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

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3
votes
1answer
75 views

Continuous non-constant function with infinite intersections with horizontal line on a compact interval?

The title might be misleading, but whether such a function exists is what boggles me about the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that for all ...
4
votes
3answers
1k views

Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
-2
votes
0answers
47 views

Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated. I got struck ...
-1
votes
0answers
26 views

A sequence of continuous functions that converge to the characteristic set of a G-Delta Set [on hold]

If $\mathcal{O} \subset I$ is an open subset of an interval then I know we can find a sequence of continuous functions $f_j \in C(I)$ such that $0 \le f_j(x) \nearrow \chi_{\mathcal{O}}(x)$ for all $x ...
8
votes
6answers
2k views

What's a natural candidate for an analytic function that interpolates the tower function?

I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
-2
votes
0answers
46 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first? [migrated]

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
0
votes
0answers
55 views

smoothness of boundary under Riemann mapping [on hold]

Suppose there is a smooth Jordan curve separating the complex plane. For complicity, assume the curve is given by a graph $(x, \phi(x))$, where $\phi(x)$ is smooth, bounded, and derivatives are ...
1
vote
0answers
141 views

How to solve $f(f(x))=x^2+x$ [duplicate]

Now I just have the equation $f(f(x))=x^2+x$. How can I find $f(x)$? I have already tried many times, but I cannot solve it by any way I know. Is a solution possible?
72
votes
9answers
20k views

solving $f(f(x))=g(x)$

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
2
votes
1answer
870 views

For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that $$ \sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty, $$ where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
5
votes
2answers
136 views

Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...
0
votes
1answer
70 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
3
votes
1answer
235 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
3
votes
1answer
112 views

Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
47
votes
9answers
8k views

Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?
4
votes
3answers
175 views

Measure of intersections in probability spaces

Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A_i)\geq \epsilon$ for all $i\in\mathbb{N}$. ...
2
votes
0answers
61 views

Inverses of probability generating functions: positivity of derivatives

Let $\mathcal{G}$ be the set of probability generating functions of random variables taking positive integer values, considered as functions on $[0,1]$. So $G\in\mathcal{G}$ can be written ...
1
vote
1answer
94 views

A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
3
votes
1answer
74 views

Is the variation of two BV functions the same in the set in which they coincide?

Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set $A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$ is it true that ...
9
votes
3answers
284 views

Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort ...
1
vote
2answers
110 views

Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set ...
0
votes
0answers
28 views

Minimum size set of partition-of-unity smooth functions with bounded support [closed]

Let D be the unit disk in $R^2$, and take fixed $\epsilon <1$. I am looking for smallest possible set ${\left\{f_i\ \right\vert{}\ {0\!\leq\!f}_i\!<\!1\}}_{i=1}^n\,\,$ with the following ...
2
votes
0answers
57 views

Asymptotic behavior of a function [migrated]

Let $n\geq 2$, and define $$\phi(t) = \int_{S^{n-1}} \cos(t \omega_1) d\omega,$$ where $S^{n-1}$ is the unit sphere in $n-$dimensional euclidean space, and $d\omega$ denote the surface area on ...
-1
votes
0answers
27 views

Total measure and Riesz theorem [migrated]

As I specified in the other question I asked, analysis is not my field, so I'm sorry if my question is trivial. Riesz representation theorem states the following: Given $X$ a locally compact ...
-4
votes
1answer
143 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
1
vote
1answer
84 views

Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
0
votes
0answers
48 views

Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
-1
votes
0answers
20 views

Is scaling mapping uniformly continuous for any metric in $R$? [migrated]

For example, is $f(x)=2x$ as a function of $R\rightarrow R$ uniformly continuous for a general metric $d(x,y)$? Or is the following true? $\forall \epsilon > 0, \exists \delta $, such that ...
3
votes
1answer
295 views

On the embedding of a function space $X$ into $L^2\cap L^4$

It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding $$ L^4({\Omega})\subset L^2({\Omega}) $$ since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...
-1
votes
0answers
11 views

How can i get real analog of complex function? [migrated]

I have a function: sin(wt-jT) (1.1), where j - complex number I transform it to function with real arguments: ...
0
votes
1answer
261 views

The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads. Let's rewrite the product in a way ...
0
votes
0answers
20 views

Correct definition of submodularity

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
2
votes
1answer
233 views

Is this a log-concave function?

Let $(a_k)$ be a log-concave positive decreasing sequence. Is $\sum\limits_{k=1}^n a_k(1-e^x)^{k-1}$ log-concave in $x<0$, for each natural $n$?
1
vote
0answers
50 views

Inverse Laplace transform of a non-negative function

Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty ...
29
votes
5answers
3k views

Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
5
votes
1answer
320 views

Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive

Is the following function real analytic in $t>0$: $$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$ where $-a$ and $b$ are positive, and $c\not=a$? ...
0
votes
0answers
47 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
2
votes
0answers
151 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral ...
0
votes
1answer
169 views

Is the limsup or liminf of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
1
vote
1answer
73 views

Do we have independence if we let the indices of the events increase?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. ...
3
votes
0answers
123 views

Extreme derivatives in Baire class 1

In the 1994 volume of "Differentiation of Real Functions" A. Bruckner poses the following problem (p.41): "Find necessary and sufficient conditions on a continuous function $F$ that its Dini ...
-2
votes
0answers
105 views

Conjecture about tail events

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \tau_{A} := \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ ...
1
vote
0answers
64 views

The real method of interpolation and operator ideals,

Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding) ...
0
votes
1answer
113 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
1
vote
2answers
672 views

on the set of numbers generated by integer linear combination of two real numbers.

Let $b > a > 0$ be two real numbers. I am interested in the set of numbers $X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$. What ...
30
votes
2answers
2k views

An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
-2
votes
1answer
89 views

Is this intergral inequality valid? [closed]

Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
0
votes
1answer
355 views

a unique solution ? iteration involving conditional distributions

consider the following mappings, G and T, $y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$ $z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$ where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in ...
-1
votes
0answers
70 views

inverse Fouier transform from partial Fourier transform

It is known that a function $u(\mathbf{x})\in C_0^{\infty}(\Omega)$ can be reconstructed from its Fourier transform using inverse Fourier transform. $$u(\mathbf{x}) = \mathcal{F}^{-1} (\widehat{u}) = ...
2
votes
1answer
118 views

Approximation of the cumulative normal distribution

As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ ...