Tagged Questions
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3
votes
1answer
150 views
Fourier expansion of Takagi-function (everywhere non differentiable function).
Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...
1
vote
2answers
54 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
224 views
Does there exist a function such that $\int_{\mathbb{R}_+^{\star} } t^nf(t)dt=0$? [on hold]
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
116 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
40 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
323 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
96 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) + ...
2
votes
1answer
393 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
79 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
67 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
125 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
86 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
490 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
398 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
96 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 ...
4
votes
0answers
79 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
51 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
42 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
93 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 ...
7
votes
1answer
372 views
+50
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
25 views
Can anyone comment on this proof for products of sequences? [migrated]
I am working on this (http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-436j-fundamentals-of-probability-fall-2008/assignments/MIT6_436JF08_hw01.pdf) and it's pretty hard. I ...
4
votes
0answers
185 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
158 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
553 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 ...
24
votes
0answers
779 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 $(-1,1)$
...
-2
votes
0answers
7 views
Definition of Fresnel Integrals $C(\zeta)$ and $S(\zeta)$ [migrated]
I'need to compute the definite Fresnel Integrals
$$ C(\zeta) = \int^\zeta_0 \cos(\frac{\pi}{2}x^2)dx $$ $$ S(\zeta) = \int^\zeta_0 \sin(\frac{\pi}{2}x^2)dx $$
This seems to be a common math/physics ...
0
votes
1answer
130 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
124 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
244 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
429 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
138 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
72 views
Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$
I want to classify the following functions:
$$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$
$$ ...
2
votes
1answer
106 views
Equivalence of negative Sobolev norm of derivative to $L^2$-norm
Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the ...
2
votes
1answer
110 views
Is there a dense rational sequence of positive separation?
Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, ...
1
vote
1answer
72 views
A question which belongs to a class of Zygmund functions
Let $f$ be an absolutely continuous, periodic with period 1 and satisfies the condition
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const}\frac{\delta}{(\log\frac{1}{\delta})^{\epsilon}}, ...
0
votes
1answer
93 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
$$\forall x\in [0,1],\quad
f(x)=\int_0^1 k(x,y) g(y) ...
2
votes
0answers
113 views
Analytic varieties for the primes and the twin primes
I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...
0
votes
1answer
75 views
Looking for methods/results for explicitly bounding iterations of rational functions
In Theorem 2.6.4 of Beardon's book, "Iteration of Rational Functions", he states the values for the first two coefficients of an iterated power series.
That is, suppose that
$$
...
1
vote
1answer
77 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
97 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 $X^{\infty}$ ...
1
vote
1answer
148 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
381 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
103 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 ...
0
votes
3answers
131 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
113 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
106 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
82 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
164 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
2answers
141 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
241 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 ...
