Piecewise monotone interpolation and approximation with Muntz polynomials

Passow, Eli; Raymon, Louis; Shisha, Oved

doi:10.1090/S0002-9947-1976-0399705-1

Piecewise monotone interpolation and approximation with Muntz polynomials
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by Eli Passow, Louis Raymon and Oved Shisha PDF

Trans. Amer. Math. Soc. 218 (1976), 197-205 Request permission

Abstract:

The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of “Muntz polynomials” on a real interval is demonstrated: (i) approximation of a continuous function by a “copositive” Muntz polynomial; (ii) approximation of a continuous function by a “comonotone” Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials—i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.

References

Joseph Bak and Donald J. Newman, Müntz-Jackson theorems in $L^{p}$ $[0,$ $1]$ and $C[0,\,1]$, Amer. J. Math. 94 (1972), 437–457. MR 310507, DOI 10.2307/2374631
William J. Kammerer, Polynomial approximations to finitely oscillating functions, Math. Comp. 15 (1961), 115–119. MR 123865, DOI 10.1090/S0025-5718-1961-0123865-3
D. Leviatan, On the Jackson-Müntz theorem, J. Approximation Theory 10 (1974), 1–5. MR 425428, DOI 10.1016/0021-9045(74)90091-4
Kim Pin Lim, Note on monotone approximation, Bull. London Math. Soc. 3 (1971), 366–368. MR 293303, DOI 10.1112/blms/3.3.366
G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, No. 8, University of Toronto Press, Toronto, 1953. MR 0057370
G. G. Lorentz and K. L. Zeller, Degree of approximation by monotone polynomials. I, J. Approximation Theory 1 (1968), 501–504. MR 239342, DOI 10.1016/0021-9045(68)90039-7
E. Passow and L. Raymon, Monotone and comonotone approximation, Proc. Amer. Math. Soc. 42 (1974), 390–394. MR 336176, DOI 10.1090/S0002-9939-1974-0336176-9
Eli Passow and Louis Raymon, The degree of piecewise monotone interpolation, Proc. Amer. Math. Soc. 48 (1975), 409–412. MR 430608, DOI 10.1090/S0002-9939-1975-0430608-4
Eli Passow, Louis Raymon, and John A. Roulier, Comonotone polynomial approximation, J. Approximation Theory 11 (1974), 221–224. MR 352807, DOI 10.1016/0021-9045(74)90013-6
John A. Roulier, Monotone approximation of certain classes of functions, J. Approximation Theory 1 (1968), 319–324. MR 236580, DOI 10.1016/0021-9045(68)90009-9
O. Shisha, Monotone approximation, Pacific J. Math. 15 (1965), 667–671. MR 185334, DOI 10.2140/pjm.1965.15.667
W. Wolibner, Sur un polynôme d’interpolation, Colloq. Math. 2 (1951), 136–137 (French). MR 43946, DOI 10.4064/cm-2-2-136-137
Sam W. Young, Piecewise monotone polynomial interpolation, Bull. Amer. Math. Soc. 73 (1967), 642–643. MR 212455, DOI 10.1090/S0002-9904-1967-11806-8