Nonexistence of best alternating approximations on subsets

Dunham, Charles B.

doi:10.1090/S0002-9939-1976-0417643-8

Nonexistence of best alternating approximations on subsets
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by Charles B. Dunham PDF

Proc. Amer. Math. Soc. 60 (1976), 203-206 Request permission

Abstract:

The existence of best Chebyshev approximations by an alternating family on closed subsets of an interval is considered. In varisolvent approximation, existence on subsets of sufficiently low density is guaranteed if the best approximation on the interval is of maximum degree. The paper studies the case in which the best approximation is not of maximum degree and shows that in many common cases, no such guarantee is possible.

References

Richard Barrar and Henry Loeb, On $N$-parameter and unisolvent families, J. Approximation Theory 1 (1968), 180–181. MR 236578, DOI 10.1016/0021-9045(68)90021-x
Charles B. Dunham, Chebyshev aproximation with respect to a weight function, J. Approximation Theory 2 (1969), 223–232. MR 252922, DOI 10.1016/0021-9045(69)90016-1
Charles B. Dunham, Chebyshev approximation by $A+B^{\ast }\ \textrm {log}\ (1+CX)$. II, J. Inst. Math. Appl. 10 (1972), 369–372. MR 338632
C. B. Dunham, Approximation by alternating families on subsets, Computing (Arch. Elektron. Rechnen) 9 (1972), 261–265 (English, with German summary). MR 312686, DOI 10.1007/bf02241601
Charles B. Dunham, Approximation by exponential-polynomial products on finite point sets, J. Inst. Math. Appl. 10 (1972), 125–127. MR 380200
Charles B. Dunham, Alternating Chebyshev approximation, Trans. Amer. Math. Soc. 178 (1973), 95–109. MR 318736, DOI 10.1090/S0002-9947-1973-0318736-8
Charles B. Dunham, Varisolvent Chebyshev approximation on subsets, Approximation theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973) Academic Press, New York, 1973, pp. 337–340. MR 0333534
Charles B. Dunham, Nonexistence of best rational approximations on subsets, J. Approximation Theory 14 (1975), no. 2, 160–161. MR 402353, DOI 10.1016/0021-9045(75)90087-8
Ch. B. Dunham, Chebyshev approximation by logarithmic families, Z. Angew. Math. Mech. 53 (1973), no. 6, 352–353. MR 387902, DOI 10.1002/zamm.19730530510
John R. Rice, The approximation of functions. Vol. 2: Nonlinear and multivariate theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0244675