## Hillam's Theorem

If (where denotes the Closed Interval from to on the Real Line) satisfies a Lipschitz Condition with constant , i.e., if

for all , then the iteration scheme

where , converges to a Fixed Point of .

References

Falkowski, B.-J. On the Convergence of Hillam's Iteration Scheme.'' Math. Mag. 69, 299-303, 1996.

Geist, R.; Reynolds, R.; and Suggs, D. A Markovian Framework for Digital Halftoning.'' ACM Trans. Graphics 12, 136-159, 1993.

Hillam, B. P. A Generalization of Krasnoselski's Theorem on the Real Line.'' Math. Mag. 48, 167-168, 1975.

Krasnoselski, M. A. Two Remarks on the Method of Successive Approximations.'' Uspehi Math. Nauk (N. S.) 10, 123-127, 1955.