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Localization of spectral expansions of distributions


  Abdumalik 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.



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