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      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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      23 views

      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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      vote
      1answer
      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\...
      1
      vote
      0answers
      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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      votes
      2answers
      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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      votes
      3answers
      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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      vote
      0answers
      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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      votes
      2answers
      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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      votes
      0answers
      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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      votes
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      20 views

      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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      votes
      2answers
      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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      votes
      1answer
      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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      0answers
      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\...
      3
      votes
      1answer
      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)...

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