $h\in ...

**6**

votes

**2**answers

185 views

### uniform approximation by a particular set of functions

Consider the interval $[0,1]$ and let $\mu_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu_1(t)<\mu_2(t)<\...

**0**

votes

**0**answers

99 views

### How to define $``\ll"$ in higher dimension?

Fix $C>0,$ we say $n \sim m $ if $|n-m| < C$ $ (n, m \in \mathbb Z)$ and $n\ll m$ if $n-m \leq C$ and $n\gg m$ if $n-m \geq C.$
Let $n_1, n_2, n_3, n_4 \in \mathbb Z$.
Assume that $|n_1-...

**0**

votes

**1**answer

47 views

### Sufficient and necessary condition for the global uniqueness of fix-points

https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if ...

**0**

votes

**1**answer

87 views

### Upper bound of a ratio of integrals

I'm wondering how to upper bound the following ratio of integrals:
$$\frac{\int_{\Delta_a}(\prod_{i=1}^n\lambda_i)^{p-1}\prod_{i<j}|\lambda_i-\lambda_j|}{\int_{\Delta_b}(\prod_{i=1}^n\lambda_i)^{p-...

**3**

votes

**0**answers

50 views

### Flow lines of a real analytic vector field convergent to a point

Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...

**2**

votes

**1**answer

81 views

### Box dimension as the critical value of the fractal content

Let $M \subseteq \mathbb{R}^n$ be bounded and $N_{\epsilon}(M)$ the minimum number of 'squares' of side $\epsilon$ with center in M necessary to cover $M$. The box dimension of M is then defined as $\...

**3**

votes

**0**answers

60 views

### Boundary behavior of $H^2_0(\Omega)$ functions

If $u \in H^2_0(\Omega)$, is it true that $$u(x) \le C\mathrm{dist}(x,\partial \Omega)^2$$ as $x$ goes to the boundary?

**2**

votes

**0**answers

77 views

### Is the Fourier transform of a measurable function as a tempered distribution necessarily a complex Borel measure?

Let $u\in\mathcal{S}'(\mathbb{R}^n)$. Suppose that $u$ is also a measurable function on $\mathbb{R}^n$. Is it true that the Fourier transform $\hat{u}$ as a tempered distribution is always a complex ...

**1**

vote

**2**answers

171 views

### $q$-factorial coefficient asymptotics

Consider the $[n]!_q = \prod\limits_{k = 1}^{n} \frac{q^k - 1}{q - 1} = \sum\limits_{k = 0}^{\binom n 2} c_k q^k$ and let $\{f_n\}_{n \in \mathbb{N}}$ be the sequence of the functions on $[0; 1]$ ...

**1**

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125 views

### About the sum of prime reciprocals

Let $B$ be the Meissel-Mertens constant., $\theta$ the Chebyshev theta. Let $S(x)=\sum_{p\le x}1/p$, $p$ prime. Let $M(x)=S(x)-\log\log x-B$. Robin, and later Diamond and Pintz, showed that $M(x)$ ...

**3**

votes

**0**answers

106 views

### A new characterization of Riemann-Integrability

Question :
Given two bounded functions $\,f:[a,b]→\mathbb{R}\,$
and $\;θ:(0,b−a]→[0,1]$.
Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $\,[a,b]$.
Let $\,Δx_k=x_k−x_{k−1}\,$ and $\,\...

**0**

votes

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102 views

### Positivity of an integral

Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...

**1**

vote

**1**answer

137 views

### Solving Fractional Laplacian Equations with Boundary Condition

I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:
$r^{+}(\nabla^s) v = f$
where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ ...

**0**

votes

**0**answers

54 views

### Signal sets that arises from a detection theory problem

Consider the discrete time signal $y(t)=\frac{d x(t)}{dt}$ where $x(t)\in[0,1]$ where $t\in\mathbb Z$ (differential is just subtraction of consecutive samples).
Suppose we make two signals $w(t)$ and ...

**1**

vote

**0**answers

81 views

### Approximating $3SAT$ by polynomials

Given $3SAT$ formula $\phi$ in $n$ variables and a $\mu\in(0,1/2)$ what is the smallest degree of polynomial $f\in\mathbb Z[x_1,\dots,x_n]$ such that for an $(a_1,\dots,a_n)\in\{0,1\}^n$ if $\phi(a_1,\...

**1**

vote

**0**answers

56 views

### Smoothness of a periodization [closed]

If you are given a smooth function, how to verify whether its priodization is well-defined and also smooth (or not)?
For instance, suppose, there is a series
$$
f(x) = \sum_{n=-\infty}^{\infty}e^{-\...

**6**

votes

**1**answer

110 views

### Kolmogorov superposition on the Hilbert Cube

A result of Kolmogorov and Arnold says that continuous functions on $\mathbb{R}^n$ can be represented as sums of the form
$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...

**2**

votes

**2**answers

77 views

### Rate of convergence for eigendecomposition

Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix
$$\Delta_D := \left( \begin{matrix} 2 &...

**1**

vote

**1**answer

119 views

### Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...

**0**

votes

**1**answer

186 views

### Is the function $f(x=\sum a_n/2^n)=\sum na_n/2^n$ continuous and nowhere differentiable? [closed]

Let $x=\sum_{1<=n<\inf}a_n/2^n$ be the binary expansion of a real number in [0,1]. Assume that infinitely many of a_n are 0 so that the expansion is unique.
Define $f(x)=\sum_{1<=n<\inf}...

**2**

votes

**0**answers

86 views

### Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...

**3**

votes

**1**answer

296 views

### Where to find the proof of this property?

I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...

**1**

vote

**0**answers

60 views

### How to see the divergence of a series is not faster than some order? [closed]

$$
\sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p}
$$
For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.

**3**

votes

**3**answers

192 views

### Existence of solution to linear fractional equation

We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...

**11**

votes

**1**answer

453 views

### Approximation of a compactly supported function by Gaussians

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...

**10**

votes

**1**answer

486 views

### A question concerning Lusin’s Theorem

We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...

**4**

votes

**0**answers

203 views

### A simple proof of Jordan curve theorem [closed]

I need a short proff of the Jordan curve theorem please.
The one I have is 16 pages long and is for a little expo, so I need one a little shorter.
Thanks

**1**

vote

**1**answer

128 views

### Equivalent notion of approximate differentiability

Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...

**1**

vote

**1**answer

121 views

### Formal justification of the Chaos game in the Sierpinski triangle

I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals.
Suppose that $(X,d)$ is a ...

**2**

votes

**0**answers

49 views

### Generalized definition of integrable condition on rough complex subbundle

Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...

**8**

votes

**0**answers

195 views

### Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...

**6**

votes

**2**answers

162 views

### Box dimension of the graph of an increasing function

This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...

**1**

vote

**0**answers

115 views

### Convergence to a $C^\infty$ function

For every $\epsilon \in [0, 1)$ and $n$ in $\mathbb N$, let $g(n, \epsilon)$ be a function $\mathbb R \to \mathbb R$.
Suppose for all $\epsilon \in [0, 1)$, $g(n, \epsilon)$ is $C^n$ smooth and that ...

**6**

votes

**1**answer

201 views

### Asympotic density of a very simple sequence

Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more ...

**7**

votes

**2**answers

315 views

### Properties of heat equation

** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...