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Herbert E. Salzer

Research and Advances Nov 1 1973

A recurrence scheme for converting from one orthogonal expansion into another

A generalization of a scheme of Hamming for converting a polynomial Pn(x) into a Chebyshev series is combined with a recurrence scheme of Clenshaw for summing any finite series whose terms satisfy a three-term recurrence formula. An application to any two orthogonal expansions Pn(x) = ∑nm=0 amqm(x) = ∑nm=0 AmQm(x) enables one to obtain Am directly from am, m = 0(1)n, by a five-term recurrence scheme.

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Research and Advances Mar 1 1973

Gray code and the ± sign sequence when ±f(±f(±f(•••±f(x)•••))) is ordered

A previous note gives a rule for determining the sequence of + and - signs to obtain the ith largest value of +ƒ(±ƒ(±ƒ(··· ±ƒ(x)···))) when ƒ(x) is positive and monotonic [1, p. 46, §2]. Shortly after publication the writer received a letter dated March 9, 1972, from John Murchland, of the Planning & Transport Research & Computation Co. Ltd., London, in which he mentioned that the conversion scheme for obtaining the ± sign sequence from i - 1 happens to be also the standard way of going from a binary number to its Gray code equivalent, when after the conversion every plus sign is replaced by 0 and every minus sign by 1.
Research and Advances May 1 1969

Chebyshev interpolation and quadrature formulas of very high degree

All the zeros x2m,i, i = 1(1)2m, of the Chebyshev polynomials T2m(x), m = 0(1)n, are found recursively just by taking n2n-1 real square roots. For interpolation or integration of ƒ(x), given ƒ(x2m,i), only x2m,i is needed to calculate (a) the (2m - 1)-th degree Lagrange interpolation polynomial, and (b) the definite integral over [-1, 1], either with or without the weight function (1 - x2)-1/2, the former being exact for ƒ(x) of degree 2m+1 - 1.

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