Successive remainders of the Newton series

Authors:
G. W. Crofts and J. K. Shaw

Journal:
Trans. Amer. Math. Soc. **181** (1973), 369-383

MSC:
Primary 30A08

DOI:
https://doi.org/10.1090/S0002-9947-1973-0320286-X

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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0320286-X

Keywords:
Newton series,
zeros of remainders,
extremal functions,
matrix transformations

Article copyright:
© Copyright 1973
American Mathematical Society