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Calculus research, capabilities and equations.
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Let a linear space. In general considerations we may assume that is a linear x, x x x x x x 20 CHAPTER! space over an arbitrary field with characteristic 0, but in fact in the following we shall restrict ourselves with linear spaces over the field C of the complex numbers. --+32. , i. e. a linear operator mapinto itself. In sect. 1 we have considered the space ~(L1) of the ping continuous functions on an interval if containing the zero point, and as an endomorphism in it we have taken the Volterra integration operator t.
A bilinear, commutative and associative operation *: xXx ~ £ is said to be a convolution of the linear operator L iff the relation x x. (1) is fulfilled for all x, Y E32. The bilinearity can be expressed in the following way: For all Xl' x 2 , Yl' Y2 E32 and for arbitrary constants al' az, Pl' pz the bilinearity rei a tion (2) (alx1 +a2x2) * (Pl Yl + P2Y2) = alPl(x1 * Yl)+alP2(x l *Y2) + a2Pl(x2 *Yl)+a2tJ2(x2 *Yz) is fulfilled. The commutativity and associativity can be expressed by the identities (3) and (4) Of course, we shall be interested in non-trivial convolutions only, i.
N-l. Thus we have taken e as the unit of the corresponding convolutional algebra. According to the Hamilton-Caley theorem, we have P(L)=O, and the powers of L with degrees ;;:;; n should be replaced by sums of linear combinations of lower degrees. Defined in such a way, the operation * is bilinear. commutative and associative. e. Lx=(Le)*x. Hence, the operation * is a convolution of L in Qn. We shall show that * is a convolution without divisors of zero in Qn. Indeed, let a=l=0' CONVOLUTIONS OF LINEAR OPERATORS.