Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

learn more… | top users | synonyms (2)

1
vote
0answers
11 views

L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$ \|q\|_1 \leq O(n) \|p\|_1 $$ where we define $\|f\|_p := (\int_{-1}^1 |f(x)|^pdx)^{1/p}$? ...
2
votes
0answers
121 views

Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here. E.g. $\cos^2x+\sin^2x=1$ has ...
-3
votes
0answers
75 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere injective? [on hold]

Let $f:\mathbb R^2 \rightarrow \mathbb R^2$, and $rank(\frac{df}{dx}) = 2$ everywhere. If $f$ is surjective $f$ necessarily injective? Also, what if $f$ maps $\mathbb R^{2+}$ (i.e. $\{x_1>0, x_2&...
1
vote
4answers
351 views

Sum and interpolation of hurwitz zeta functions

$$f(a,x)=\sum_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$ Can I apply Euler-Maclauren formula to this sum? where $a\in(0,0.5)$, p is a natural number, and $x$ is a ...
0
votes
0answers
67 views

some strange sums ramanujan type

i found sum as $$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...
2
votes
0answers
90 views

Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
11
votes
6answers
694 views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
1
vote
0answers
71 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 $\...
10
votes
7answers
4k views

a^b = b^a when a is not equal to b.

I am attempting to find a closed form solution or a nice series for $x^{x+1}=(x+1)^{x}$. First of all I looked at $a^b=b^a$. Fixing a, this means finding out when $a^{1/a} = x^{1/x}$. $f(x)=x^{1/x}$ ...
9
votes
3answers
2k views

erfc lower bound

I've seen the following lower bound for the complementary error function (erfc) but I haven't been able to prove it. Does anyone know how to establish the following? $$erfc(x) > \frac{ x \exp(-x^...
0
votes
0answers
48 views

Min-max theorem - Subspaces over $H_0^1$ [closed]

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem $$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\...
0
votes
1answer
201 views

The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
5
votes
1answer
62 views

Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that $$ \lvert u\rvert_s = \...
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 ...
13
votes
1answer
416 views

Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...
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 $...
3
votes
0answers
49 views

Riesz basis of Paley-Wiener space

Let us consider the Paley-Wiener space: $$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$ Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...
6
votes
1answer
122 views

The “Peano phenomenon” for differential equations

Consider the following statement: If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$ the "Peano phenomenon" can arise only at those values ...
3
votes
1answer
172 views

On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...
7
votes
1answer
335 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by $...
7
votes
1answer
287 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
5
votes
3answers
275 views

Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem: $$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$ where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
3
votes
0answers
81 views

Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...
3
votes
2answers
352 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
0
votes
1answer
76 views

Behavior of a Solution of a Nonlinear ODE

In my work, I encountered the following equation: $$ (a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x). $$ I would like to know as much as possible about the solution. More particularly, I would ...
14
votes
2answers
307 views

Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...
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} ...
3
votes
1answer
496 views

Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent") $$\lim_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$ To simplify, lets ...
9
votes
2answers
771 views

On the average of continuous functions $f:\mathbb{R}^2\rightarrow[0,1]$

Is it true that if the average of a continuous function $f:\mathbb{R}^2\rightarrow[0,1]$ over a unit circle centered around $(x,y)$ is $f(x,y)$ for all $(x,y)\in\mathbb{R}^2$, then $f$ is necessarily ...
2
votes
1answer
223 views

Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Generalizations to broader ...
0
votes
1answer
247 views

Uniformly continuous functions and Borel hierarchy in the compact-open topology

Let $\Omega\subset\mathbb{R}^n$ be open, $\mathscr{C}(\Omega,\mathbb{R})$ the Fréchet space of real-valued continuous functions on $\Omega$ endowed with the compact-open topology, and $\mathscr{C}_u(\...
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 ...
3
votes
0answers
172 views

Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty ...
17
votes
3answers
2k views

Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
5
votes
0answers
66 views

Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only): Theorem: if $f \in H^s(\mathbb{R}^3)$ with $ s > 3/2$ is compactly supported, then $$ \left| \int_{\mathbb{R}^3} f - \...
2
votes
1answer
82 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
0
votes
0answers
24 views

diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...
29
votes
3answers
3k views

What is the standard notation for a multiplicative integral?

If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum_{i=1}^n f(t_i^*) dt_i$, or ...
2
votes
1answer
177 views

Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ...
6
votes
1answer
214 views

Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that $$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|...
2
votes
1answer
150 views

Ramanujan-type sum

Could you show that $$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(...
7
votes
1answer
150 views

$C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference. Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
9
votes
1answer
484 views

Find functions such that f(f(x))=f(x)e^x

Are there monotonically increasing functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = e^x f(x)$?
1
vote
0answers
103 views

Ramanujan sum type

I try to show $$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...
1
vote
0answers
45 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...
7
votes
4answers
4k views

Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is: $$ x^2 y'' + a x y' + b y = 0 $$ Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...
1
vote
0answers
56 views

History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of - $\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
3
votes
1answer
76 views

Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
19
votes
12answers
3k views

Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...