Let \(L^ 2\) denote the Banach algebra of Lebesgue-integrable functions \(f: [0,\omega)\to {\mathbb{C}}\) with convolution product ''*'', \(\Omega\) a nonfinite compact subset of the right half-plane Re s\(>0\) and the space of generalized functions \(X_{\Omega}\) the completion of \(L^ 1\) with respect to the norm \[ | x|_{\Omega}=\max_{s\in \Omega}| \hat x(s)|,\quad \hat x(s):=\int^{\infty}_{0}e^{-st}x(t)dt. \] The space \(W_{\Omega}\) of all locally integrable functions, where \(\hat w(\)s) converges on some open half-plane containing \(\Omega\), can be embedded in \(X_{\Omega}\). The author proves the theorem: Let \(f(s)=\sum^{\infty}_{n=0}\alpha_ ns^ n\) be any power series whose open disc of convergence contains \(\Omega\). Then \(f| \Omega\) is the Laplace transform of a generalized function a, and for all \(x\in X_{\Omega}\), \(x*a=\alpha_ 0x+\alpha_ 1x^{[1]}+\alpha_ 2x^{[2]}+...(x^{[k]}\) is the kth order generalized derivative). If \(a\in W_{\Omega}\) and \(x\in C^{(\infty)}_{0,\Omega}:=\{x\in C^{\infty}[0,\infty)| x^{(k)}(0)=0\), \(x^{(k)}\in W_{\Omega}\), \(k=0,1,2,...\}\) then \(\int^{t}_{0}x(u)a(t-u)du=\alpha_ 0x(t):\alpha_ 1x'(t)+\alpha_ 2x''(t)+....\) He gives some examples concerning the Euler-Maclaurin sum and the Poisson summation formula.