On existence of a best uniform approximation of a function in two variables by the sums ϕ(x) + Ψ(y)

Garkavi, A. L.; Medvedev, V. A.; Havinson, S. Ya.

doi:10.1007/bf02107327

Cited by 8 publications

(9 citation statements)

References 5 publications

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“…Diliberto-Straus [6], Flatto [8], Garkavi-Medvedev-Khavinson [10], von Golitschek [11,12], von Golitschek-Light [13], Light-Cheney [16,17] and Rivlin-Sibner [19]. They mainly deal with the question of proximinality, non-uniqueness as well as with an algorithmic approach to best uniform approximants.…”

Section: Introduction and Main Resultmentioning

confidence: 99%

Best one-sided L1-approximation of bivariate functions by sums of univariate ones

2000

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Let f P C 1Y1 À1Y 1 2 , d 2 f adxdy^0. We characterize the unique best onesided L 1 -approximant h Ã to f from above (resp. h Ã from below) with respect to the subspace B 1Y1 which consists of all bivariate functions which are sums of univariate functions. h Ã resp. h Ã are constructed by a Hermite type interpolation on the diagonal ftY t X t P Ig resp. the anti-diagonal ftY Àt X t P Ig, where I X À1Y 1. Introduction and main result.In various situations the characterization of best L 1 -approximants can be given in terms of interpolation with respect to canonical point sets.Interpolation with respect to such sets yields best L 1 -approximants for a large class of functions. As classical examples we mention the Markov Theorem (see p. 87]) and the best one-sided L 1 -approximation of monomials by polynomials of lower degree. These examples deal with univariate approximation by finite dimensional spaces of approximating functions.On the other hand, there are characterization results for multivariate L 1 -approximants by interpolation with respect to canonical point sets where the spaces of approximating functions are infinite-dimensional. For example, if the space of approximating functions consists of harmonic functions on the unit ball in R n , then we have a concentric sphere as a canonical point set (see [2] for details). If one considers L 1 -approximation by blending functions defined by

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Section: Introduction and Main Resultmentioning

confidence: 99%

Best one-sided L1-approximation of bivariate functions by sums of univariate ones

2000

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“…The set S satisfies all conditions of Theorem 3 of [15] on existence of the best approximation ϕ 0 (x)+ψ 0 (y). By Havinson's result [5], E(f, S) = sup l⊂S {|r(f, l)|}.…”

Section: Remarkmentioning

confidence: 95%

Methods for computing the least deviation from the sums of functions of one variable

Ismailov

2006

Sib Math J

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“…Diliberto and Straus [6] were the first to establish a formula for E( f ,R), where R is a rectangle with sides parallel to coordinate axes. Their formula contains the supremum over all closed lightning bolts (for this terminology, see [1,8,9,11,13]). Later the same formula was established by other authors differently in rectangular case (see [13]) and for more general sets (see [9,11]).…”

Section: Approximation By Sums Of Univariate Functionsmentioning

confidence: 99%

“…This has several reasons, which get clear through the proof of Theorem 2.1. Here we are able to explain one of these reasons: by [8,Theorem 3], a continuous function f (x, y) defined on a polygon with sides parallel to coordinate axes has an extremal element, the existence of which is required by our method. Now, let K be a rectangle (not speaking about polygons) with sides not parallel to coordinate axes.…”

Section: Approximation By Sums Of Univariate Functionsmentioning

confidence: 99%

On error formulas for approximation by sums of univariate functions

Ismailov

2006

International Journal of Mathematics and Mathematical Sciences

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The purpose of the paper is to develop a new method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions. It should be remarked that formulas of this type have been known only for functions defined on a rectangle with sides parallel to coordinate axes. Our method, based on a maximization process over closed bolts, allows the consideration of functions defined on a hexagon or octagon with sides parallel to coordinate axes.

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On existence of a best uniform approximation of a function in two variables by the sums ϕ(x) + Ψ(y)

Cited by 8 publications

References 5 publications

Best one-sided L1-approximation of bivariate functions by sums of univariate ones

Best one-sided L1-approximation of bivariate functions by sums of univariate ones

Methods for computing the least deviation from the sums of functions of one variable

On error formulas for approximation by sums of univariate functions

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