When isf(x,y) = u(x) + v(y)?

Cowsik, R.; Kłopotowski, Andrzej; Nadkarni, M. G.

doi:10.1007/bf02837767

Cited by 17 publications

(18 citation statements)

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“…These objects were first introduced by Diliberto and Straus [7] (in [7], they are called "permissible lines"). They appeared further in a number of papers with several different names such as "bolts" (see, e.g., [2,20,28]), "trips" (see [29]), "links" (see, e.g., [6,22,23]), etc. Paths with respect to two directions a 1 and a 2 were exploited in some papers devoted to ridge function interpolation (see, e.g., [3,13]).…”

Section: Equioscillation Theorem For Ridge Functionsmentioning

confidence: 99%

A note on the equioscillation theorem for best ridge function approximation

Ismailov

2017

Expositiones Mathematicae

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Abstract. We consider the approximation of a continuous function, defined on a compact set of the d-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.Mathematics Subject Classifications: 41A30, 41A50, 46B50, 46E15

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Section: Equioscillation Theorem For Ridge Functionsmentioning

confidence: 99%

A note on the equioscillation theorem for best ridge function approximation

Ismailov

2017

Expositiones Mathematicae

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“…These objects were first introduced by Diliberto and Straus [5] (in [5], they are called "permissible lines"). They appeared further in a number of papers with several different names such as "paths" (see, e.g., [17,18]), "trips" (see [20,21]), "links" (see, e.g., [3,15,16]), etc. The term "bolt of lightning" was due to Arnold [1].…”

Section: Consider the Iterationsmentioning

confidence: 99%

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Asgarova

Ismailov

2017

Proc Math Sci

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In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto-Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto-Straus's original result holds for a large class of compact convex sets.Mathematics Subject Classifications: 41A30, 41A65, 46B28, 65D15

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“…C) If f is a Borel function on S, can one choose functions u and v on X and Y respectively which satisfy (1) to be Borel measurable? D) If S is a good set then, as proved in [7,3], we can write S as a union of two graphs G and H of functions defined on subsets of X and Y respectively. Can one choose the graphs G and H to be Borel sets if we know in addition that S is a Borel set?…”

Section: 2mentioning

confidence: 99%

“…Such sets occur in connection with some problems in ergodic theory and have a description [3]. It turns out that some very natural questions about good Borel sets have their answers in the facts about Borel sets with doubleton sections and one of the purposes of this paper is to exhibit this connection.…”

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confidence: 96%

Sets with doubleton sections, good sets and ergodic theory

Kłopotowski¹,

Nadkarni

Sarbadhikari

et al. 2002

Fund. Math.

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Abstract.A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.Introduction. Given non-empty sets X and Y , a subset of X × Y is called a matching if it is the graph of a one-to-one map of X onto Y . The question as to which subsets S of X × Y contain a matching has been of combinatorial and set-theoretic interest [12]. By a theorem of D. König [11], if each section S x = {y ∈ Y : (x, y) ∈ S}, x ∈ X, and S y = {x ∈ X : (x, y) ∈ S}, y ∈ Y , consists of exactly n elements (n finite; for infinite n see

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When isf(x,y) = u(x) + v(y)?

Cited by 17 publications

References 1 publication

A note on the equioscillation theorem for best ridge function approximation

A note on the equioscillation theorem for best ridge function approximation

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Sets with doubleton sections, good sets and ergodic theory

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