Convergence Of Continued Fractions Research Articles

Continued fractions of operator-valued analytic functions

The subject of this work is the convergence of infinite continued fractions whose coefficients are analytic functions which take their values in the space of bounded linear operators on a complex Hilbert space. By appropriately combining a prior result of Fair with Vitali's theorem, we show that convergence under the uniform operator topology occurs on certain angular regions of the complex plane whenever the operator coefficients are commutative, invertible, and satisfy certain conditions on their numerical ranges.

Noncommutative Continued Fractions

A number of theorems are proved concerning the convergence of continued fractions whose entries are linear operators on a Banach space. These theorems are analogues of some of the well-known results for ordinary continued fractions.

The convergence of continued fractions

The convergence of continued fractions

Continued Fraction Convergents as a Source of Fibonacci and Lucas Identities

Continued Fraction Convergents as a Source of Fibonacci and Lucas Identities

Numerical Evaluation of Continued Fractions

Numerical Evaluation of Continued Fractions

Note on a nonlinear eigenvalue problem

1. V. F. Cowling, Walter Leighton and W. J. Thron, Twin convergence regions for continued fractions, Bull. Amer. Math. Soc. 50 (1944), 351-357. 2. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. 3 (1952), 904-913. 3. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. 67 (1949), 368-380. 4. H. S. WVall, Some convergence problems for continued fractions, Amer. MvIath. Monthly 64 (1957), 95-103.

A Characterization of QF-3 Algebras

1. D. F. Dawson, Concerning convergence of continued fractions, Proc. Amer. Math. Soc. 11 (1960), 640-647. 2. , Continued fractions with absolutely convergent even or odd part, Canad. J. Math. 11(1959), 131-140. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. 47 (1940), 155-172. 4. H. S. Wall, Partially bounded continued fractions, Proc. Amer. Math. Soc. 7 (1956), 1090-1093.

Concerning convergence of continued fractions

with sequence of approximants {Fp} has property (A) means there exists a complex number v such that each of the sequences { F2p-1I and { F2p} contains an infinite subsequence convergent to v. Here, c denotes the complex number sequence { cp }I P and d denotes the complex number sequence { dp } p0. When we speak of F(c, d) convergent absolutely at least in the wider sense we shall mean that there exists a positive integer n such that the continued fraction

Quasi-barrelled locally convex spaces

1. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 904-913. 2. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. vol. 67 (1949) pp. 368-380. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 155-172. 4. H. S. Wall, Analytic theory of continued fractions, New York, D. Van Nostrand Company, Inc., 1948.

Convergence of continued fractions of Stieltjes type

One result presented here is an extension of a result mentioned in [3 ].1 In [3] it is stated that if { g2p-1 ( 1 g2p )converges absolutely and I g2p } (9 g2p-1 } )converges, then (1.1) converges if the series E I b2p-l | (the series E I b2p i) diverges. The extension of this result is that if { g2p-1 I ( g2p I )converges absolutely and { g12p ( g2p-1 I ) converges, then (1.1) converges if, and only if, b satisfies one of the following conditions:

Absolute convergence of continued fractions

wheref1 is a number, {a,, a2, a3, } is a sequence of nonzero numbers, and { b1, b2, b3, * } is a sequence of numbers. We obtain conditions necessary and sufficient for (1.1) to converge absolutely, and we indicate their relationship to older sufficient conditions. We find a new characterization of positive definite continued fractions, whose importance is emphasized by the fact (Theorem 4.2) that if (1.1) converges, then there is a positive definite continued fraction which is a contraction of (1.1). We also obtain new sufficient conditions for absolute convergence of positive definite continued fractions.

Convergence of continued fractions in parabolic domains

Convergence of continued fractions in parabolic domains

The convergence and values of periodic continued fractions

The convergence and values of periodic continued fractions

On the convergence of algebraic continued fractions whose coefficients have limiting values

On the convergence of algebraic continued fractions whose coefficients have limiting values

Errata: “On the convergence of continued fractions with complex elements” [Trans. Amer. Math. Soc. 2 (1901), no. 3, 215–233; 1500565

Errata: “On the convergence of continued fractions with complex elements” [Trans. Amer. Math. Soc. 2 (1901), no. 3, 215–233; 1500565

On the convergence of continued fractions with complex elements

Up to the present time few theorems of a general character for the conivergence of continued fractions with complex elements have been obtained, and these few are of very recent date. In the first section of this paper such theorems upon the subject as are known to the writer are brought together for the purpose of indicating the present state of our knowledge, and the scope of the paper is also explained. Some new criteria for convergence are then deduced in the succeeding sections. The results obtained are summed up in theorems 1-10, which may be read independently of the rest of the paper. The demonstration of these theorems is based upon certain equations, Nos. 3-8, 11, and 12, which seem to be new and of a fundamental character.