Monotone approximation

Shisha, Oved

doi:10.2140/pjm.1965.15.667

Cited by 75 publications

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“…II and III also have affirmative answers, as established by Wolibner [12], Kammerer [2] and Young [13]. Furthermore, quantitative estimates have been obtained for the degree «(e) of the polynomial that is sufficient for monotone and comonotone approximation [11], [10], [6], [4], [9], [7], and for the degree n(X, Y) of the polynomial that is sufficient for piecewise monotone interpolation [8]. (iii) 2(1/X,) = oo.…”

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confidence: 53%

Piecewise monotone interpolation and approximation with Muntz polynomials

Passow¹,

Raymon²,

Shisha³

1976

Trans. Amer. Math. Soc.

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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 fcth difference by a Muntz polynomial with a monotone fcth 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.The following related questions have been settled: I iMonotone Approximation). Let fix) he a continuous function with the property that the /th difference u¿f> 0 on [0, 1] where / is some nonnegative integer. Must there be for a given e > 0 a corresponding polynomial p(x) with p0)(x) > 0 on [0, 1] such that ||/-p|| = sup \f(x)-p(x)\

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confidence: 53%

Piecewise monotone interpolation and approximation with Muntz polynomials

Passow¹,

Raymon²,

Shisha³

1976

Trans. Amer. Math. Soc.

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“…Let (e0, e,,..., eq) be a (a+l)-tuple with elements equal to ± 1 and let {k,}U0 t>e a fixed set ofq+l integers satisfying 0<ko<k,< ■ -■ <kq^n-1. "Monotone approximation" in C [a,b] by the set N={pePn_1; etp^'W^Q on [a, b], i = 0, 1,..., a} has been studied by O. Shisha [13], G. G. Lorentz and K. L. Zeller [6], R. A. Lorentz [7], and J. A. Roulier [10].…”

Section: Definitionmentioning

confidence: 99%

A unified approach to uniform real approximation by polynomials with linear restrictions

Chalmers¹

1972

Trans. Amer. Math. Soc.

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Abstract. Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

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“…The topic of monotone approximation started in [6] has become a major trend in approximation theory. A typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥ 0 by polynomials having this property.…”

Section: Introductionmentioning

confidence: 99%

Univariate right fractional polynomial high order monotone approximation

Anastassiou

2016

Demonstratio Mathematica

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Abstract. Let f P C r pr´1, 1sq, r ≥ 0 and let L˚be a linear right fractional differential operator such that L˚pf q ≥ 0 throughout r´1, 0s. We can find a sequence of polynomials Qn of degree ≤ n such that L˚pQnq ≥ 0 over r´1, 0s, furthermore f is approximated right fractionally and simultaneously by Qn on r´1, 1s. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f prq .

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Monotone approximation

Cited by 75 publications

References 0 publications

Piecewise monotone interpolation and approximation with Muntz polynomials

Piecewise monotone interpolation and approximation with Muntz polynomials

A unified approach to uniform real approximation by polynomials with linear restrictions

Univariate right fractional polynomial high order monotone approximation

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