Abstract
We show that if a function bounded in the unit disk gives rise to a limit periodic Schur algorithm with |γ|=1, Σd n ,<∞ then the function can be continued analytically to a meromorphic function in the entire plane except, possibly, for an essential singularity at z=−1. The lpS provides a very good approximation to the function.
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J. Schur, Über Potenzreihen die im Inneren des Einheitskreises beschränkt sind, J. reine u. angewandte Math. 147 (1917) 205–232; 148 (1918/19) 122–145.
W.J. Thron, Some results on separate convergence of continued fractions,Computational Methods and Function Theory, eds. St. Ruscheweyh, E.B. Saff, L.C. Salinas and R.S. Varga, Lecture Notes in Mathematics 1435 (Springer, Berlin/Heidelberg) pp. 191–200.
W.J. Thron, Truncation error for limit periodic Schur algorithms, SIAM J. Math. Anal.
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This research was supported in part by the US National Science Foundation under Grant DMS-9103141.
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Thron, W.J. Limit periodic Schur algorithms, the case 441-1441-1441-1. Numer Algor 3, 441–450 (1992). https://doi.org/10.1007/BF02141951
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DOI: https://doi.org/10.1007/BF02141951