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Questions tagged [real-analysis]

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

4
votes
0answers
91 views

Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2. We now define for each $k \geq 2$ ...
4
votes
1answer
171 views

Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
3
votes
0answers
198 views

Eudoxus real numbers

I recently remembered the eudoxus construction of the real numbers. Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction? Clearification: The ...
2
votes
1answer
138 views

Chain rules for Dini Derivative

Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
4
votes
0answers
70 views

Anderson Localization and Homogenization theory

I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here. The question is mostly related to homogenization theory in mathematical physics. $\textbf{...
3
votes
1answer
432 views

Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples? This seems to be a well-known result, but I can ...
6
votes
2answers
451 views

Summing Bernoulli numbers

Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers. It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$. I am curious about the ...
-2
votes
1answer
120 views

Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support. What is the relationship between $$ \widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
0
votes
1answer
87 views

How to prove that the following function is monotonically decreasing?

$f(x) = \frac{\int_{\alpha x}^{x} e^{-t} t^{b+1}\ dt}{x \int_{\alpha x}^{x} e^{-t} t^b\ dt}:\ ]\ 0,+\infty\ [\ \to \mathbb{R}$ where $\ 0<\alpha<1\ $ and $\ b>0$
2
votes
1answer
98 views

Direct proof a property of hyperstonean spaces

First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
0
votes
0answers
66 views

Integral Representation of Analytic Functions

Lang's Complex Analysis 3ed, Lemma XV 1.1 (pg. 392): Let $I$ be an interval of real numbers, and $U$ an open set of complex numbers. Let $f(t,z)$ be continuous on $I \times U$. Suppose further that ...
2
votes
2answers
66 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
6
votes
2answers
148 views

Bound on sum of coefficients of polynomials w.r.t a weighted integral

Fix $k\in\mathbb{N}$ and assume $f(x)$ is a real polynomial of degree $n$ such that we have the normalization $$\int_{-1}^1f(x)^2\,(1-x)^kdx=1.$$ I am interested in the optimal size of the sum of the ...
4
votes
1answer
98 views

a compact set with nonempty convex sections

Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space. For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$. Given a set $Y \subseteq X$ ...
7
votes
1answer
229 views

Descartes' rule of signs for infinite series

Consider the function given by $$f(x)=1-a_1x-a_2x^2-a_3x^3-\cdots$$ where each $a_k\geq0$ and some $a_j>0$. If $f(x)$ is a polynomial then Descartes' Rule of signs tells us there is exactly one ...
3
votes
2answers
122 views

Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$

I am working in data science and I have to deal with the following problem for which I would like to find a simplification: We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...
7
votes
1answer
510 views

Can the supremum of continuous functions be discontinuous at every point of an interval?

Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued ...
3
votes
1answer
113 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
4
votes
1answer
216 views

Complex Structure on Manifold of Maps

Suppose $M$ is a compact smooth manifold and $V$ is a compact complex manifold. I want to show that the spaces $C^{k,\alpha}(M,V)$ and $W^{k,p}(M,V)$ (the latter for $kp>\dim M$) are complex ...
4
votes
1answer
125 views

Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad. I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
0
votes
0answers
140 views

What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?

I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking, in that the residuals from the ...
1
vote
1answer
136 views

About a Dirichlet series [closed]

I would like to know if the following assertion is true: Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
13
votes
1answer
513 views

Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?

Willie Wong asked here (MO) and here (MSE) very interesting question. As he phrased it: Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$...
8
votes
0answers
145 views

Infinite series identities in search of a proof

This comes in relation to the Fishburn numbers. I stumbled on the following relation for which I ask a proof if true. Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then $$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
16
votes
0answers
437 views

Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
3
votes
1answer
100 views

Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?

Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$. Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...
0
votes
0answers
83 views

What is a relation between energy space and $L^p_s-$Sobolev spaces?

We define energy space $$E= \left\{ f\in \mathcal{S}'(\mathbb R^d): \|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$ and Sobolev spaces $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^...
50
votes
3answers
3k views

Does every real function have this weak continuity property?

In my research I came across the following question : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
1
vote
1answer
84 views

Orthogonal complement vector space

Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study $X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$ and $X^{\perp_{H^{-...
5
votes
2answers
458 views

Dominated convergence 2.1?

After this question : Dominated convergence 2.0? I want to know, what about the case when $h\in L^1([0,1])$. The completed question : Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging ...
1
vote
0answers
57 views

Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given. ...
2
votes
1answer
152 views

Riemann-Stieltjes integral as a limit of Riemann integrals

Let us suppose that $f, g:(A, B)\to \mathbb{R}$ are both continuous on $(A, B)$ and for $[a, b]\subset (A, B)$, suppose that $g$ is of bounded variation on $[a, b]$ (we may add, if necessary, that ...
1
vote
0answers
115 views

Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
6
votes
2answers
287 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
7
votes
1answer
387 views

Dominated convergence 2.0?

During my research, I came across the following question. Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that: $\forall n\in\mathbb N, f_n''<h$, ...
1
vote
0answers
53 views

Linear dependence of solution?

Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...
0
votes
0answers
105 views

Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]

Recall a very famous theorem due to Hartogs for complex analytic functions of several variables. Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
2
votes
1answer
96 views

small perturbation of BV function

consider an interesting real analysis question: define average operator on $[0,1]$: $A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $ ( may clarify ...
4
votes
1answer
109 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
4
votes
1answer
73 views

Convergence of a real sequence (stochastic approximation)

Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$ I would like ...
1
vote
0answers
880 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
6
votes
0answers
167 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
0
votes
1answer
55 views

A question on existence of a Sobolev Hilbert space, where convergence implies uniform convergence [closed]

Is there a Sobolev Hilbert space $H^k(\Omega)$($\Omega$ open subset of $\mathbb{R}^m$, with a smooth boundary), for some $k \in \mathbb{N}$, such that, any sequence in the space $C^0(\bar{\Omega})\cap ...
3
votes
0answers
87 views

inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...
2
votes
0answers
47 views

Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
8
votes
2answers
296 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
0
votes
0answers
121 views

A problem on integrability of derivatives

Let $$f : (0,1) \to \mathbb{R}$$ and $$g(x) = |f(x)|^{r-1} f(x)$$$r \in \mathbb{N}$. It is known that $g\in \mathcal{L}^2(0,1)$ and the $r^{th}$ weak derivative, $ g^{(r)} \in \mathcal{L}^2(0,1)$. I ...
1
vote
1answer
276 views

A simple two variable analytic inequality, inspired by probability

I'm trying to prove the following inequality: $$ bf_1g_1 + (x-b)f_1g_0 + (y-b)f_0g_1 + (1-x-y+b)g_0f_0 \le (|f_1|^p x + |f_0|^p (1-x))^{1/p} (|g_1|^p y + |g_0|^p (1-y))^{1/p} $$ where $0\le xy\le b\le ...
2
votes
0answers
72 views

A variant of the optimal transport

Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows: $$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$ where the inf is taken ...
1
vote
0answers
246 views

Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?