Localization of spectral expansions of distributionsAbdumalik Rakhimov Depaertment of Applied Mathematics of Tashkent Divition of Moscow University UZBEKISTAN given at strobl07 (22.06.07 11:30) id: 541 length: 25min status: accepted type: talk LINK-Presentation: http://univie.ac.at/nuhag-php/dateien/talks/541_Rakhimov's talk abstract.pdf ABSTRACT: TeX: \textbf{1. Introduction.} Classical analysis deals with the smooth or piecewise smooth functions. But many phenomena in nature require for its description either "bad" functions or even they can not be described by regular functions. Therefore, one has to deal with distributions that describe only integral characteristics of phenomena. Application of the modern methods of mathematical physics in the spaces of distributions, leads to the convergence and sumability problems of spectral expansions of distributions. The convergence and sumability problems of spectral expansions of distributions, associated with partial differential operators, connected with the development of mathematical tools for modern physics. Especially simple and important example is Fourier series of Dirac's delta function, partial sum of which is well known Dirichlet's kernel. From the classic theory of trigonometric series it is known that Dirichlet's kernel is not uniformly approximation of delta function. So spectral expansions of Dirac's delta function is not convergent in any compact set out of the support of the distribution. But arithmetic means of the partial sum of Fourier series of Dirac's delta function coincides with Fejer's kernel and in one dimensional case it uniformly convergent to zero in any compact set where delta function is equal to zero. In multidimensional case the problem become more complicated. \textbf{2. Spectral expansions connected with partial differential operators.} Let \quad $\Omega$ \quad - an arbitrary \quad $N$ \quad- dimensional domain. Consider a differential, elliptic, half bounded and symmetric operator \quad $ A(x,D) = \sum_{\alpha\leq2m} a_{\alpha}(x)\cdot D^{\alpha} $ \quad in Hilbert's space \quad $L_{2}(\Omega)$ \quad with domain of definition of \quad $C_{0}^{\infty}(\Omega)$ , \quad here \quad $\alpha$ \quad is \quad $N$ \quad - dimensional vector with non negative integer coordinates \quad $\alpha=(\alpha_{1}, \alpha_{2},...., \alpha_{N})$, \quad $| \alpha | = \alpha_{1} + \alpha_{2} +....+ \alpha_{N}$ , \quad $D_{j}=\frac{1}{i}\frac{\partial}{\partial x_{j}}$ \quad and \quad $D^{\alpha}=D^{\alpha_{1}} \cdot D^{\alpha_{2}}\cdot\cdot\cdot\cdot\cdot\cdot D^{\alpha_{N}}$ . \quad Let \quad $\hat{A}$ \quad some selfadjoint extension of this operator in \quad $L_{2}(\Omega)$ \quad and \quad $\{E_{\lambda} \}$ \quad corresponding spectral family of projections. The projections \quad $\{E_{\lambda} \}$ \quad are integral operators with the kernels \quad $\Theta(x, y, \lambda)$ : \begin{equation} E_{\lambda}f (x) = \int_{\Omega} f(y)\Theta(x, y, \lambda) dy, \quad \quad f \in L_{2}(\Omega). \end{equation} Function \quad $\Theta(x, y, \lambda)$ \quad is called spectral function of operator \quad $\hat{A}$, \quad and integral (1) is called spectral expansions of \quad $f$ \quad corresponding to operator \quad $\hat{A}$. \textbf{3. Problems of summability and localization of spectral expansion.} \quad One can study convergence and summability problems of spectral expansions of distributions in classical means in the domain where they coincide with regular functions. But singularities of the distribution still will be essential for convergence problems even at regular points as it was mentioned above in case of Delta function. For spectral expansions one can apply Riesz's method of sumability or other regular methods (for instant Chezaro method). Reisz's means of order \quad $s \geq 0$, \quad of spectral expansion \quad $E_{\lambda}f (x)$ \quad define by equality: \begin{equation} E_{\lambda}^{s}f (x) = \int_{\mu}^{\lambda}\big{(}1-\frac{t}{\lambda}\big{)}^{s}dE_{t}f (x). \end{equation} We study summability problems for spectral expansions in different topologies and in different functional spaces. In particular we consider the problems of localization of spectral expansions. The problem of localization can be formulated in following way: \emph{Let \quad $f$ \quad an infinite differentialable function in neighborhood of a point \quad $x_{0}.$ \quad What is the influence of smoothness (or non smoothness) of function \quad $f$ \quad in some other points for the convergence of \quad $E_{\lambda}^{s}f (x)$ \quad in small neighborhood of the point \quad $x_{0}.$} \textbf{4. Main results.} Main results of the present work are obtaining sharp conditions for summability and localization of spectral expansions connected with elliptic partial differential operators in different spaces of distributions. It is obtained sharp relation between order of summation and smoothness of the distributions in Hilbert spaces. In case of Banach spaces sharp results established for multiple Fourier series and expansions connected with Laplace-Beltrami operator on sphere. \textbf{5. Some references.} Below it is given a list of some papers devoted investigations of these problems. |