Convolution Equations on a Finite Interval

Let k ∈ L₁r × r (−a, a), where a is a given positive real number and r is a positive integer. Then for any real number p ≥ 1 the formula 0.1$$\begin{array}{*{20}{c}} {(Kf)(t) = \int_0^a {k(t - s)f(s)ds} }&{(0 \leqslant t \leqslant a)} \end{array}$$ defines an operator on L _p r × r and on L _p r × r (0,a) and on L _p r (0, a). Here L _p r (0, a) denotes the Banach space of all functions f =(f₁f₂...f_r)T on (0, a ) with f₁,...,f_r in L _p (0, a ) and with $${\left\| f \right\|_{L_p^r(0,a)}} = \sum\limits_{j = 1}^r {{{\left\| {{f_j}} \right\|}_{{L_p}(0,a)}}}$$ We will abbreviate this norm to ‖f ‖ p. L _p r × r is defined similarly. An operator in the form of (0.1) we will refer to simply as a convolution operator on a finite interval.