Banach Gelfand Triples and Rigged Hilbert SpacesHans G. Feichtinger (NuHAG, Faculty of Mathematics, Univ. Vienna) given at Palermo (16.09.19) id: 3672 length: 45min status: type: LINK-Presentation: https://nuhagphp.univie.ac.at/dateien/talks/3672_PalermoFei19A.pdf ABSTRACT: TeX: The purpose of this talk is to point out, that very often {\it Hilbert spaces $\Hilb$ are surrounded by a family of similar function spaces} of a similar nature. It also occurs quite often, that the intersection of all these spaces forms itself a topological vector space, let us write $\Scsp$ , often called a space of {\it test functions}, whose dual (the space of continuous linear functionals on this space of test functions, its elements are often called distributions or ultra-distributions) contains not only the Hilbert space but also all the ``surrounding'' Banach spaces. We write $\ScPsp$. The triple $(\ScGTr)$ is then called a Gelfand Triple. For the case of the Euclidean space on can start with $H = L^2(R^d)$ and consider the collection of all (polynomially) weighted $L^p$-spaces as well as the corresponding Sobolev space $H^s(R^d)$. Their intersection gives the classical Schwartz space $\Scsp(R^d)$ of rapidly decreasing functions on $R^d$, with the dual space $\ScPsp(R^d)$ of {\it tempered distributions}. The whole class of Besov-Triebel-Lizorkin spaces is contained between $\Scsp(R^d)$ and $\ScPsp(R^d)$. This is what makes up the ``theory of function spaces'' in the spirit of H.~Triebel. % \end{document} Here as well in many other situations one can describe the space of test functions naturally as the common domain of the powers/iterates of some unbounded (self-adjoint) operator on $\Hilb$. In the concrete case one might take the Schr\"odinger operator and describe all some of the relevant spaces, including the so-called Shubin classes, via Hermite expansions and the corresponding decay resp.\ growth of the corresponding coefficients. This is usually the way to prove a so-called {\it kernel theorem}, which allows to characterize the vector space of all continuous linear operators from $\Scsp$ into $\ScPsp$ by (distributional integral) kernels in $\ScPsp$ (over the product domain). As we know this concrete Banach Gelfand triple also has a nice description in terms of wavelet expansions and in particular using Gabor expansions. In fact, replacing the continuous wavelet transform (CWT) by the STFT (short-time Fourier transform) one ends up by looking at the more and more used family of {\it modulation spaces} (introduced in 1983), among them the now classical spaces $M^s_{p,q}(R^d)$, among the unweighted modulation spaces $M_{p,q}(R^d)$. By focussing on the simple {\bf Banach Gelfand Triple} % $\SOGTrRd$ $(S_0,L_2,S_0')(R^d)$, (with $S_0(R^d) = M^1(R^d) = M_{1,1}(R^d)$ and $S_0'(R^d) = M_{\infty,\infty}(R^d)$) we have a technically much more simple situation, which allows to explain a number of concepts using only basic concepts from functional analysis, including Banach spaces, bounded linear functionals and operators, and the $w^* $-topology. % \end{document} We will focus on a couple of important results that can be obtained in this way, in particular the role of the perhaps somewhat surprising existence of a kernel theorem (despite the obvious absence of any nuclear Frechet space). Within classical Fourier analysis the first such Banach Gelfand Triple was introduced in principle by N.~Wiener, when he used $A(T)$, the algebra of absolutely convergent Fourier series, sitting inside $\Hilb = L^2(T)$, and contained in the space $PM$ of pseudo-measures. The identification % PMTN does not exist of this BGTr with the {\it canonical} BGTr $(\ell^1,\ell^2,\ell^\infty)$ can be realized by extending the classical Fourier series expansion (e.g. using duality, or distributional Fourier coefficients). In some sense the discussed BGTr allows to transfer this situation to the setting of general LCA groups. |
