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

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

Anti-delta function?

Did anyone ever consider a “function” or “distribution” $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\infty F(x)\,dx=1$? It definitely can be seen as a limit of smooth functions, and its Fourier transform will be a finction, equal to $1$ at $0$ and otherwise $0$. I think such … Read more