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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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 ...