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

Covergence of fractional Taylor series

Let f(x) be a function that is continuous and infinitely smooth on entire R. Let’s consider Taylor-Maclaurin series for this function: f(x)=∞∑0fn(x0)(x−x0)nn! Where fn(x) is n-th derivative of f(x), and fn(0) is correspondingly the value of n-th derivative f(x) at 0. We can create continuous generalization of this series based on integral: g(x)=∫∞0Dtf(x0)Γ(1+t)(x−x0)tdt Where Dt … Read more

The space of periodizable tempered distribution

The periodization operator Per is defined for a Schwartz function φ∈S(R) as Per{φ}(x)=∑n∈Zφ(x−n),∀x∈R. The sum in (1) is of course well-defined pointwise due to the rapid decay of φ and we then have that Per{φ} is an infinitely smooth 1-periodic function. More generally, it is possible to define the periodization operator Per over rapidly decaying … Read more

Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with L∞ coefficient and distributional right hand side: ΔΔw+u(x,y)(α2∂w∂t+β2w)+γ2∂2w∂t2=P[θ(t)−θ(t−τ)]δ(x−x0(t))δ(y−y0(t)),  (x,y,t)∈(−l1,l1)×(−l2,l2)×(0,T), subject to w=∂2w∂x2=0,  x=−l1;l1,  for all (y,t)∈[−l2,l2]×[0,T], w=∂2w∂y2=0,  y=−l2;l2,  for all (x,t)∈[−l1,l1]×[0,T], and w(x,y,0)=w0(x,y),  ∂w∂t|t=0=w10(x,y),  for all (x,y)∈[−l1,l1]×[−l2,l2]. Here Δ is the Laplacian, θ(t) is the Heaviside`s function, δ(t) is that of Dirac, u∈L∞[−l1,l1]×[−l2,l2], x0, y0 are continuous, w0 and w10 are, at least, piecewise continuous and in consistency with boundary conditions, … Read more

Colombeau generalized functions

I’m currently reading some aspects of Colombeau generalized functions, and in almost all of his examples he discuss aspects of Quantum Field Theory, but then I go to some “standard” texts on QFT and I can not find any information (Ziedler only point out some similarities with Hörmander wave front sets; I can’t find anything … Read more

Generalized functions on a product of two manifolds

Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map $$T\colon C^{-\infty}(X\times Y)\to Bil(C^\infty(X),C^\infty(Y)),$$ where the target is the space of continuous bilinear functionals $C^\infty(X)\times C^\infty(Y)\to \mathbb{C}$. By definition $(T\Phi)(f,g)=\Phi(f\otimes g)$. It is well known that $T$ … Read more

The abstract kernel theorem implies Schwartz kernel theorem. How exactly?

Let me first give a little rapid background prior to formulating the question. Let D be a Schwartz space of infinitely differentiable functions and D′ is the space of distributions acting on D. The Schwartz Kernel Theorem states that for any linear continuous operator A:D→D′ there exists a distribution K∈[D×D]′ such that (Af,φ)=K(f⊗φ), where f⊗g … Read more

What are all the stationary and pointwise independent random processes?

In the 60’s, I. Gel’fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process Φ, he defines the concepts of stationarity (Φ(φ) and Φ(φ(⋅−t0)) have the same law) and of independence at every point (the random variable Φ(φ1) and Φ(φ2) are independent … Read more

Sequential continuity of linear operators

Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces. Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is formally weaker that the usual topological continuity in the case of non-metrizable spaces.) Let $L_0\subset L$ be a topologically dense linear subspace. Assume that … Read more

Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of functional analysis Answer While I do not know much about current development of the general theory of distributions, I can say … Read more