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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Dual space of continuous functions

Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
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How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?

In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
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Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$

Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
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Elliptic regularity and Sobolev spaces

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...
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Closed formula for series $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$

What can be said about $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$ (for $|x|>1$ and $|y|>1$ and $x\neq y$)? Is there a kind of closed formula for this? By comparing to the geometric series, this sum ...
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Maximum of a function

Let $p,q\in\Bbb N$ with $p\not=q$. Put $$M=\sup_{x\in[0,1]} \left|\cos(2 p\pi x)-\cos(2 q\pi x)\right|.$$ What is the value of $M$. Thanks
zoran  Vicovic's user avatar
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All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)

I have asked this question on MSE, but this is a better place. The heat equation and the heat kernel. Consider the heat equation on $\mathbb R$: $$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
Lorenzo Pompili's user avatar
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Quantifying the degree of continuity of a function via perturbations

Let $f: \mathbb R \to \mathbb R$ be a measurable function. Define the perturbation operator $T_f$ on measurable functions $g: \mathbb R \to \mathbb R$ by $$T_f (g)(x) := f(x + g(x)) - f(x).$$ Observe ...
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Regularity of the spherical mean of a compactly-supported function

The problem Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$. Then, consider ...
Gilles Mordant's user avatar
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Weaker versions of the Riemann series theorem in constructive mathematics

The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real ...
Madeleine Birchfield's user avatar
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Is there a natural finitely additive measure for which Vitali sets have measure zero?

Vitali sets are nonmeasurable and in particular are not null sets. But all Vitali sets are in some sense small, as described below. Let $V$ be any Vitali set and let $k \in \mathbb{N}$. For each $i \...
Aaron Hill's user avatar
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Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
Tom Copeland's user avatar
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Nature of function as $x\rightarrow\infty$

I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
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How far can a continuous, almost everywhere differentiable function be from being a Sobolev function?

Let $\Omega$ be the open unit ball in $\mathbb R^n$. Consider the set $\mathcal D$ of continuous functions $f:\Omega \to \mathbb R$ that are differentiable a.e, and with $|\nabla f| \leq 1$ wherever $...
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What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?

Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence. Question: What is a minimal topological condition on $X$ ...
user141240's user avatar
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Does there exist a “fat” Thomae’s function?

Definitions and some motivation: Thomae’s function, also known as the raindrop function has several curious properties. One of which is the following - both its discontinuity set and continuity set ...
Nate River's user avatar
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Inequality for functions on $[0,\infty)$

Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions: $$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;...
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Minimal growth condition for a rearrangement

Let $\sigma: \mathbb{N}\to\mathbb{N}$ be bijective such that there is a sequence $(n_k)_{k\ge 0}$ in $\mathbb{N}$ satisfying $|\sigma(n_k)−n_k|\to\infty$ for $k\to\infty$. Question: Is there a (...
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Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?

While solving a problem in calculus of variations, I came to the following question: Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...
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Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?

In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
Nilotpal Kanti Sinha's user avatar
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Is Sobolev limit of bijective maps surjective?

This is a cross-post. Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps ...
Asaf Shachar's user avatar
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Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
Guy Fsone's user avatar
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A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?

In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
UserA's user avatar
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Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
Tony419's user avatar
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Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?

Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
Rajesh D's user avatar
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Minimizing total variation

Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by $$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
H A Helfgott's user avatar
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Useful notion for locally convex spaces - well known?

In my current work the following property of maps between locally convex spaces showed up at several places and proved to be useful. It seems quite elementary to me, so I would like to know whether it ...
Jan Bohr's user avatar
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Points where singular sum is small

We consider $x_1,..,x_N$ points in the plane $\mathbb{R}^2.$ We define the sum $$F(x):=\frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} \vert x_i-x_j \vert^{-2}.$$ I am looking for a statement of the ...
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Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?

This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO. For every ...
Nilotpal Kanti Sinha's user avatar
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0 answers
255 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
Arrow's user avatar
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Is the ratio of a number to the variance of its divisors injective?

The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
Nilotpal Kanti Sinha's user avatar
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452 views

Partitioning $\mathbb{R}^n$ into closed sets

Let $n$ be a positive integer. It is well-known that $\mathbb{R}^n$ cannot be non-trivially partitioned into open sets, since it is connected. Let $\frak P$ be a partition of $\mathbb{R}^n$ into ...
Dominic van der Zypen's user avatar
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Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace

Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$. Is there a characterization of the set of projections of $f$...
BigbearZzz's user avatar
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A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...
Ali Taghavi's user avatar
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Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
Romeo's user avatar
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Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
Elliott's user avatar
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An integral trigonometric inequality

Problem 1. Suppose that $\xi>0$ and $\sin(2\xi)<0$. Let $$b_\nu=(N-v+1)\tfrac{\pi}{\xi}\quad\mbox{for}\quad\nu=1,\dots,N:=\big[\tfrac{\xi}{\pi}\big].$$ Prove that $$\mathrm{sgn}(\sin \xi)\...
Lviv Scottish Book's user avatar
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Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. ...
Ceeerson's user avatar
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A subadditive bijection on the positive reals

I posed some time ago this question on MSE, which I am proposing also here since we got no definitive answer. Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)...
Paolo Leonetti's user avatar
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global estimate for biharmonic function

My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...
Paul's user avatar
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What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?

Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
Vesselin Dimitrov's user avatar
5 votes
0 answers
317 views

Tietze extension theorem for lower semi continuous functions

On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
M. Reza. K's user avatar
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0 answers
202 views

The boundary integral of a harmonic function

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary and $f$ be a harmonic function on $\Omega.$ It is known that $$ \limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...
Han Ju's user avatar
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Proving that a certain function (related to a volume of a region) has a bounded derivative

Let $F$ be a homogeneous form in $n$ variables with integer coefficients. Let $D$ be a closed box in $\mathbb{R}^n$ (product of closed and bounded intervals). Assume that the partial $\partial F/\...
Johnny T.'s user avatar
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Uniqueness of a SDE with non-negativity constraint

I am working on the following SDE (but we will dealing only with deterministic object: $\omega\in\Omega$ is fixed): \begin{equation}\label{sde}%sde x_t=\underbrace{\xi_0+\int_0^tb(s,x_s)\,ds+\int_0^t\...
Joe's user avatar
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How to solve this operator equation numerically?

I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form $$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$ So even if there ...
Zinkin's user avatar
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0 answers
382 views

Partition of the unit interval into uncountably many sets of full outer measure

Is it possible to construct an uncountable partition $(A_\delta)_{\delta\in[0,1]}$ of the unit interval $[0,1]$ such that $\mu (A_\delta)=1$ for each $\delta\in[0,1]$? ($\mu$ stands for the outer ...
Oleg's user avatar
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operation on Ord., Exp., Dri. generating functions

The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by $$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n\...
T. Amdeberhan's user avatar
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0 answers
116 views

For $f$ a polynomial, does strict convexity of $\log f(e^s)$ imply that the second derviative of $\log f(e^s)$ has no zeros?

Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e. $f(s^2) \cdot f(t^2) > f(st)^2$ for all $s, t \...
user avatar
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375 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
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