Generalized Limits in General Analysis.
نویسنده
چکیده
In a previous paper of the same titlet I have developed the fundamental principles of a general theory which includes as particular instances the theories of Cesàro and Holder summability of divergent series and divergent integrals. I further made use of these fundamental principles to prove a general theorem which includes as special cases several important theorems in the above mentioned special theories. In the present paper the general theory referred to above is extended to the case of multiple limits and the theorem mentioned is likewise generalized. The theorem thus obtained includes as special cases the extension to multiple series of the Knopp-Schnee-Ford theorem^ on the equivalence of Cesàro and Holder summability for divergent series, the extension to multiple integrals of the analogous theorem of Landauf for the case of divergent integrals, and the extension to partial derivatives of a corresponding theorem with regard to the equivalence of certain generalized derivatives. Once the principles of the theory are set forth, the proof of this general theorem is fully as simple as the proofs of any of the special theorems would be. Thus we have exhibited the greater power of the methods of General Analysis as compared with the methods of classical analysis. The basis of our general theory may be indicated as follows :
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عنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 8 10 شماره
صفحات -
تاریخ انتشار 2010