The parabola theorem for continued fractions over a vector space

Author:
F. A. Roach

Journal:
Proc. Amer. Math. Soc. **28** (1971), 137-146

MSC:
Primary 40.12

DOI:
https://doi.org/10.1090/S0002-9939-1971-0275004-4

MathSciNet review:
0275004

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Abstract: In a recent paper, we defined a type of reciprocal for points of a real inner product space and considered continued fractions based on this reciprocal. These continued fractions were analogous to ordinary continued fractions in which each partial numerator is unity. In the present paper, we develop a type of continued fraction which is analogous to an ordinary continued fraction of the form in which each partial denominator is unity. The main result is a convergence theorem for such continued fractions which is a direct extension of a theorem by W. T. Scott and H. S. Wall (the Parabola Theorem).

**[1]**J. Findlay Paydon and H. S. Wall,*The continued fraction as a sequence of linear transformations*, Duke Math. J.**9**(1942), 360–372. MR**0006386****[2]**F. A. Roach,*Continued fractions over an inner product space*, Proc. Amer. Math. Soc.**24**(1970), 576–582. MR**0412654**, https://doi.org/10.1090/S0002-9939-1970-0412654-5**[3]**W. T. Scott and H. S. Wall,*A convergence theorem for continued fractions*, Trans. Amer. Math. Soc.**47**(1940), 155–172. MR**0001320**, https://doi.org/10.1090/S0002-9947-1940-0001320-1

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0275004-4

Keywords:
Continued fractions

Article copyright:
© Copyright 1971
American Mathematical Society