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1
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2answers
87 views

a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
3
votes
1answer
304 views

Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [closed]

Let $f\in C([a,b],\mathbb{R})$ such that $\displaystyle\int_{a}^{b} t^nf(t)dt=0$ for all integer n. We know that $f\equiv 0$. It's call Hausdorff theorem. This theorem is wrong on $\mathbb{R^+}$, a ...
1
vote
2answers
153 views

Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
3
votes
1answer
47 views

Complete classification of complexity classes / infinite approaching sequences

http://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities For complexity as seen in the above link, complexity classes can be log, polynomial, exp, or composition of any of these ...
0
votes
0answers
364 views

Given an even function how to obtain the most close odd function and vise versa?

Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$? By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference ...
3
votes
1answer
207 views

About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality $f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...
3
votes
1answer
434 views

Is there a direct proof of the following real analysis fact?

I want to prove the following fact without using topological degree theory or related algebraic topology Let $h:\overline{B}(0,1)\to \mathbb{R}^n$ be a continuous map such that $|h(x)-x|\leq \delta$ ...
0
votes
1answer
184 views

Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros? More precisely, suppose that $f_i: R^n \to R^m$ is a ...
5
votes
0answers
84 views

Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$. By this I mean that there exists a polynomial $P$ with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!) such that $P(\phi) = 0$ Let ...
5
votes
3answers
163 views

Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
2
votes
2answers
120 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
2
votes
3answers
685 views

Assessing effectiveness of (epsilon, delta) definitions [closed]

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
7
votes
3answers
429 views

Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$). We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...
0
votes
1answer
117 views

How to perturb a function to separate points

Consider two smooth functions $f,g\in C^\infty(\Omega)$ with $\partial \Omega$ smooth and $\Omega\subset \mathbb{R}^3$. Assume that $f=g$ on $\partial \Omega$. For any given $\varepsilon>0$, how ...
5
votes
0answers
90 views

Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$? ...
2
votes
1answer
65 views

radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke: Teorem. Let $f : U ⊂ \mathbb{R}^n ...
0
votes
0answers
58 views

How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
3
votes
0answers
112 views

Tauberian theorem wanted

At least, I think it might deserve to be called a Tauberian theorem, inasmuch as it would generalize the Tauberian theorem mentioned by Liviu Nicolaescu in his reply to my question Using a quadratic ...
9
votes
1answer
540 views

Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence \begin{equation} a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1), \end{equation} with $a(1,1)=1$ and ...
4
votes
0answers
207 views

Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by \begin{equation} ...
3
votes
1answer
185 views

Error of midpoint method for functions that are not twice-differentiable

All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
16
votes
2answers
611 views

Felix Klein on 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 ...
26
votes
0answers
959 views

Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$ ...
0
votes
1answer
315 views

Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by ...
3
votes
1answer
203 views

When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem). Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
3
votes
1answer
281 views

An elementary inequality: reference request

Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$. Now ...
19
votes
2answers
513 views

Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
1
vote
1answer
155 views

There is a horseshoe with positive measure

Here is a theorem by Bowen : My question is about the highlighted part in the picture. why there such a function $g$ exist?
0
votes
0answers
79 views

Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$

I want to classify the following functions: $$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$ $$ ...
3
votes
1answer
209 views

Equivalence of negative Sobolev norm of derivative to $L^2$-norm

Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the ...
2
votes
1answer
118 views

Is there a dense rational sequence of positive separation?

Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, ...
1
vote
1answer
127 views

A question which belongs to a class of Zygmund functions

Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, ...
0
votes
1answer
103 views

Positive kernel property

Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let $f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that $$\forall x\in [0,1],\quad f(x)=\int_0^1 k(x,y) g(y) ...
2
votes
0answers
120 views

Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say about the primes and twin primes. According to Wikipedia analytic variety is defined locally as the set of common zeros of finitely many analytic ...
0
votes
1answer
86 views

Looking for methods/results for explicitly bounding iterations of rational functions

In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series. That is, suppose that $$ ...
1
vote
1answer
86 views

On weak linear continuous functions

This is what I have first asked in SE but I think it is more suitable for here. I am interested in the set of all continuous functions $f: (0, \infty) \longrightarrow \Bbb{R}$ with the following ...
-1
votes
1answer
124 views

Is the countably infinite product of locally convex topological vector spaces locally convex?

Let $(X,\tau)$ be a locally convex topological vector space and denote the product space $$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$ If we endow $X^{\infty}$ ...
2
votes
1answer
185 views

Inequality in the Sobolev space $H^1$

I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...
4
votes
1answer
423 views

A generalization of a theorem of Grothendieck

In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$. Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$. Assume that $S$ is a subvector space ...
0
votes
1answer
178 views

Constructing a continuous matrix valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
1
vote
3answers
194 views

Formalism for moving from a metric space into a vector space for mathematical/statistical modeling given a data

I have a metric space $(X,d)$. I have a physical situation (data) where each physical entity corresponds to an $x \in X$. I want to do some mathematical/statistical modeling of this data, but the ...
2
votes
0answers
123 views

Improving a bound from Taylor's Theorem

For this problem, suppose $g:\mathbb{R}\rightarrow\mathbb{R}$ is such that $g\in\mathcal{C}^{k}(\mathbb{R})$, and there exists $\epsilon>0$ such that \begin{align*} ...
3
votes
2answers
115 views

series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
3
votes
1answer
84 views

Number of small projections

Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of ...
1
vote
1answer
225 views

Is the space of test functions separable? [closed]

Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...
0
votes
3answers
232 views

dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
10
votes
3answers
339 views

The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
9
votes
1answer
197 views

Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points and a positive integer $n$, is there ...
1
vote
1answer
126 views

Pohozaev result for equations with weights

I am interested in nonnegative solutions of $-div( e^{-\gamma(x)} \nabla u(x)) = e^{-\gamma(x)} u(x)^p$ in $\Omega$ with $ u=0$ on $ \partial \Omega$. Or instead the equation $ -\Delta u + ...
3
votes
1answer
213 views

Julia sets without Montel's theorem

Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...