# 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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### Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...

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26 views

### Reverse Loomis-Whitney Inequality for funcctions

I was wondering if the reverse Loomis-Whitney inequality holds for general functions:
Let $n\geq 2$. Let $(X_i,\mu_i)$, $1\leq i\leq n$ be measure spaces. Write $x=(x_1,\dots,x_n)$ and for each $1\...

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55 views

### How to see the divergence of a series is not faster than some order? [on hold]

$$
\sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p}
$$
For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.

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**2**answers

195 views

### Sums of entire surjective functions

Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...

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138 views

### Existence of solution to linear fractional equation

We consider the equation
$$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...

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123 views

### Lower bound for the $L^2$ norm of a polynomial / hypergeometric function

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial
\begin{equation}
\varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1),
\...

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256 views

### Can we use Ramanujan's parameterization of Klein's quartic to solve Klein's septic?

I. Klein
In "On the Order-Seven Transformation of Elliptic Functions" (pp. 287-331), he discusses in p. 298 what we now call the Klein quartic,
$$\lambda^3\mu+\mu^3\nu+\nu^3\lambda= 0\tag1$$
and in ...

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18 views

### linear system of differential equations with variable coefficient [closed]

Any application about linear system of differential equation with variable coefficient which is related to modelling or other fields and mehods to solve them as well.

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### Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems?

From what I read of time-scale calculus literature, most results of continuous-time and discrete-time systems can be generalized to arbitrary time-scales by considering the generalized derivative ...

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272 views

### Properties of heat equation

** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...

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57 views

### Algebraic Riccati and WKB [closed]

It's a one-liner to show that the algebraic Riccati equation (ARE) and the lowest order form of WKB for a linear ode are the same. But I've looked all over the web and there does not seem to be a ...

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**1**answer

121 views

### Brascamp-Lieb inequalities on the sphere

In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...

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57 views

### Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following:
Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...

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**1**answer

168 views

### On convex functions which are non constant on every segment

I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...

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29 views

### Unique solution for a difference ODE?

Any idea how to find general solution
$$a'_{n}(t)= (n+\alpha )a_{n}(t) + \beta a_{n+1}(t) + \gamma a_{n+2}(t)$$
for some coefficients $\alpha, \beta, \gamma$?, Where $a'_{n}(t)=\frac{d}{dt} a_{n}(t)...