A new theory for Fourier analysis. IV: Basic multiplication and convolution on the dual space
Continuing his previous investigations [Parts I--III: J. Math. Anal. Appl. 168, No. 2, 342-350 (1992; Zbl 0776.42012); 173, No. 2, 419-429 (1993; Zbl 0776.42013); 175, No. 1, 257-267 (1993; Zbl 0776.42014)], the author defines and investigates multipliers in his test space \({\mathcal G}\) and its dual \({\mathcal G}'\), and the convolution on \({\mathcal G}'\times {\mathcal G}\). Space \({\mathcal G}\) was defined and investigated in Parts I, II and space \({\mathcal G}'\) in Part III. There are given some equivalent formulae for the convolution and there are proved its properties. There are also introduced and investigated elementary convolvers for \({\mathcal G}'\), i.e., \(f\in {\mathcal G}'\) such that \(F* \psi\in {\mathcal G}\), whenever \(\psi\in {\mathcal G}\) and the mapping \(\psi\to f* \psi\) is linear and continuous from \({\mathcal G}\) to \({\mathcal G}\).
scientific article; zbMATH DE number 562694