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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
2answers
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,
$$ f(A):=\sum f(\lambda)...
4
votes
1answer
136 views
Real analytic function of one variable with given set of values
Given two strictly increasing bounded sequences of reals $x_n$ and $y_n$. What is known about existence of real analytic function $f$ with property $f(x_{n_k})=y_{n_k}$ for some subsequence $x_{n_k}$ ?...
1
vote
0answers
118 views
Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
4
votes
1answer
296 views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
1
vote
1answer
108 views
Comparison of two integrals
Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)...
2
votes
1answer
174 views
Hausdorff dimension of the graph of a BV function
Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
Update.
In an answer to this post, it ...
0
votes
1answer
168 views
Integration by parts formula for the double Riemann-Stieltjes integral
In my research the following integration by parts formula for the double Riemann-Stieltjes integral
$$\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)...
1
vote
1answer
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
2answers
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
0answers
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
1answer
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
1answer
87 views
Upper bound of a ratio of integrals
I'm wondering how to upper bound the following ratio of integrals:
$$\frac{\int_{\Delta_a}(\prod_{i=1}^n\lambda_i)^{p-1}\prod_{i<j}|\lambda_i-\lambda_j|}{\int_{\Delta_b}(\prod_{i=1}^n\lambda_i)^{p-...
3
votes
0answers
50 views
Flow lines of a real analytic vector field convergent to a point
Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...
2
votes
1answer
81 views
Box dimension as the critical value of the fractal content
Let $M \subseteq \mathbb{R}^n$ be bounded and $N_{\epsilon}(M)$ the minimum number of 'squares' of side $\epsilon$ with center in M necessary to cover $M$. The box dimension of M is then defined as $\...
3
votes
0answers
60 views
Boundary behavior of $H^2_0(\Omega)$ functions
If $u \in H^2_0(\Omega)$, is it true that $$u(x) \le C\mathrm{dist}(x,\partial \Omega)^2$$ as $x$ goes to the boundary?
2
votes
0answers
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
2answers
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
vote
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
$$ f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^n\phi_{p,q}(x_p)\...
2
votes
2answers
77 views
Rate of convergence for eigendecomposition
Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix
$$\Delta_D := \left( \begin{matrix} 2 &...
1
vote
1answer
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
1answer
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
0answers
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
$$...
3
votes
1answer
296 views
Where to find the proof of this property?
I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...
1
vote
0answers
60 views
How to see the divergence of a series is not faster than some order? [closed]
$$
\sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p}
$$
For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.
3
votes
3answers
192 views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
11
votes
1answer
453 views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
10
votes
1answer
486 views
A question concerning Lusin’s Theorem
We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$.
Let $f$ be measurable. For every $e$ in $...
4
votes
0answers
203 views
A simple proof of Jordan curve theorem [closed]
I need a short proff of the Jordan curve theorem please.
The one I have is 16 pages long and is for a little expo, so I need one a little shorter.
Thanks
1
vote
1answer
128 views
Equivalent notion of approximate differentiability
Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...
1
vote
1answer
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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 ...
