A remark on best L1-approximation by polynomials

Brass, Helmut

doi:10.1016/0021-9045(88)90049-4

Cited by 4 publications

(7 citation statements)

References 4 publications

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“…Further, we extend these estimates to product rules which include among their nodes also the endpoints ±1 . Our results of §3 generalize some of those presented in [3,4,8,9], and can be used, for…”

Section: Introductionsupporting

confidence: 81%

“…in Theorems 1 and 3 above we set v = 0 we obtain that the L,-errors of tm and qm are of the same order as the Lx -errors associated with the corresponding best L,-approximation polynomial (see [1,3]). Notice, however, that our polynomials tm and qm , which in general are not the best Lx -approximations, satisfy the extra properties (2.5), (2.6), (2.14) and (2.15), which are essential in our proofs of the following Theorems 5-8.…”

Section: Lemma 1 the Integralmentioning

confidence: 99%

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Convergence properties of a class of product formulas for weakly singular integral equations

Criscuolo¹,

Mastroianni²,

Monegato³

1990

Math. Comp.

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Abstract.We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the typewhere the kernel K(x, y) is weakly singular and the function f{x) has singularities only at the endpoints ± 1 . In particular, when K(x , y) = log \x -y\, K(x ,y) = \x -y\", v > -1, and f(x) has algebraic singularities of the form ( 1 ± x)a , a > -1 , we prove that the uniform rate of convergence of the rules is 0{m~ ~ "log m) in the case of the first kernel, and 0(m~ ~ "~ "logm) if v < 0, or 0(m~ ~ " \o%m) if t> > 0, for the second, where m is the number of points in the quadrature rule.

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Section: Introductionsupporting

confidence: 81%

Section: Lemma 1 the Integralmentioning

confidence: 99%

Convergence properties of a class of product formulas for weakly singular integral equations

Criscuolo¹,

Mastroianni²,

Monegato³

1990

Math. Comp.

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“…4], we know that when n is an even integer we have (1.5)). This allows us to derive an explicit expression for f − p L1 n 1 by using an explicit formula for f − p cheb n 1 [8]. By applying the formula in [8] to √ 1 − x 2 , we find that…”

Section: It Provides Theoretical Justification That P L1mentioning

confidence: 96%

“…14]. In particular, when N = n, the polynomial interpolant of f at the points in (1.4), i.e., [8,26].…”

Section: Yuji Nakatsukasa and Alex Townsendmentioning

confidence: 99%

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Error Localization of Best $L_{1}$ Polynomial Approximants

Nakatsukasa¹,

Townsend²

2021

SIAM J. Numer. Anal.

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An important observation in compressed sensing is that the 0 minimizer of an underdetermined linear system is equal to the 1 minimizer when there exists a sparse solution vector and a certain restricted isometry property holds. Here, we develop a continuous analogue of this observation and show that the best L 0 and L 1 polynomial approximants of a polynomial that is corrupted on a set of small measure are nearly equal. We demonstrate an error localization property of best L 1 polynomial approximants and use our observations to develop an improved algorithm for computing best L 1 polynomial approximants to continuous functions.

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Best L1-approximation by polynomials, II

Fiedler¹,

Jurkat²

1990

Journal of Approximation Theory

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A remark on best L1-approximation by polynomials

Cited by 4 publications

References 4 publications

Convergence properties of a class of product formulas for weakly singular integral equations

Convergence properties of a class of product formulas for weakly singular integral equations

Error Localization of Best $L_{1}$ Polynomial Approximants

Best L1-approximation by polynomials, II

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