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

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0
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1answer
114 views

Application of the inverse theorem function on singular points

This question is related to this one A question about the inverse theorem function in $\mathbb{R}^n$. In the response I received, I was told that the issue was related to the singularities, as I do ...
-22
votes
1answer
2k views

Can Poisson Summation formula break?

The Poisson summation formula states if $f: \mathbb{R} \to \mathbb{R}$ then $\displaystyle \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n) $ where $$\hat{f}(\xi) = ...
0
votes
2answers
214 views

A question about the inverse theorem function in $\mathbb{R}^n$

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ ...
4
votes
2answers
232 views

Limit of a double integral

What is the $\varepsilon\to 0$ limit of the following double integral $$\int\limits_{-1}^1d\tau\;\sqrt{1-\tau^2}\;\tau\int\limits_0^\infty dq\;q^2e^{iq(\tau+i\varepsilon)}\;?$$ I was asked about this ...
3
votes
0answers
88 views

Meaning of fractional Fourier transform with imaginary iteration count

(I'm reposting this from math.stackexchange as I didn't get an answer there and thought it might be "advanced" enough for this site.) As one may know, the Fourier Transform $$F[f](\nu) = ...
1
vote
2answers
98 views

Bound deg 3 partial differential operator on Laplace eigenfunction?

I am no expert on PDE and analysis but I am looking for certain technique from PDE. Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or ...
10
votes
3answers
818 views

Convergent subsequence of $\sin n$

It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such ...
10
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2answers
291 views

Is it possible to have the set $f^{-1}(\lbrace x \rbrace)$ perfect for every $x$?

There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup_n ...
6
votes
1answer
299 views

Convergence and Closed Form of an Integral Involving Bell Numbers

1. Does the following integral converge ? $$\int_0^\infty \frac{b(x)}{B(x)} dx$$ where $$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum_{n=1}^\infty \frac{n^x}{n!}$$ 2. ...
4
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0answers
199 views

decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
1
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0answers
278 views

fourier decomposition of white noise

I found in some lecture notes that Brownian motion is defined by its Fourier series: $$ B(t) = \sum_{m \in \mathbb{Z},\; m \neq 0} \frac{a_m}{m} e^{2\pi i \,mt} $$ Then I would get that its ...
0
votes
0answers
118 views

multivariate integral calculation in closed form

I am looking for a closed form for the below integral but since I don't have the necessary backgrounds I am not able to solve it: i know the final solution is in the form of modified Bessel functions ...
6
votes
1answer
238 views

Certain asymptotics involving double infinite sum

Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow ...
2
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2answers
163 views

Ewald's generalized theta function

Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-( Die Berechnung optischer und ...
0
votes
1answer
75 views

An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = ...
7
votes
1answer
266 views

Kakeya and Nikodym maximal functions

I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...
20
votes
3answers
438 views

In what ways is the standard Fourier basis optimal?

Is there any convincing sense in which the standard trigonometric basis for the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is optimal among all the orthonormal bases? (If ...
1
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0answers
84 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

I was able to reduce a research problem relating to normal numbers to analyzing the solutions of the following family of nonlinear systems of equations: \begin{align*} c_0+c_1+\cdots+c_{t-1}&=2t\\ ...
6
votes
0answers
152 views

Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
4
votes
1answer
292 views

On the convergence of the the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$

Let $f$ be a smooth real function defined around origin. If we informally differentiate from the series $\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$ term by term we get ...
2
votes
1answer
166 views

Roots of the XXZ Bethe Ansatz equation

The XXZ spin chain Bethe Ansatz equations are a complicated system of rational function equation: \[ \left(\frac{\lambda_j + i/2}{\lambda_j - i/2} \right)^N = \prod_{l=1, l \neq j}^M ...
10
votes
3answers
550 views

Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?

I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover. ...
1
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1answer
121 views

Where find proof of such theorem about uniform convergence of differences

Where to find a proof of theorem which says that: if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesque measure or on the set of second category with Baire propert ...
2
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0answers
154 views

Differential Equations vs Difference Equations

My question is: Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain? I mean if I know a solution to the difference ...
0
votes
1answer
109 views

The solvability of a Hölder ODE [closed]

The problem is follow, I want to know weather there is a function $u\in \text{C}^{0}\left((-1,1)\right)$ but not $\text{C}^1((-1,1))$, such that $\lim\limits_{h\rightarrow ...
2
votes
0answers
119 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
6
votes
1answer
204 views

Laurent expansion of inverse of vandermonde determinant

I wish to find the coefficients of the Laurent expansion of the inverse of the Vandermonde determinant, that is, the Laurent expansion at 0 of $$\prod_{1\leq i<j \leq n}(x_j-x_i)^{-1}.$$ We can ...
4
votes
0answers
132 views

semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix. \[ \int ...
11
votes
1answer
314 views

Is there something wrong with Hörmander's theorem on stationary phase method

It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - ...
4
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0answers
129 views

Integrate the exponential of sum of circular differences?

Given positive integer $N$ and parameters $T>0$, $a$, $b$, what is $\int_{t_1=0}^T \cdots \int_{t_N=0}^T e^{a(t_1+\cdots+t_N)+b(|t_1-t_2|+\cdots+|t_{i-1}-t_i|+|t_N-t_1|)} dt_1 \cdots dt_N$ ? Any ...
2
votes
1answer
151 views

Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form \begin{equation*} f(t)=\sum_{k=-n}^n c_k ...
2
votes
1answer
217 views

bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$ \begin{equation*} f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})} ...
1
vote
0answers
82 views

Conditions on a measure to satisfy certain relation on moments.

Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$. I'd like to impose some conditions on $\mu$ so the function $$f:p\to \frac{\int_0^\infty ...
3
votes
2answers
819 views

Monotone functions are measurable

Hello there, This is probably very easy but I can't find an argument. Call a function $f: R^n \to R$ monotone increasing if $x_i \le y_i$ for each $1 \le i \le n$ implies $f(x) \le f(y)$. I'd like ...
0
votes
1answer
179 views

Can any compactly supported continuous function be written as a linear combination of functions with small support

Does anyone have a reference for the following result? I am pretty sure that it is true, and should not be hard to prove, but i would surprise me if it is not already proven in many places: Let ...
3
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0answers
205 views

The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is ...
1
vote
2answers
409 views

Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$, $$ \mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
2
votes
1answer
410 views

Existence of Geodesics in continuous metrics

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...
10
votes
4answers
715 views

Calculus book in the spirit of the 18th century

I have been reading a lovely 18th century book by John Rowe called An Introduction to the Doctrine of Fluxions. It presents calculus completely motivated by geometric questions - finding tangents to ...
8
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0answers
143 views

Inertia group vs. differential equations

The tame quotient of the inertia group of $\mathbf Q_p$, say, is the profinite group generated by the Frobenius $\sigma$ and the monodromy $\tau$, subject to the relation $\tau^{p-1} [\tau, \sigma] = ...
2
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0answers
146 views

Fourier : from analyticity to exponential decay ; what prevents optimal decay ?

Hello everyone ! In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) : if ...
1
vote
2answers
184 views

An almost orthogonality principle for L^p

I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it goes: If two ...
5
votes
3answers
725 views

On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$. I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...
3
votes
1answer
201 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
2
votes
2answers
202 views

“Then obviously…” (a bound on strongly differential functions on an interval)

On the fourth page of their 1967 paper Local Behavior of Solutions of Quasilinear Parabolic Equations, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real ...
1
vote
1answer
174 views

Convergence of a sum to the integral

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property : ...
2
votes
1answer
126 views

Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book "Understanding Analysis" by Abbott is stated in page 128 ...
2
votes
1answer
119 views

determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximation for $\log$ function and it had negative sign. Is there any $m_0$ such that for all $n>m>m_0$, the following ...
3
votes
2answers
183 views

Elliptic Harnack inequality for 1D Schrodinger operator?

For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is: There exist $C_{H} > 0$ and ...
11
votes
1answer
191 views

An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series: $$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$ It can be expressed in terms of a hypergeometric function: ...