Successive remainders of the Newton series
Authors:
G. W. Crofts and J. K. Shaw
Journal:
Trans. Amer. Math. Soc. 181 (1973), 369383
MSC:
Primary 30A08
MathSciNet review:
0320286
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Abstract: If is analytic in the open unit disc and is a sequence of points in converging to 0, then admits the Newton series expansion , where is the th divided difference of with respect to the sequence . The Newton series reduces to the Maclaurin series in case . The present paper investigates relationships between the behavior of zeros of the normalized remainders of the Newton series and zeros of the normalized remainders of the Maclaurin series for . Let be the supremum of numbers such that if is analytic in and each of , has a zero in , then . The corresponding constant for the Maclaurin series ( , where ) is called the Whittaker constant for remainders and is denoted by . We prove that , for all , and, moreover, if . In obtaining this result, we prove that functions analytic in have expansions of the form , where , for all , and is a polynomial of degree determined by the conditions .
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 J. D. Buckholtz and J. L. Frank, Whittaker constants, Proc. London Math. Soc. 23 (1971), 348370. MR 0296297 (45:5358)
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 J. D. Buckholtz and J. K. Shaw, Zeros of partial sums and remainders of power series, Trans. Amer. Math. Soc. 166 (1972), 269284. MR 0299762 (45:8810)
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 P. J. Davis, Interpolation and approximation, Blaisdell, New York, 1963. MR 28 #393. MR 0157156 (28:393)
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 M. M. Dragilev, On the convergence of the AbelGončarov interpolation series, Uspehi Mat. Nauk 15 (1960), no. 3 (93), 151155. (Russian) MR 22 #4839. MR 0114009 (22:4839)
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 E. Hille, Analytic function theory. Vol. 1, Introduction to Higher Math., Ginn, Boston, Mass., 1959. MR 21 #6415. MR 0107692 (21:6415)
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 A. L. Perissini, Concerning the order structure of Köthe sequence spaces. II, Michigan Math. J. 11 (1964), 357364. MR 0166604 (29:3877)
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 M. Pommiez, Sur les différences divisées successives et les restes des séries de Newton généralisées, Ann. Fac. Sci. Univ. Toulouse (4) 28 (1964), 101110. MR 33 #7559b. MR 0199414 (33:7559b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719730320286X
PII:
S 00029947(1973)0320286X
Keywords:
Newton series,
zeros of remainders,
extremal functions,
matrix transformations
Article copyright:
© Copyright 1973 American Mathematical Society
