Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers

摘要

The classical sampling theorem has often been attributed to E.T. Whittaker, but this attribution is not strictly valid. One must carefully distinguish, for example, between the concepts of sampling and of interpolation, and we find that Whittaker worked in interpolation theory, not sampling theory. Again, it has been said that K. Ogura was the first to give a properly rigorous proof of the sampling theorem. We find that he only indicated where the method of proof could be found; we identify what is, in all probability, the proof he had in mind. Ogura states his sampling theorem as a “converse of Whittaker’s theorem”, but identifies an error in Whittaker’s work. In order to study these matters in detail we find it necessary to make a complete review of the famous 1915 paper of E.T. Whittaker, and two not so well known papers of Ogura dating from 1920. Since the life and work of Ogura is practically unknown outside Japan, and there he is usually regarded only as an educationalist, we present a detailed overview together with a list of some 70 papers of his which we had to compile. K. Ogura is presented in the setting of mathematics in Japan of the early 20th century. Finally, because many engineering textbooks refer to Whittaker as a source for the sampling theorem, we make a very brief review of some early introductions of sampling methods in the engineering context, mentioning H. Nyquist, K. Küpfmüller, V. Kotel’nikov, H. Raabe, C.E. Shannon and I. Someya.