# Generalization of the Notion of Convolution

Chapter

## Abstract

In Chapter 11 we considered
where (
where (

*H*-convolutions of generalized*H*-transforms. These convolutions are bilinear, commutative and associative operations in the linear space and possess the following property$$
\left( {H\left( {f\mathop {*g}\limits^a } \right)} \right)\left( x \right) = x^a \left( {Hf} \right)\left( x \right)\left( {Hg} \right)\left( x \right)
$$

(1)

*Hf*)(*x*) is the generalized*H*-transform (4.1). In S.B. Yakubovich (1990), S.B. Yakubovich and Nguyen Thanh Hai (1991), Nguyen Thanh Hai and S.B. Yakubovich (1992), the generalization of the*H*-convolution was introduced and investigated. This generalized convolution is not commutative and associative operation in general case and, consequently, it is not convolution in the usual sense. The main property of the generalized convolution * is the following$$
\left( {H_1 \left( {f*g} \right)} \right)\left( x \right) = \sqrt x \left( {H_2 f} \right)\left( x \right)\left( {H_3 g} \right)\left( x \right),
$$

(2)

*H*_{ i }*f*)(*x*),*i*= 1, 2, 3 are some generalized*H*-transforms. If (*H*_{ i }*f*)(*x*) ≡ (*Hf*)(*x*),*i*= 1, 2, 3 and*a*= 1/2, then the generalized convolution coincides with the considered*H*-convolution (11.5). As it was showed by S.B. Yakubovich and Yu.F. Luchko (1991a), (1991b), (1991c), Nguyen Thanh Hai and S.B. Yakubovich (1992), H.M. Srivastava et al. (1993) the generalized convolution has some important applications, in particular, new Leibniz type rules and theirs integral analogues which we will consider in Chapter 13.## Keywords

Integral Equation Asymptotic Behaviour Special Function Linear Space Important Application
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media Dordrecht 1994