On the Dini calculus. II
In an earlier note (Part I) in this Journal 118, 255-262 (1986; Zbl 0606.42012), we introduced the notion of the Dini calculus and developed it completely in the 1-dimensional case. At the end of that article, we offered a conjecture on how things should go in general. Our purpose here will be to confirm that conjecture. In essence, the main theme is the abstract Dini theorem. It is a theorem of existence, thus it is qualitative in nature. We have also included a vanishing theorem that provides the justification for a claim made elsewhere by us [Pac. J. Math. 140, No.2, 263-352 (1989; Zbl 0697.22021), Section 10]. While of independent interest, the main immediate application of these results is to the derivation of the Selberg trace formula [the authors, Trans. Am. Math. Soc. 324, No.2, 623-653 (1991)].
scientific article; zbMATH DE number 4177839