Some Chebyshev approximations by polynomials in two variables

Gearhart, W. B.

doi:10.1016/0021-9045(73)90009-9

Cited by 27 publications

(15 citation statements)

References 8 publications

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“…In [3], the function t~*t 4 was approximated on the unit circle by bivariate polynomials of total degree 7. From [12,18], the optimal error in (tl, t2) -=(cos ~b, sin 4)) is 2 -7 cos 8 q5 so that the same solution is obtained on a discrete set containing the extrema defined by q~ =jrc/8 (0 <j< 15). This occurs with m points equally spaced from 1 where m is a multiple of 16, which yields Ax ~ b with A e P," • 36 and b e R".…”

Section: -1mentioning

confidence: 96%

Strict Chebyshev approximation for general systems of linear equations

Thiran

Thiry

1987

Numer. Math.

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Summary. An ascent exchange algorithm for computing the strict Chebyshev solution to general systems of linear equations is presented. It uses generalized exchange rules to ensure convergence and splits up the entire system into subsystems by means of a canonical decomposition of a matrix obtained by Gaussian elimination methods. All updating procedures are developed and several numerical examples illustrate the efficiency of the algorithm.

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Section: -1mentioning

confidence: 96%

Strict Chebyshev approximation for general systems of linear equations

Thiran

Thiry

1987

Numer. Math.

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“…Для размерности два для широкого класса а.м. из A 2 q найдены [2], [3] точные значения (1). Дана [2] классификация мономов x ν1 1 x ν2 2 , ν 1 + ν 2 = q, для которых имеется единственность а.м.…”

Section: математические заметкиunclassified

“…из Φ n q−1 на шаре B n произвольной размерности n 3. В [2] показано, что E q−1 (f ) = 1, n = 2. Поскольку B n ⊃ B 2 , то E q−1 (f ) 1 для n 3.…”

Section: математические заметкиunclassified

О Многочленах Наилучшего Приближения

Юдин¹,

Yudin²

2007

Mat. Zametki

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“…While for the unit disc, i.e., for d = 2, for a wide class of homogeneous polynomials of degree N , min-max polynomials are known for every N ∈ N (see [6,8]), for the ball and the sphere in dimension d, d ≥ 3, only correspondingly modified Chebyshev polynomials are known to be min-max polynomials for every N ∈ N, as T 2N (‖x‖) and T N (⟨a, x⟩), where a = (a 1 , . .…”

Section: Introductionmentioning

confidence: 99%

An explicit class of min–max polynomials on the ball and on the sphere

Moale

Peherstorfer

2011

Journal of Approximation Theory

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Let Π d n+m−1 denote the set of polynomials in d variables of total degree less than or equal to n + m − 1 with real coefficients and let P(x), x = (x 1 , . . . , x d ), be a given homogeneous polynomial of degree n + m in d variables with real coefficients. We look for a polynomial p * ∈ Π d n+m−1 such that P − p * has least max norm on the unit ball and the unit sphere in dimension d, d ≥ 2, and call P − p * a min-max polynomial. For every n, m ∈ N, we derive min-max polynomials for P of the form P(x) = P n (x ′ )x m d , with x ′ = (x 1 , . . . , x d−1 ), where P n (x ′ ) is the product of homogeneous harmonic polynomials in two variables. In particular, for every m ∈ N, min-max polynomials for the monomials x 1 . . . x d−1 x m d are obtained. Furthermore, we give min-max polynomials for the case where P n (x ′ ) = ‖x ′ ‖ n T n (⟨a ′ , x ′ ⟩/‖x ′ ‖), a ′ = (a 1 , . . . , a d−1 ) ∈ R d−1 , ‖a ′ ‖ = 1, and T n denotes the Chebyshev polynomial of the first kind.

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Some Chebyshev approximations by polynomials in two variables

Cited by 27 publications

References 8 publications

Strict Chebyshev approximation for general systems of linear equations

Strict Chebyshev approximation for general systems of linear equations

О Многочленах Наилучшего Приближения

An explicit class of min–max polynomials on the ball and on the sphere

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