Neutrices and convolution products of distributions. (English) Zbl 0868.46029

Starting from some results of D. S. Jones [Q. J. Math., Oxf. II. Ser. 24, 145-163 (1973; Zbl 0256.46054)], the author, in a lot of papers [see for example Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 16, No. 2, 119-135 (1986; Zbl 0639.46041) and – together with L. C. Kuan – ibid. 23, No. 1, 13-27 (1993; Zbl 0821.46050)], have obtained extensions concerning the convolution of distributions. In this sense, in the present paper, these results are used to obtain a commutative neutrix convolution product of two distributions \(f,g\in{\mathcal D}'(\mathbb{R})\), \((f*g,\varphi)=\lim_{n\to \infty}(f_n*g_n,\varphi)\), \(\forall\varphi\in{\mathcal D}(\mathbb{R})\) (if the neutrix limit exists) with \(f_n(x)= f(x)k_n(x)\), \(g_n(x)=g(x)k_n(x)\) and \(k_n(x)=\begin{cases} 1, & x\geq -n\\ k(n^nx+ n^{n+1}), & x<-n\end{cases}\) where \(k\) is an infinitely differentiable function satisfying \(0\leq k(x)\leq 1\), \(k(x)=1\) for \(-{1\over 2}\leq x<\infty\) and \(k(x)=0\) for \(x\leq -1\). A lot of examples are considered in the sense of this definition.


46F10 Operations with distributions and generalized functions
44A35 Convolution as an integral transform