This is the abstract of the paper "Convolution Kernels and Generalized Functions" by B. Baeumer, G. Lumer, and F. Neubrander . For the whole paper send the command get evolve-l 98-00007 to: "listserv@rz.uni-karlsruhe.de". Instead you may get the paper at "http://ma1serv.mathematik.uni-karlsruhe.de/evolve-l/index.html" too. Abstract: The operation of convolution $k\star f: t\mapsto \int_{-\un}^{\un} k(t-s)f(s)\,ds$ is basic in classical analysis. It generalizes derivatives $\de^{(n)}\star f$, antiderivatives $ {t^{n}\over n!}\chi_{[0,\un)}\star f$ and arises when two Laplace or Fourier integrals are multiplied together. The linearity and time invariance of the convolution operator (i.e., convolution commutes with translation) puts it in the center of signal processing applications and the theory of Markov processes. Moreover, convolution defines a zero divisor free multiplication of functions with appropriate support (see below). Thus, the totality of complex valued functions with support in some interval $[a,\un)$ extends to a field of generalized functions. This field provides a mathematically clear and technically simple basis for an ``operational calculus'' applicable to linear ordinary differential equations, difference equations, and integral equations. Concerning the convolution transform, two results are of particular importance. One is E.~C.~Titchmarsh's ``Injectivity Theorem'', the other is C. Foia\c{s}' ``Dense Range Theorem''. The aim of this paper is to extend these results to (a) Banach space valued functions $f$, (b) defined on intervals or point inflection invariant bounded open sets $\Om\sb \RR^{n}$, and (c) scalar or semigroup valued kernels $K$. For Banach space valued functions $f$ on real intervals we give outlines of two proofs of Titchmarsh's and Foia\c{s}' results: an elegant proof by contradiction following the original proofs in the numerical case and a new, constructive proof yielding a sequence $f_{n}$ of smooth functions ``converging'' to the generalized solution $f$ of the convolution equation $k\star f=g$ (for given $k,g$). We extend Foia\c{s}' result to functions defined on regions $\Om\sb \RR^{n}$ that are bounded and symmetric to a point; i.e., we show that the injectivity of the convolution transform as a mapping from $C(\Om;X)\to C(\Om;X)$ implies that the closure of the image contains $C_{0}(\Om;X)$. As example of an operator valued kernel we consider strongly continuous semigroups $K$ and show that the convolution operator $Tf:=\intt K(t-s)f(s)\,ds$ is injective and has dense range in $C_{0}([0,a];X)$.