The products on the unit sphere and even-dimension spaces (Q1771383)

Algebras of generalized functions introduced new possibilities to use the product of generalized functions. However, the papers on some products of distributions which are applicable in solving or defining mathematical models in physics are worthy of attention. The results in this paper corresponds to the product \(f(r)\delta^{(k)}(r-1)\) on the unit sphere, where \(f\) is a differentiable function at \(x= 1\). These results have been applied to two concrete functions: \(f(r)= (r-1)^n\) and \(f(r)= \sin r\). In spaces of even dimensions, the author proves that \(\delta^2(x)= 0\) and \(\Delta^k(r^{2k-m}\ln r)\delta(x)= 0\) if \(2k >m\).