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On discrete analog of singular integral operators in terms of D-calculus and its application
by
S.A. Azizov
Baku State University, Baku, 370069 Azerbaijan, Rasim Mukhtar str. 10
In the given work, in terms of many-dimensional D -calculus is áonstructed discrete analog of singular integral operators such as A.P.Calderon and A.Zygmund [1, 2]. It is shown, that the discrete singular integrals can be applied for study properties of a solutions of discrete analog of the Poisson equation.
D -calculus is a finite-difference calculus, constructed because of specified conditions and expressed in terms of continuous analysis. One-dimensional D -calculus for a unit grid is considered by the author [3]. In this work, for preparation of theoretical base for construction of discrete analogs of singular integrals, were accepted conditions, which generalize one-dimensional D-calculus for many-dimensional case, when the grid on everyone variable has an arbitrary pitch.
Let's accept the following. Let N be the set of positive, Z
the set of integers, n Î N, Rn-the n-dimensional Euclidean
space with the standard basis (e1, ..., en),
hk=(h1k1, ..., hnkn ) Î (0, ¥)n,
k=(k1, ..., kn) Î Zn,
Zhn:={xk; xk=(x1k1, ..., xnkn ) Î Rn, \xiki=åj=0ki-1hij, xi(-ki )=-xiki, \i=1, ..., n}. For j: Zhn ® R we put
Dij(xk): = (j(xk+hikiei) - j(xk))/hiki . Let in the set F:={j: Zhn® R}
is defined usual operation of addition (+) and operation
multiplication (°) satisfying to a condition
| (1) |
Based on these conditions is constructed many-dimensional D -calculus
of the
grid functions. Some elements of the D -calculus are reduced below.
Property (1 ) for power functions xk[m]:=(x1k1[m], ..., xnkn[m])
is reduced to the formula
Dixiki[m]=mxiki[m-1] , because of that the
essence of the power functions is uncovered. For m ³ 0 we have:
xiki[m]=m!Tikim, where
Tikim = åj1=0ki-1åj2=j1+1ki-1¼åjm=jm-1+1ki-1hij1hij2¼hijm, Tiki0=1 and Ti0m=0
for m ¹ 0. \
xiki[-m-1]=mikim/Õj=0mxi(ki+j), where mikim is some performance of a grid and for grids with a constant
pitch this parameter is equal to unit. For nonuniform grids we have:
mikim+1=(mikimxi(ki+m)-mi(ki+1)mxiki)/mhiki, miki1=1.
The many-dimensional definite integral of D-calculus is defined as the
integral sum used in the analysis for definition appropriate continuous
definite integrals.
On Zh n this integral will be look so:
ò Zhn j(xk) Dxk º åk Î Znj(xk)hk.
Let's enter norm in F as follows: ||·||=(ò Zhn|·|[p] Dxk)[1/p]. Let's designate the obtained space as lp( Zhn, F) .
Let's consider a discrete singular integral operator I of the following
aspect:
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Theorem. If j(·) Î lp( Zhn, F), where 1 < p < ¥, then Ij(·) Î lp( Zhn, F) and ||Ij||lp( Zhn) £ C||j||lp( Zhn), where C depends only on p and dimension n .
As an application it is possible to show a solution of discrete analog of the Poisson equation: åi=0nDii2u(xk)=j(xk). It is easily checked, that P (xk) is a fundamental solution of the equation åi=0nDii2u(xk)=0. Then a solution of discrete analog of the Poisson equations will be: u(xk)=ò Zhn P(xk-yn)°j(yn) Dyn. The second order mixed partial differences of this solution reduces to singular integral operator I . Then the reduced theorem is some a priori estimation of a solution of discrete analog of the Poisson equation.
References
1. A.P.Calderon and Zygmund, On the existence of certain singular
integrals, Acta Math. 88 (1952), 85-139.
2.A.P.Calderon and Zygmund, On singular integrals, Amer. J. Math.
78 (1956), 289-309.
3.S.A.Azizov, Axiomatic construction of the one-dimensional discrete
analysis , Baku, 1999, 229 p. ( Russian )
Date received: June 11, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahv-00.