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

learn more… | top users | synonyms

2
votes
1answer
62 views

Variation of Radon transform for probability measures on $\mathbb C$

Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...
6
votes
1answer
248 views

A question about composition of functions

Recently, I heard this question: are there two functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is strictly increasing on $\mathbb{R}$ and $g\circ f$ is ...
0
votes
1answer
115 views

Methods to tackle this series and get to the limit?

Take a look at the averaging sum $$\frac{\pi}{n}\sum_{k=1}^n\;\exp{(-\sin\theta_k)}\cdot \sin(\theta_k +\cos\theta_k)\, \qquad\text{where }\;\theta_k=(2k-1)\frac{\pi}{2n}$$ depending on $n\in\...
4
votes
0answers
50 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
6
votes
1answer
202 views
+50

Density of convolution

Let $\{X_i\}$ be i.i.d random variables uniform on a measurable, symmetric set $A$ contained in $[-1,1]$. Let $g_{n}$ be density of $X_1+\ldots + X_n$. Question (general): Is there any non-trivial ...
1
vote
0answers
69 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). Let $\...
0
votes
0answers
65 views

Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function? [migrated]

[Q.] Is there a semicontinuous function, which has its discontinuous set with non-zero measure? Remark: Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...
0
votes
0answers
35 views

Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function. We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...
1
vote
0answers
45 views

How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
1
vote
1answer
130 views

Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
0
votes
0answers
18 views

Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy, $$ \int_{\Omega}|\nabla u |^2\, -\, \int_{\...
0
votes
1answer
39 views

characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
0
votes
1answer
128 views

Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question. Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
8
votes
1answer
239 views

Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
0
votes
0answers
56 views

Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence: \...
3
votes
0answers
39 views

Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions. We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
18
votes
2answers
1k views

Felix Klein on mean value theorem and infinitesimals

This is a reference request prompted by some intriguing comments made by Felix Klein. In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
2
votes
0answers
31 views

Well-definedness for a singular integral

Let $T_\alpha$ be a singular integral operator defined by $$ T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds $$ for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$. ...
5
votes
1answer
95 views

Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set $\...
2
votes
2answers
83 views

Continuous upper envelope of upper semicontinuous function

Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by $$A = \{\phi \in C(K): \phi \ge u\}.$$ [Q.] Is the following ...
3
votes
1answer
345 views

About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
-1
votes
0answers
12 views

Continuous function on compact topological space [migrated]

I came across the following statement. Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \...
2
votes
1answer
79 views

Step 2 of The Strichartz's Estimates in Cazenave's Book

My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates. The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...
5
votes
3answers
159 views

Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that $$ \|u\|_{H^s}^2=\sum_{k\...
1
vote
0answers
31 views

Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...
4
votes
1answer
69 views

weak convergence in $H_0^1$ and strong convergence in $L^2$

I'm reading a hand-waving argument in a proof of Chapter 7 of the Navier-Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly. Let $\Omega\subset{\mathbb{R}^n}$ ...
0
votes
0answers
39 views

Equivalent Definitions of the Gaussian Surface Measure for Regular Sets

I wonder if the following definitions of the Gaussian surface measure are equivalent. First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....
8
votes
1answer
248 views

Sequence of nested sets in $[0, 1]$ with bound on gaps

What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that $$ d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N} $$ for all $N\in ...
5
votes
0answers
75 views

Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$. The $L^p$ norm on $M$ is given by \begin{...
2
votes
1answer
962 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 $f(x)=e^{-|x|}$, ...
10
votes
1answer
870 views

The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known $...
0
votes
0answers
74 views

Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
29
votes
1answer
3k views

Is the following identity true?

Calculation suggests the following identity: $$ \lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}. $$ I have verified this identity for $n$ up to $5000$ ...
3
votes
1answer
157 views

Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
6
votes
1answer
211 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)...
5
votes
1answer
150 views

Domain of Laplacian

Let $L$ be an operator on $C^2(\mathbb R)$, defined by $$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$ for a measure $\nu(dy) = |y|^{-2} ...
4
votes
0answers
66 views

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ ...
1
vote
1answer
117 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
4
votes
1answer
166 views

Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in $$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$ ...
1
vote
1answer
96 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
1
vote
2answers
243 views

How can we obtain the $-\frac{4\pi}3\mu(x)$ term?

Given the expression $$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$ where $\mathscr X=\mathbb R^3$, how does one derive the expression \begin{align} ...
-4
votes
2answers
178 views

Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]

There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$? I guess what I mean is ...
0
votes
0answers
49 views

Bounds on the positive roots of a bivariate polynomial

It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
1
vote
1answer
59 views

Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations: $$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$ $$ ...
4
votes
1answer
407 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 $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...
10
votes
2answers
822 views

Witness to a failure of Fubini/Tonelli

Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue ...
2
votes
2answers
181 views

Proving the non-degeneracy of the critical points of the potential function for a certain vector field with $ n $ point-singularities

This question is an expansion of another question that I asked over at Math Stack Exchange. In what follows, $ \alpha \in \mathbb{R}_{> 1} $ is a constant, $ n $ a fixed integer $ \geq 2 $, and $ [...
0
votes
0answers
128 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...