On the causal and anticausal n-dimensional convolution equation related to the Diamond kernel of Marcel Riesz

TRIONE, SUSANA ELENA

INTERNATIONAL JOURNAL OF MATHEMATICS AND MATHEMATICAL SCIENCES

Lugar: new york eeuu; Año: 2006 p. 1 - 8

0161-1712

{\leftskip=1,3cm \rightskip=1,3cm\noindent \ssmc ABSTRACT.~\rm In this Note we extend the Theorem 4.1 ([3], p. 37) due to A. Kananthai which says that ``Given the lineardifferential equation of the form$$(e^{\alpha t}\diam^k\del)* u(t)=L^ku(t)=\del; \eqno\rm(I,1)$$then$$u(t)=e^{\alpha t}(-1)^k S_{2k}*R_{2k}(t), \eqno\rm (I,2)$$is an elementary solution of (I,1) or, equivalently, of the Diamond kernel of Marcel Riesz of (I,1), where $S_{2k}(t)$ and$R_{2k}(t)$ are defined, respectively by (2,1) and (2,3) of  [1] with $\gam=2k$. \noi Our main result is the Theorem V.1, formula (V,3) which expresses that: ``Given the linear partial differentialequation of the form$$ (e^{\alpha t}\diam^k\del)*(P\pm i0)^{\alpha-n\over 2}=L^k(P\pm i0)^{\alpha-n\over 2}=\del.\eqno\rm (I,3)$$Here $L$ is the partial differential operator of Diamond type defined by (IV,2). Then$$(P\pm i0)^{2k-n\over 2}=e^{\alpha t}(-1)^k \cdot S_{2k}(P'\pm  i0)*R_{2k}(P\pm i0),\eqno\rm (I,4)$$is an elementary solution of (I,3) where $S_{2k}(P'\pm i0)$ and $R_{2k}(P\pm i0)$ are defined by (II,10) and (II,7),respectively.\par } \newdimen\normalbaselineskip\normalbaselineskip=16pt\normalbaselines