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International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 4, Pages 737-748
http://dx.doi.org/10.1155/S0161171293000924

An orthonormal system on the construction of the generalized cantor set

Mathematics Department, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt

Received 16 January 1992; Revised 22 June 1992

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a new complete orthonormal system of functions defined on the interval [0,1] and whose supports shrink to nothing. This system related to the construction of the Cantor ternary set. We defined the canonical map ξ and proved the equivalence between this system and the Walsh system. The generalized Cantor set with any dissection ratio is established and the constructed system is defined in the general case.