Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over distributions. Are there any applications of the former which cannot be obtained using the latter? Any important examples? I was told one general property of hyperfunctions … Read more

Dynamics of an inequality

The dynamics D∋(ri,ri+1)↦(ri+1,ri+2)∈D on the set D:={(x,y)∈R2:x>0,y>x2/2} is given by the recurrence ri+2=r2i+12+1r3i+1(ri+1−r2i2) for i=0,1,…. Questions: Is it true that the only periodic sequence (ri) here is the constant one with ri=1 for all i? Take any natural n. Suppose that the sequence (ri) is periodic with period n and r0⋯rn−1=1. Does it then always … Read more

Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let n≥1 be an integer, let F(x,y)=∫∞0un(x+y)(Kx−y(u))ndu for x,y≥0. When n=1, this is just Mellin transform of the Bessel K function. When n=2, F(x,y) has an explicit form in product of Gamma functions, given by the Parseval formula for Mellin transform. For general n, I expect some Stirling formula type estimation for F(x,y). I tried … Read more

continuous linear recurrent relations

For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation $$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$ for some constant real coefficients $a_{0},a_1,\dots,a_{n-1}$. Assume also that $f$ is continuous. What are conditions for polynomial $t^n+a_{n-1}t^{n-1}+\dots +a_0$ which allow to conclude that $f$ is linear? Answer AttributionSource : Link , Question Author : Fedor … Read more

Complex L^1 spaces; reference request

I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. And it is not the transform itself that is my interest but rather specific functions under the transform. I … Read more

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let f be analytic and consider the one-dimensional (the multidimensional case is more intricate, but can be done in a similar fashion³) differential equation ˙x=f(x), x(0)=x0. By defining the variables ϕk=xk, k=1,2,… … Read more

Fefferman’s article: Pointwise convergence of Fourier series, II

I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917 I got stuck in Chapter 6, Lemma 5. In the proof he split the P′ into three subcollections P′k, Pk″, \mathcal P”’_k. The first and third subcollections were estimated. However, for the second, he imposed a new assumption: \varphi_k(\xi_0′)=0, where \xi’_0 is … Read more

Properties of solution to Schrödinger equation

Given a Schrödinger equation with, let’s say continuous, periodic potential −y″ where V(x+1)=V(x) and V is even, i.e. for x \in (0,\frac{1}{2}) we have V(x+\frac{1}{2})=V(\frac{1}{2}-x) then it seems that the two solutions y_1,y_2 with initial conditions y_1(\lambda,0)=1, y_1′(\lambda,0)=0 and y_2(\lambda,0)=0,y_2′(\lambda,0)=1 have the property that the function f(\lambda):=y_1(\lambda,1)+\alpha y_2(\lambda,1), where \alpha \in \mathbb{R}, has non-vanishing derivative … Read more

Origins of the generalized shift operator exp(t*g(z)d/dz)

Charles Graves in the 1850s investigated iterated operators of the form g(x)ddx (see page 13 in The Theory of Linear Operators … (Principia Press, 1936) by Harold T. Davis). Graves published “A generalization of the symbolic statement of Taylor’s theorem” in the Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853) (when Sophus Lie was … Read more

Analytically continuing Euler’s partition function

Author’s note: This question might be a little hopeless, but maybe someone has some form of good feedback. It’s a long one because I tried to be very thorough. I tried to explained all the odds and ends I had at approaching this. It is a preliminary fact when studying modern analytic number theory that … Read more