How to Extend A Calculus

Abstract

Fifty years ago while writing my doctor’s thesis 1 was already strongly under the influence of Landau although I had never met him as yet. The fact was that in his correspondence he gave as well sharp criticism as warm encouragement which altered the contents considerably. Especially the period, in which I was fortunately enough to work with him at Göttingen, contributed beyond measure to my mathematical development. The present paper would perhaps never have been written without him, even if it diverges to a great extent from the method followed always by him in thought and intent. He preferred to devote his attention to special problems which were characteristic for extensive domains. Whenever as a pioneer he had found a solution for these problems, it was made easy for his followers to generalize the results. The opposite way is taken in this article in which the general question is treated whether and in which measure a given calculus can be extended. In view of the extensiveness of the subject 1 must restrict myself to some indication. The method developed herewith can be applied in many diversified branches, a.o. in the analytic theory of numbers and for this reason it may find a place in a volume dedicated to Edmund Landau. Had this publication appeared fourty years earlier he certainly would have read, accepted and applied it.