Integral Transform and Generalized Sampling Theorem
Éi Iti Takizawa, Jenn-Lin Hwang · Hokkaido University Collection of Scholarly and Academic Papers (Hokkaido University) · 1972
A generalization of the sampling theorem is presented under the consideration of the reciprocity relation of integral transforms.New sampling formuiae, being based on the generalized sampling theorem presented here, are also given, one of which includes Shannon's sampling theorem as a special case.ttt Zusammenfassung Unter BerUcksichtigung der Reziprozitatsrelation der Integraltransformierten ist eine Verallgemeinerung des Abtast-theorems (sampling-theorems) angegeben.Auf Grund des hieran angegebenen verallgemeinerten Abtast-theorems sind neuen Abtast-formeln auch prasentiert, eine von denen das Shannon'sche Abtasttheorem als einen besonderen Fall enthtilt. gl. PreliminariesThe generalization of the sampling theorem and the reconstruction of a bandlimited function from its sampled values and derivatives were made by Kohlenbergi), Foge12), Jagerman and Foge13), Bond and Cahnˋ), and Linden and Abramson5).The sampling theorem was also generalized by Balakrishnan6) to the case of a continuous-parameter stochastic process.On the other hand, it was pointed out that the sampling intervals need not be uniformly distributed').In the previous paper8), the authors presented a generalized sampling theorem along the line of consideration of the reciprocity relation of integral transforms and gave some of its examples, which include Someya-Shannon's sampling theo-rem9}'-i') as a special case.There the sampling intervais were not uniformly distributed.Recently the authors were informed that the sampling theorem was also generalized by Isomitii2) in connection with the generalized frequency domain.Some of his results agreed with those given by the authors8).Here in the present paper, the authors wish to discuss the series-expansion of functions in an orthogonal set of functions, and also to give some of the new sampling formulae.gZ. Generalization of Sampling Theorem Let f