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Term by Term Dyadic Differentiation

Published online by Cambridge University Press:  20 November 2018

Charles H. Powell  and
William R. Wade
Charles H. Powell
Affiliation:
The University of Tennessee, Knoxville, Tennessee
William R. Wade
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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Let ψ0, ψ1, … denote the Walsh-Paley functions and let ∔ denote the group operation which Fine [5] defined on the interval [0, 1). Thus, if k ≧ 0 is an integer and if u, t are points in the interval [0, 1) then

(where αk = 0 or 1 represents the kth coefficient of the binary expansion of t), and

A real-valued function ƒ, is said to be dyadically differentiable at a point x ∈ [0, 1) if ƒ is defined at x and at x ∔ 2–n–1, n = 0, 1, …;, and if the sequence

(1)

converges as N → ∞.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 247 - 256
Copyright
Copyright © Canadian Mathematical Society 1981

References

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