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Optimal Error Bounds for Convergents of a Family of Continued Fractions

โœ Scribed by Yair Shapira; Avram Sidi; Moshe Israeli

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
134 KB
Volume
197
Category
Article
ISSN
0022-247X

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โœฆ Synopsis

Let F F be the family of continued fractions K a r1 , where a s yg , a s

g x , ps2, 3, . . . , with 0 F g F 1, g fixed, and x F 1, p s py 1 p p p p p 2, 3, . . . . In this work, we derive upper bounds on the errors in the convergents of ลฝ . K a r1 that are uniform for F F, and optimal in the sense that they are attained by p some continued fraction in F F. For the special case g s g -1r2, i s 1, 2, . . . , this i bound turns out to be especially simple, and for g s g s 1r2, i s 1, 2, . . . , the i known best form of the theorem of Worpitzki is obtained as an immediate corollary.

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