# Questions tagged [ca.classical-analysis-and-odes]

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

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### Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$

Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that \begin{align*} \int_\Omega \left(\rho_{1} \...
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### A kind of reflection formula for the logarithmic derivative of the zeta function

So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
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### Functions of $\mathbb{R}^d$ preserving convexity of sets

Consider a function $f : \mathbb{R}^d \to \mathbb{R}^d$, with $d\geq 2$, such that: $f$ is injective, For any convex set $A$ of $\mathbb{R}^d$, $f(A)$ is also convex. What can we say about $f$ ? In ...
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### Generating function of product of binomial coefficients

Let $m, n\in \mathbb{N}$ and $|x| < 1$. I look for hints to derive an analytic formula for $$f_{m,n}(x) = \sum_{k \in \mathbb{N}} {n + k \choose k} {m + k \choose k} x^{k}.$$
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### Reproducing kernel Hilbert space of Matérn kernels

I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv) On the top of ...
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### On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
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### How to compute this limit involving the associated Legendre function?

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
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### Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$

Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$. Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$: \...
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### Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?

This seems such a simple question that I fear I must have missed some elementary maths. I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real....
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### Turán–Nazarov's lemma for algebraic polynomials?

Nazarov proved a version of Turán's lemma in Complete Version of Turan’s Lemma for Trigonometric Polynomials on the Unit Circumference, which is now known by the name Nazarov–Turán's lemma. A special ...
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### Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n$$ where $$a_n = \prod_{i=1}^{n}G((i-1)h)$$ We can conclude that the series satisfies a Linear differential ...
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### A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction: $$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$ where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
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### How to understand the following limit exist: $\lim_{n\to \infty} \frac{1}{n} f(X)^Tf(X)=?$
I read one paper about Approximate Message Passing algorithm and I am confused about the notations. If I have a matrix $X\in\Bbb R^{n\times d}$ and a function \$f: \Bbb R^{n\times d} \to \Bbb R^{n\...