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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Inequality for functions on [0,1], continued

Let $0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set $$f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$$ Question. Is it true that, ...
Deepti's user avatar
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1 answer
274 views

Rademacher type theorem for Alexandrov spaces

The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere. I am wondering if this result can be generalized as follows. ...
asv's user avatar
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6 votes
1 answer
345 views

Harmonic maps are light

Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected? I hope that the answer is yes. But actually I ...
Anton Petrunin's user avatar
6 votes
2 answers
215 views

Subsets $X$ such that their Hausdorff outer measure is not finite

Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the ...
Paolo Leonetti's user avatar
6 votes
1 answer
178 views

On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
Taras Banakh's user avatar
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6 votes
3 answers
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Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators. So my question is something like this: Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
QuantumTheory's user avatar
6 votes
1 answer
1k views

Jackson's theorem for partial sum of Fourier series

There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies $$ |f(x) - ...
Kurisuto Asutora's user avatar
6 votes
2 answers
3k views

Multivariable monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...
Kurisuto Asutora's user avatar
6 votes
1 answer
364 views

An inequality for a concave function $f(x)=x^{p/2}$

Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
MathArt's user avatar
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1 answer
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Log-convexity of determinant

Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
António Borges Santos's user avatar
6 votes
1 answer
127 views

Small shifts of weakly converging sequences in $L^1$

$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
Iosif Pinelis's user avatar
6 votes
2 answers
693 views

Upper bounds for Bessel functions

Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$ See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...
mape's user avatar
  • 63
6 votes
1 answer
340 views

Existence of smooth function that characterizes boundary and interior of set

It is well known that every closed set $A \subset \mathbb{R}^{n}$ is the zero level set of some smooth function. It follows that every closed set is also the zero sublevel set of some smooth function, ...
node's user avatar
  • 329
6 votes
1 answer
126 views

Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that: $$ \lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0 $$ where $\Vert \cdot \Vert_1$ is the $L_1$ ...
Tomer's user avatar
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6 votes
1 answer
546 views

Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
neverevernever's user avatar
6 votes
1 answer
213 views

F-sigma subset of plane meeting every circle at 3 points

Is there an $F_{\sigma}$-set (countable union of closed subsets of plane) $S \subseteq \mathbb{R}^2$ that meets every circle at 3 points?
Ashutosh's user avatar
  • 9,771
6 votes
2 answers
3k views

Upper semicontinuity of set-valued maps with open values

Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as: Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at ...
flyingwith's user avatar
6 votes
1 answer
257 views

bounding derivative of a sequence

I've been banging my head against the wall on this one ... define a sequence of polynomials $q_n$ by $q_0 = 0$ and $$q_{n+1} = q_n + .5(t^2 - q_n^2).$$ If $q_n \leq t$ on $[0,1]$ then $$.5(t^2 - q_n^2)...
Nik Weaver's user avatar
6 votes
1 answer
2k 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 $\...
Andrew's user avatar
  • 559
6 votes
2 answers
2k views

How to prove the Hahn-Banach constructively

I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space. Thanks in advance for any helpful answers.
q.g's user avatar
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2 answers
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Continuity of a convolution (Version 2)

Hello, This problem bothers me for some time. Suppose that $\mu$ is a non-negative Radon measure (or positive linear functional of the space of continuous functions with compact support); $\psi$ is ...
6 votes
6 answers
4k views

existence of antiderivatives of nasty but elementary functions

In discussing with my honors calculus class the fact that some continuous elementary functions do not have an elementary antiderivative, I realized I was unsure whether every discontinuous elementary ...
James Propp's user avatar
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6 votes
1 answer
191 views

Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm?

Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by $$m_\varepsilon f(x) := \inf_B \frac1{|B|} \int_B |f| ,$$ where the infimum is taken over ...
Nate River's user avatar
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6 votes
1 answer
188 views

Characterization of sums of periodic functions over the real line

Is there any known characterization of the functions $\mathbb{R \to R}$ that can be written as a sum of (a finite family of) periodic functions? Not assuming any regularity condition (not even ...
Lê Thành Dũng 'Tito' Nguyễn's user avatar
6 votes
2 answers
325 views

On frequency decay of an integral transform of a function

Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that $$ \bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$ for all $\tau \...
Ali's user avatar
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6 votes
1 answer
211 views

Set where the speed of convergence is uniform in Lebesgue's density theorem

Let $B \subset \mathbb R^n$ be the unit ball. Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$). Then, Lebesgue's density theorem, says that ...
HHN's user avatar
  • 363
6 votes
1 answer
171 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
Ali's user avatar
  • 4,077
6 votes
1 answer
327 views

Optimal constant in Sobolev embedding

It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\...
Delio Mugnolo's user avatar
6 votes
1 answer
463 views

Holomorphic extensions of a non-vanishing real-analytic function

Let f(z) be a holomorphic function defined on an open neighborhood $R$ of the interval $I=[0,1]\subset \mathbb{R}$. Assume $f$ does not vanish on $I$. Then $g(x) = |f(x)|$ is a real-analytic function ...
H A Helfgott's user avatar
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6 votes
2 answers
669 views

How to control Wasserstein distance in terms of characteristic function

Let $\mathcal P(\Omega)$ be the set of probability measures supported on some compact subset $\Omega\subset\mathbb R^d$. For $\mu\in\mathcal P(\Omega)$, denote by $F_{\mu}$ its characteristic function,...
user avatar
6 votes
1 answer
180 views

Minimum of $z:\mathbb{R}^n \to \mathbb{R}$ along paths implies local minimum of $z$

Suppose we are given a smooth function $z: \mathbb{R}^n \to \mathbb{R}$, a point $x_0 \in \mathbb{R}^n$ and a set $\mathcal{F}$ consisting of certain paths in $\mathbb{R}^n$, i.e. $f: [0,1] \to \...
Pierre's user avatar
  • 63
6 votes
1 answer
282 views

What is the growth rate of the products of binomial coefficients?

Question 1: Are the following empirically observed relationships true $$ {n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a} \sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg) $$ ...
Nilotpal Kanti Sinha's user avatar
6 votes
1 answer
134 views

Second derivative of integral function

Let $f: \mathbb R^2 \rightarrow \mathbb R$ be a smooth strictly convex function with unique minimum at $0$ such that all level sets $A_x:=\left\{z ; f(z) \le x \right\}$ are compact. Imagine something ...
user avatar
6 votes
2 answers
239 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)<\...
Ali's user avatar
  • 4,077
6 votes
1 answer
181 views

Reference request: A collection of topologies on $\mathbb{N}$ formed via series

First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
Jason DeVito - on hiatus's user avatar
6 votes
1 answer
344 views

Double Series involving Gamma function

Does anyone have any ideas on howto verify $$\sum_{n,m=0}^\infty \frac{\Gamma(n+m+3x)}{\Gamma(n+1+x)\Gamma(m+1+x)}\cdot \frac{1}{3^{n+m+3x-1}} = \Gamma(x)$$ for $x>0$? I posted this question also ...
maliesen's user avatar
  • 284
6 votes
2 answers
656 views

integral depending on a parameter

Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy the following estimate $$ \left|\int_0^1 f(t) e^{st}dt\right|\le Cs^{\frac12},\quad s>1, $$ where the constant $C$ ...
CooLee's user avatar
  • 375
6 votes
3 answers
440 views

Exercise related to log-Sobolev inequalities

This is essentially what Exercise 5.4 in Boucheron, Lugosi, Massart Concentration Inequalities boils down to: For real $a,b$ and $0<p<1$, \begin{align*} &pa^2\log( \frac{a^2}{b^2+pa^2-pb^2}...
Aryeh Kontorovich's user avatar
6 votes
1 answer
304 views

In the plane, does complement of Brownian path have infinitely many connected components?

Let $d = 2$. Do we have that with $P_x$—probability $1$, for every $T> 0$ the complement $W[0, T]^c$ of the Brownian path up to time $T$ has infinitely many connected components? I had seen this ...
Edward Hoenn's user avatar
6 votes
1 answer
909 views

Legendre transform and Lipschitz approximation

Let $(X,d)$ be a compact metric space and $f:X \to \mathbb{R}$ a real valued continuous function. Let us agree that a modulus of continuity means concave, nondecreasing, uniformly continuous function $...
truebaran's user avatar
  • 9,140
6 votes
1 answer
142 views

Terminology for sequences/functions that approach each other

What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ...
Toby Bartels's user avatar
  • 2,644
6 votes
1 answer
600 views

Arbitrary small positive lower semi continuous functions

This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way. Def: Let $(X,\tau)$ be a Tychonoff ...
Ali Reza's user avatar
  • 1,778
6 votes
3 answers
1k views

functional subrings

I should recall the notion of maximal subring of a commutative unitary ring $R$. Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...
Ali Reza's user avatar
  • 1,778
6 votes
2 answers
2k views

non-maximal prime ideal in the ring of continuous functions

Let $A=C(0,1)$ be the ring of continuous real valued functions on the open interval $(0,1)$. It is not too difficult to show that if $\mathfrak{m}\subseteq A$ is a maximal ideal with residue field $A/\...
Hugo Chapdelaine's user avatar
6 votes
1 answer
182 views

Oscillatory integrals with a decaying factor in the integrand

Lemma 4.5 of Titchmarsh's book The Theory of the Riemann Zeta function says (slightly rephrased): Let $F$ be a twice differentiable real function such that $ F''(x) \geq r > 0$ for all $x$ in $[a,b]...
Evandra's user avatar
  • 63
6 votes
1 answer
495 views

Rate of decrease of the Fourier transform of standard mollifiers

What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$, $$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$ and $$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
Iosif Pinelis's user avatar
6 votes
2 answers
493 views

When is $\lVert f*g\rVert_\infty=\lVert f\rVert_1\lVert g\rVert_\infty$?

If $1\leq p<\infty$, it is easy to find nice necessary and sufficient equality conditions for the convolution inequality $$\lVert f*g\rVert_p\leq\lVert f\rVert_1\lVert g\rVert_p\qquad (f\in L^1(\...
apanpapan3's user avatar
6 votes
1 answer
234 views

Perron-Frobenius and Markov chains on countable state space

The following question naturally arises in the theory of Markov chains with countable state space to which I would be curious to know the answer: Let $A:\ell^1 \rightarrow \ell^1$ be a contraction, i....
Landauer's user avatar
  • 173
6 votes
1 answer
451 views

A smooth function $\mathbb{R}\to\mathbb{R}$ agrees with an analytic function on a bounded infinite set

Fix a smooth function $f:\mathbb{R}\to\mathbb{R}$. Do there exist real numbers $a<b$, an infinite set $S\subset (a, b)$ and an analytic function $g$ defined on $(a-\epsilon, b+\epsilon)$ for some $\...
user avatar
6 votes
1 answer
196 views

Circular sequences continuous?

I noticed something interesting when playing around with Mathematica. Consider the sum $$x(N)= \frac{1}{N^2} \sum_{i=1}^{N-1} \frac{1}{1-\cos(2\pi i/N)}$$ this sequence will converge to $1/6$ as $N$...
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