Computations of the Hill functions of higher order
A. J. Jerri
Math. Comp. 31 (1977), 481-484
Primary 65D05; Secondary 41A15
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Abstract: In this note, we express the hill function of an order n as a Fourier cosine series which is of simple form that allows proving the function's basic properties. For the hill functions of higher order the form of the coefficients makes the series "essentially" self-truncating. For such high order hill functions, this truncated series (with thirty terms) computes the hill function with the same accuracy as the method of Legendre polynomials with local coordinates, but without the latter required coefficients which are to be computed in advance. The preliminary time analysis indicates that the time for the two methods starts to be the same at , changes slightly for the cosine series for and varies roughly as for the localized Legendre polynomial method. In comparison with the most recent efficient methods which require a storage of order n, this note's method required a storage of the order 25-40 for , executed with almost the same speed and accuracy and stayed stable as long as the above methods did.
A. Ditkin and A.
P. Prudnikov, Integral transforms and operational calculus,
Translated by D. E. Brown. English translation edited by Ian N. Sneddon,
Pergamon Press, Oxford-Edinburgh-New York, 1965. MR 0196422
Approximation by hill functions, Comment. Math. Univ. Carolinae
11 (1970), 787–811. MR 0292309
Segethová, Numerical construction of the hill
functions, SIAM J. Numer. Anal. 9 (1972),
199–204. MR 0305552
de Boor, On calculating with 𝐵-splines, J.
Approximation Theory 6 (1972), 50–62. Collection of
articles dedicated to J. L. Walsh on his 75th birthday, V (Proc. Internat.
Conf. Approximation Theory, Related Topics and their Applications, Univ.
Maryland, College Park, Md., 1970). MR 0338617
C. de BOOR, Package for Calculating with B-Splines, Tech. Report #1333, Math. Research Center, Univ. of Wisconsin, Madison, 1973.
D. Helms and J.
B. Thomas, Truncation error of sampling-theorem expansions,
Proc. IRE 50 (1962), 179–184. MR 0148199
- V. A. DITKIN & A. P. PRUDNIKOV, Integral Transforms and Operational Calculus, Fizmatgiz, Moscow, 1961; English transl., Pergamon Press, New York, 1965, pp. 177-178. MR 33 #4609. MR 0196422 (33:4609)
- I. BABUŠKA, "Approximation by Hill functions," Comment. Math. Univ. Carolinae, v. 11, 1970, pp. 787-811. MR 45 #1396. MR 0292309 (45:1396)
- J. SEGETHOVÁ, "Numerical construction of the Hill functions," SIAM J. Numer. Anal., v. 9, 1972, pp. 199-204. MR 46 #4682. MR 0305552 (46:4682)
- C. de BOOR, "On calculating with B-splines," J. Approximation Theory, v. 6, 1972, pp. 50-62. MR 49 #3381. MR 0338617 (49:3381)
- C. de BOOR, Package for Calculating with B-Splines, Tech. Report #1333, Math. Research Center, Univ. of Wisconsin, Madison, 1973.
- H. D. HELMS & J. B. THOMAS, "Truncation error of sampling-theorem expansions," Proc. IRE, v. 50, 1962, pp. 179-184. MR 26 #5707. MR 0148199 (26:5707)
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