Canonical Sets of Best 𝐿_𝟏-Approximation

Abstract

In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best 𝐿₁-approximation with emphasis on multivariate interpolation and best 𝐿₁-approximation by blending functions. The best 𝐿₁-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate 𝐿₁-approximation by sums of univariate functions. Explicit constructions of best one-sided 𝐿₁-approximants give rise to well-known and new inequalities.

1. Canonical Interpolation Sets of Best 𝐿₁-Approximation

We start with the notion of a canonical set of best 𝐿₁-approximation. Let 𝜇 be a positive Borel measure defined on the compact set 𝐾⊂𝐑^𝑑 such that the quantity ‖𝑓‖₁∶=∫_𝐾|𝑓|𝑑𝜇 represents a norm in the linear space 𝐶(𝐾) of functions that are continuous on 𝐾. Let 𝑈   be a linear subspace of 𝐶(𝐾).

1.1. The Problem of Best 𝐿₁(𝜇)-Approximation

Given a function 𝑓∈𝐶(𝐾), then we have the following.(A)Best 𝐿₁(𝜇)-approximation of 𝑓 by elements of 𝑈: find a function 𝑢_𝑓∈𝑈 such that 𝑓−𝑢_𝑓1≤𝑓−𝑢₁∀𝑢∈𝑈. (1.1) (B)Best one-sided from above 𝐿₁(𝜇)-approximation by elements of 𝑈: find a function 𝑢^∗_𝑓∈𝑈, 𝑢^∗_𝑓≥𝑓 on 𝐾, such that 𝑓−𝑢^∗_𝑓1≤𝑓−𝑢₁∀𝑢∈𝑈,𝑢≥𝑓on𝐾. (1.2)(C)Best one-sided from below 𝐿₁(𝜇)-approximation by elements of 𝑈: find a function 𝑢_∗𝑓∈𝑈, 𝑢_∗𝑓≤𝑓 on 𝐾, such that 𝑓−𝑢_∗𝑓1≤𝑓−𝑢₁∀𝑢∈𝑈,𝑢≤𝑓on𝐾. (1.3)Any solution 𝑢_𝑓 of the approximation problem (A) is called a best 𝐿₁(𝜇)-approximant to 𝑓 from 𝑈. Any solution of the approximation problem (B) is called a best one-sided from above 𝐿₁(𝜇)-approximant to 𝑓 from 𝑈; respectively from below for the approximation problem (C). When 𝜇 is the usual Lebesgue measure, we use the notation 𝐿₁-approximation, respectively, 𝐿₁-approximant.For qualitative results on best 𝐿₁(𝜇)-approximation from finite-dimensional subspaces see [1].

1.2. Lagrange Interpolation Problem

Given a function 𝑓∈𝐶(𝐾). We call a subset 𝑋⊂𝐾 unisolvent for 𝑈 if the interpolation problem 𝑢(𝑥)=𝑓(𝑥),𝑥∈𝑋 (1.4) possesses a unique solution 𝑢∈𝑈 for every 𝑓∈𝐶(𝐾).

1.3. Canonical Sets of Best 𝐿₁-Approximation

Let 𝒞⊂𝐶(𝐾) be a class of functions. We say that a set 𝑋⊂𝐾, which is unisolvent for 𝑈, is a canonical set of best 𝐿₁(𝜇)-approximation to 𝒞, if for all 𝑓∈𝒞 the solution of the interpolation problem (1.4) is a best 𝐿₁(𝜇)-approximant to 𝑓 from 𝑈. Analogously, a unisolvent set 𝑋^∗⊂𝐾 for 𝑈 (resp., 𝑋_∗⊂𝐾) is called a canonical set of best one-sided from above (resp., from below) 𝐿₁(𝜇)-approximation to 𝒞, if for all 𝑓∈𝒞 the solution of the interpolation problem (1.4) with 𝑋 replaced by 𝑋^∗ (resp., 𝑋_∗) is a best one-sided from above (resp., from below) 𝐿₁(𝜇)-approximant to 𝑓 from 𝑈. In the case of one-sided 𝐿₁(𝜇)-approximation the interpolation problem (1.4) is considered in a broader Lagrange-Hermite interpolation sense: for all points 𝑥∈𝑋 which are interior for 𝐾 we require matching not only the values of 𝑢 and 𝑓 but also the values of their first (partial) derivatives.

2. Canonical Sets of Univariate Best 𝐿₁-Approximation

For details on the results in the present section see [1–6]. In the univariate case, when the set 𝐾 is an interval [𝑎,𝑏] and the subspace 𝑈 is finite-dimensional, the problem of existence and characterization of a canonical set is well studied and is closely related to the notion of a Haar-Chebyshev system (see [1] for details).

2.1. Haar-Chebyshev System of Order 𝑛

A set of 𝑛 functions {𝑢₁,𝑢₂,…,𝑢_𝑛}⊂𝐶[𝑎,𝑏] is a Haar-Chebyshev system (𝑇-system) of order 𝑛 on the interval [𝑎,𝑏] if each nontrivial linear combination of 𝑢₁,𝑢₂,…,𝑢_𝑛 has at most 𝑛−1 zeros in [𝑎,𝑏]. In other words, a set of 𝑛 functions {𝑢₁,𝑢₂,…,𝑢_𝑛} is a Haar-Chebyshev system of order 𝑛 on the interval [𝑎,𝑏] if it is linearly independent with respect to an arbitrary chosen set of 𝑛 distinct points in [𝑎,𝑏]. We say that an 𝑛-dimensional subspace 𝑈 of 𝐶[𝑎,𝑏] is a Haar-Chebyshev space (T-space) of order 𝑛 on [𝑎,𝑏] if every 𝑢∈𝑈, 𝑢≢0, has no more than 𝑛−1 distinct zeros in [𝑎,𝑏]. Equivalently, 𝑈 is a 𝑇-space of order 𝑛 on [𝑎,𝑏], if for every basis 𝑢₁,𝑢₂,…,𝑢_𝑛 of 𝑈 there exists an 𝜀∈{−1,1} such that 𝜀det{𝑢_𝑖(𝑥_𝑗)}^𝑛_𝑖,𝑗=1>0 for all 𝑎≤𝑥₁<𝑥₂<⋯<𝑥_𝑛≤𝑏. It is also often said that the basis functions 𝑢₁,𝑢₂,…,𝑢_𝑛 constitute a 𝑇-space of order 𝑛 on [𝑎,𝑏].

2.2. Characterization of Best 𝐿₁(𝜇)-Approximation by Canonical Interpolation Sets

Following [1], throughout this section we suppose that 𝜇 is a finite positive nonatomic measure. The problem of best 𝐿₁(𝜇)-approximation by a Haar-Chebyshev space has an elegant solution via interpolation on a canonical set. Let 𝑈⊂𝐶[𝑎,𝑏] be a Haar-Chebyshev space of order 𝑛 and let 𝑢₁,𝑢₂,…,𝑢_𝑛 be a basis of 𝑈. Consider the convex cone 𝒞⁺(𝑈) defined as 𝒞⁺(𝑈)∶=𝑢_𝑛+1∶det𝑢_𝑖𝑥_𝑗^𝑛+1_𝑖,𝑗=1≥0 (2.1) for all 𝑎≤𝑥₁<𝑥₂<⋯<𝑥_𝑛<𝑥_𝑛+1≤𝑏.

Theorem 2.1. Let 𝑈⊂𝐶[𝑎,𝑏] be a Haar-Chebyshev space of order 𝑛 on [𝑎,𝑏]. Then the following holds true.(a)Uniqueness of the Canonical Set. There is a unique set of points 𝑎=𝑥₀<𝑥₁<⋯<𝑥_𝑛+1=𝑏 such that ^𝑛+1_𝑗=1(−1)^𝑗𝑥𝑗_𝑥𝑗−1𝑢𝑑𝜇=0,∀𝑢∈𝑈. (2.2)(b)Best 𝐿₁(𝜇)-Approximation via Interpolation. Let 𝑓∈𝒞⁺(𝑈). Then the unique solution 𝑢_𝑓∈𝑈 of the interpolation problem 𝑢_𝑓𝑥_𝑗=𝑓𝑥_𝑗,𝑗=1,2,…,𝑛 (2.3) is the unique best 𝐿₁(𝜇)-approximant to 𝑓 from 𝑈.

Remark 2.2. Following Theorem 2.1, the set {𝑥_𝑘}^𝑛_𝑘=1 is a canonical set of best 𝐿₁(𝜇)-approximation to the convex cone 𝒞⁺(𝑈) of functions from the 𝑇-space 𝑈. In many cases the approximating subspace 𝑈 can be a kernel of a certain differential operator 𝔇, that is, 𝑈=Ker𝔇 and the convex cone 𝒞⁺_𝔇(𝑈) of functions to be approximated can be defined by 𝒞⁺_𝔇(𝑈)∶=𝑓∈𝐶[𝑎,𝑏]∶𝔇𝑓(𝑥)≥0for𝑥∈[𝑎,𝑏]. (2.4)

Remark 2.3. Note that if a canonical set of best 𝐿₁-approximation exists for a functional set, then the nonlinear problem of best 𝐿₁-approximation becomes a linear interpolation problem in this functional set.

2.3. Best 𝐿₁-Approximation by Algebraic Polynomials

Denote by 𝜋_𝑛−1 the linear space of polynomials of degree ≤𝑛−1. Taking into account that the polynomial basis 𝑢_𝑘=𝑥^𝑘−1,𝑘=1,…,𝑛 of 𝜋_𝑛−1 is a 𝑇-system of order 𝑛 on any interval [𝑎,𝑏], the best polynomial 𝐿₁-approximant to a given function 𝑓 from 𝜋_𝑛−1 can be characterized as a polynomial interpolant to 𝑓 with respect to a canonical set if 𝑓 belongs to an appropriately chosen convex cone of functions. In this direction we formulate a result due to S. Bernstein [2, 3].

Theorem 2.4. Let 𝑓∈𝐶^𝑛[−1,1] and 𝑓^(𝑛)(𝑥)≥0 for 𝑥∈[−1,1]. Then, the polynomial 𝑝_𝑓 of degree ≤𝑛−1 is the unique polynomial of best 𝐿₁-approximation to 𝑓 from 𝜋_𝑛−1 on [−1,1] if and only if 𝑝_𝑓𝑥_𝑘=𝑓𝑥_𝑘,1≤𝑘≤𝑛, (2.5) where the interpolation nodes 𝑥_𝑘=cos[𝑘𝜋/(𝑛+1)], 𝑘=1,…,𝑛 are the zeros of the 𝑛-degree Chebyshev polynomial of second kind 𝑈_𝑛(𝑥)=sin[(𝑛+1)arccos(𝑥)]√1−𝑥². (2.6)

Remark 2.5. By Theorem 2.4, the interpolation points 𝑥_𝑘,𝑘=1,…,𝑛 form a canonical set of best 𝐿₁-approximation from 𝜋_𝑛−1 to the convex cone 𝑓∈𝐶^𝑛[−1,1]∶𝑓^(𝑛)(𝑥)≥0,𝑥∈[−1,1]. (2.7)

Note that a canonical set may change with respect to the choice of the approximating space. For example, following Theorem 2.4, the canonical set of best polynomial 𝐿₁-approximation depends on the degree of the polynomial approximant.

Corollary 2.6. The 𝑛-degree polynomial 𝑈_𝑛(𝑥)2^𝑛=sin[(𝑛+1)arccos(𝑥)]2^𝑛√1−𝑥² (2.8) is the unique monic (with leading coefficient 1) polynomial of degree 𝑛 which has a minimal 𝐿₁-norm (minimal 𝐿₁-deviation) on [−1,1].

We formulate a result on best 𝐿₁-approximation by algebraic polynomials according to A. Markov (see [2, 3] for details) that is based on the notion of a Haar-Chebyshev system.

Theorem 2.7. Let 𝑓∈𝐶[−1,1] be such that the set 1,𝑥,𝑥²,…,𝑥^𝑛−1,𝑓 (2.9) is a Haar-Chebyshev system of order 𝑛+1 on [−1,1]. Then the unique polynomial Lagrange interpolant 𝑝_𝑓 to 𝑓 from 𝜋_𝑛−1 with respect to the zeros 𝑥_𝑘,𝑘=1,…,𝑛 of the 𝑛-degree Chebyshev polynomial of second kind is the unique best 𝐿₁-approximant to 𝑓 from 𝜋_𝑛−1.

Remark 2.8. In the particular case 𝑓∈𝐶^(𝑛)[−1,1] and 𝑓^(𝑛)(𝑥)>0,𝑥∈[−1,1] of Theorem 2.4, the set of functions {1,𝑥,²,…,𝑥^𝑛−1,𝑓} is a Haar-Chebyshev system of order 𝑛+1 in [−1,1]. Hence in this particular case, Theorem 2.4 is a corollary by Theorem 2.7. The notion of Haar-Chebyshev system implies that 𝑓−𝑝 cannot have more than 𝑛 zeros on [−1,1], where 𝑝∈𝜋_𝑛−1.

2.4. Best 𝐿₁-Approximation by Trigonometric Polynomials

Let 𝑇_𝑛 be the linear space of trigonometric polynomials of degree ≤𝑛. Consider the normed linear space 𝐿₁[0,2𝜋] of 2𝜋-periodic functions whose absolute value has a finite integral on [0,2𝜋] and equipped with the 𝐿₁-norm 𝑓₁∶=₀^2𝜋𝑓(𝜃)𝑑𝜃. (2.10) The linear space 𝑇_𝑛 is a finite dimensional subspace of 𝐿₁[0,2𝜋]. For a given 𝑓∈𝐿₁[0,2𝜋], there exists a best 𝐿₁-approximant 𝑡∈𝑇_𝑛 (see [2] for details). Note that 𝑇_𝑛 is a Haar-Chebyshev space of order 2𝑛+1 on [0,2𝜋). Let 𝐶^2𝑛+1[0,2𝜋) denote the linear space of 2𝜋-periodic functions having continuous derivatives of order 2𝑛+1 on [0,2𝜋). The next theorem [7] is a canonical set characterization of the best trigonometric 𝐿₁-approximants from 𝑇_𝑛 to functions from the convex cone 𝐶₊^2𝑛+1∶=𝑓∶𝑓∈𝐶^2𝑛+1[0,2𝜋),𝔇^(2𝑛+1)𝑓(𝜃)≥0for𝜃∈[0,2𝜋), (2.11) where the differential operator 𝔇^(2𝑛+1) is defined as 𝔇^(2𝑛+1)∶=𝐷^𝑛_𝑘=1𝐷²+𝑘²𝑑,𝐷=.𝑑𝑥 (2.12)

Theorem 2.9. Let 𝑓∈𝐶₊^2𝑛+1. Then, the unique Lagrange trigonometric interpolant 𝑡_𝑓 from 𝑇_𝑛 to 𝑓 at the interpolation nodes 𝑘𝜋/(𝑛+1),𝑘=1,…,2𝑛+1 is the unique best 𝐿₁-approximant to 𝑓 from 𝑇_𝑛.

2.5. Characterization of Best One-Sided 𝐿₁(𝜇)-Approximation

Best one-sided 𝐿₁(𝜇)-approximation is related to the principal representations of the measure 𝜇, that is, to the so-called quadrature formulae of Gaussian, Lobatto, and Radau type (for details see [1, 3–6]).

2.5.1. Quadrature Formulae of Gaussian, Lobatto, and Radau Type

Let 𝑈 be a Haar-Chebyshev space of order 𝑛.(a)If 𝑛=2𝑚, then there exist unique sets of points 𝑎<𝑥^𝐺₁<𝑥^𝐺₂<⋯<𝑥^𝐺_𝑚<𝑏 and 𝑎=𝑥^𝐿₀<𝑥^𝐿₁<⋯<𝑥^𝐿_𝑚=𝑏, and unique sets of positive numbers {𝑎^𝐺_𝑗}^𝑚_𝑗=1 and {𝑎^𝐿_𝑗}^𝑚_𝑗=0 such that the quadrature formulae ^𝑏_𝑎𝑓𝑑𝜇≈𝑄^𝐺𝑓∶=^𝑚_𝑗=1𝑎^𝐺_𝑗𝑓𝑥^𝐺_𝑗Gaussiantypequadratureformula,_𝑏𝑎𝑓𝑑𝜇≈𝑄^𝐿𝑓∶=^𝑚_𝑗=0𝑎^𝐿_𝑗𝑓𝑥^𝐿_𝑗Lobattotypequadratureformula (2.13) are exact for all 𝑢∈𝑈.(b)If 𝑛=2𝑚+1, then there exist unique sets of points 𝑎=𝑥₀^𝑅−<𝑥₁^𝑅−<⋯<𝑥_𝑚^𝑅−<𝑏 and 𝑎<𝑥₀^𝑅+<𝑥₁^𝑅+<⋯<𝑥_𝑚^𝑅+=𝑏, and unique sets of positive numbers{𝑎_𝑗^𝑅−}^𝑚_𝑗=0 and {𝑎_𝑗^𝑅+}^𝑚_𝑗=0 such that the quadrature formulae ^𝑏_𝑎𝑓𝑑𝜇≈𝑄^𝑅−𝑓∶=^𝑚_𝑗=0𝑎_𝑗^𝑅−𝑓𝑥_𝑗^𝑅−leftRadautypequadratureformula,_𝑏𝑎𝑓𝑑𝜇≈𝑄^𝑅+𝑓∶=^𝑚_𝑗=0𝑎_𝑗^𝑅+𝑓𝑥_𝑗^𝑅+rightRadautypequadratureformula (2.14) are exact for all 𝑢∈𝑈.

Let 𝑢₁,𝑢₂,…,𝑢_𝑛 be a basis of 𝑈 such that det{𝑢_𝑖(𝑥_𝑗)}^𝑛_𝑖,𝑗=1>0 for 𝑎≤𝑥₁<𝑥₂<⋯<𝑥_𝑛≤𝑏. Hence, the set of functions {𝑢₁,𝑢₂,…,𝑢_𝑛} is a Haar-Chebyshev system (𝑇-system) of order 𝑛 on [𝑎,𝑏] or in other words 𝑈 is an 𝑛-order 𝑇-space. The following theorem holds true.

Theorem 2.10. Let 𝑈⊂𝐶[𝑎,𝑏] be a 𝑇-space of order 𝑛. Then, for every 𝑓∈𝒞⁺(𝑈) there exist elements of best one-sided 𝐿₁(𝜇)-approximation from below 𝑢_∗𝑓∈𝑈 and from above 𝑢^∗_𝑓∈𝑈. Moreover, the best one-sided approximants 𝑢_∗𝑓∈𝑈, 𝑢^∗_𝑓∈𝑈 to 𝑓 and the function 𝑓 satisfy (a)𝑄^𝐺[𝑢_∗𝑓]=𝑄^𝐺[𝑓] and 𝑄^𝐿[𝑢^∗_𝑓]=𝑄^𝐿[𝑓], if 𝑛=2𝑚, and(b)𝑄^𝑅−[𝑢_∗𝑓]=𝑄^𝑅−[𝑓] and 𝑄^𝑅+[𝑢^∗_𝑓]=𝑄^𝑅+[𝑓], if 𝑛=2𝑚+1.

In order to guarantee the uniqueness of the best one-sided 𝐿₁(𝜇)-approximant to a given function we need the following restriction on 𝑈.

2.5.2. Extended Haar-Chebyshev System of Multiplicity 2

A set of 𝑛 functions {𝑢₁,𝑢₂,…,𝑢_𝑛}⊂𝐶¹[𝑎,𝑏] is an 𝑛-order extended Haar-Chebyshev system of multiplicity 2 (𝑛-order ET-system of multiplicity 2) on the interval [𝑎,𝑏] if each nontrivial linear combination 𝑢≢0 of 𝑢₁,𝑢₂,…,𝑢_𝑛 has at most 𝑛−1 zeros in [𝑎,𝑏], provided that the common zeros of 𝑢 and 𝑢^′ are counted twice (counting multiplicities 2). In other words, 𝑈 is an 𝑛-order extended Haar-Chebyshev space of multiplicity 2 (𝑛-order ET-space of multiplicity 2) on the interval [𝑎,𝑏], if 𝑈⊂𝐶¹[𝑎,𝑏], dim(𝑈)=𝑛 and each nontrivial 𝑢∈𝑈,𝑢≢0 has at most 𝑛−1 zeros in [𝑎,𝑏], provided that the common zeros of 𝑢 and 𝑢^′ are counted twice (counting multiplicities 2).

2.5.3. Construction of Best One-Sided 𝐿₁(𝜇)-Approximants via Interpolation on Canonical Sets

For details on the results presented here, see [1, 3, 4] and the references given there.

Theorem 2.11. Let 𝑈⊂𝐶¹[𝑎,𝑏] be an 𝑛-order ET-space of multiplicity 2 on the interval [𝑎,𝑏]. Then, for every 𝑓∈𝒞⁺(𝑈)∩𝐶¹[𝑎,𝑏] one has the following.(a)If 𝑛=2𝑚, then the unique solutions 𝑢_∗𝑓 and 𝑢^∗_𝑓 of the Lagrange-Hermite interpolation problems 𝑢_∗𝑓𝑥^𝐺_𝑗=𝑓𝑥^𝐺_𝑗,𝑢^′_∗𝑓𝑥^𝐺_𝑗=𝑓^′x^𝐺_𝑗,𝑗=1,2,…,𝑚,𝑢_∗𝑓𝑢∈𝑈,_∗𝑓𝑥^𝐿_𝑗=𝑓𝑥^𝐿_𝑗,𝑢^∗_𝑓^′𝑥^𝐿_𝑗=𝑓^′𝑥^𝐿_𝑗𝑢,𝑗=1,2,…,𝑚−1_∗𝑓𝑥^𝐿₀=𝑓𝑥^𝐿₀,𝑢^∗_𝑓𝑥^𝐿_𝑚=𝑓𝑥^𝐿_𝑚,𝑢^∗_𝑓∈𝑈, (2.15) are the unique best one-sided from below, respectively, from above 𝐿₁(𝜇)-approximants to 𝑓 from 𝑈.(b)If 𝑛=2𝑚+1, then the unique solutions 𝑢_∗𝑓 and 𝑢^∗_𝑓 of the Lagrange-Hermite interpolation problems 𝑢_∗𝑓𝑥_𝑗^𝑅−=𝑓𝑥_𝑗^𝑅−,𝑢^′_∗𝑓𝑥_𝑗^𝑅−=𝑓^′𝑥_𝑗^𝑅−𝑢,𝑗=1,2,…,𝑚,_∗𝑓𝑥₀^𝑅−=𝑓𝑥₀^𝑅−,𝑢_∗𝑓𝑢∈𝑈,_∗𝑓𝑥_𝑗^𝑅+=𝑓𝑥_𝑗^𝑅+,𝑢^∗_𝑓^′𝑥_𝑗^𝑅+=𝑓^′𝑥_𝑗^𝑅+𝑢,𝑗=0,1,…,𝑚−1,_∗𝑓𝑥_𝑚^𝑅+=𝑓𝑥_𝑚^𝑅+,𝑢^∗_𝑓∈𝑈, (2.16)are the unique best one-sided from below, respectively, from above 𝐿₁(𝜇)-approximants to 𝑓 from 𝑈.

Best one-sided 𝐿₁-approximant to a given function exists and is unique under some conditions. Next theorem (for details and similar results see [3, 4, 8]) is an example in this direction.

Theorem 2.12. Let 𝑓 be a differentiable, 2𝜋-periodic function and let 𝑇_𝑛 denote the linear space of all trigonometric polynomials of degree at most 𝑛. Then, the best one-sided 𝐿₁-approximant from above, respectively, from below to 𝑓 from 𝑇_𝑛 exists and is unique.

Remark 2.13. The set of trigonometric polynomials 𝑡_∗(𝜃)=𝛼cos(𝑛𝜃), where −1≤𝛼≤1 are best one-sided from below 𝐿₁-approximants to the continuous 2𝜋-periodic function 𝑓(𝜃)=|cos(𝑛𝜃)| from 𝑇_𝑛. Hence, continuity is not enough for uniqueness of the best one-sided 𝐿₁-approximant to be claimed.

The following theorem (see [7] for details) gives a characterization of the best one-sided 𝐿₁-approximants via canonical sets. It can be considered as a refinement of Theorem 2.12 for functions from the convex cone 𝐶₊^2𝑛+1.

Theorem 2.14. Let 𝑓∈𝐶₊^2𝑛+1. Let 𝑡_∗𝑓∈𝑇_𝑛 be the unique interpolant from 𝑇_𝑛 to 𝑓 at the interpolation nodes 2𝑘𝜋/(𝑛+1),𝑘=1,2,…,𝑛 with multiplicities 2 and 𝑡_∗𝑓(0)=𝑓(0). The interpolant 𝑡_∗𝑓 to 𝑓 from 𝑇_𝑛 is the unique best one-sided from below 𝐿₁-approximant to 𝑓 from 𝑇_𝑛.
Let 𝑡^∗_𝑓∈𝑇_𝑛 be the unique interpolant from 𝑇_𝑛 to 𝑓 at the interpolation nodes 2𝑘𝜋/(𝑛+1),𝑘=1,2,…,𝑛 with multiplicities 2 and 𝑡^∗_𝑓(2𝜋)=𝑓(2𝜋⁻). The interpolant 𝑡^∗_𝑓 to 𝑓 from 𝑇_𝑛 is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝑇_𝑛.

In order to proceed with the multivariate case we remark that the above-stated univariate results are constituted by the following prerequisites.(1)The domain 𝐾 where the functions are defined is an interval.(2)The approximating space 𝑈 in most of the cases is a kernel of a linear ordinary differential operator 𝔇, that is, 𝑈=Ker𝔇.(3)The set on which certain interpolation problem is unisolvent for 𝑈 consists of a finite number of points.(4)The convex cone 𝒞⁺_𝔇(𝑈) of functions to be approximated is defined by 𝒞⁺_𝔇(𝑈)∶=𝑓∶𝔇𝑓existsand𝔇𝑓≥0on[𝑎,𝑏]. (2.17)(5)The best (one-sided) 𝐿₁-approximants are characterized as Lagrange-Hermite interpolants on canonical sets.

Conditions (3)–(5) are due to the fact that 𝑈 is a 𝑇-space or 𝐸𝑇-space of multiplicity 2. It is known that there are no point-wise Haar-Chebyshev systems in the multivariate case [9, 10] provided that the interpolation set is of finite number of points and the interpolating space 𝑈 is finite-dimensional. This fact is not surprising taking into account that the kernels of linear partial differential operators are infinite-dimensional linear spaces. Therefore, the natural interpolation sets in the multivariate case 𝐾⊂𝐑^𝑑 should be nondenumerable point sets, and more precisely, (𝑑−1)-dimensional manifolds. Hence, an appropriate transfinite interpolation on lower dimensional manifolds can be a basis for a canonical set characterization of the best 𝐿₁-approximants in the multivariate case. However, unlike the finite number of points in an interval [𝑎,𝑏] which are topologically equivalent, there is a countless variety of possible interpolation sets (interpolation grids) in the multivariate case and one cannot expect that there would be a simple characterization corresponding to that in the univariate case. Instead, one should consider pairs (𝑈,𝑋) consisting of an approximation space 𝑈 and a set 𝑋⊂𝐾 such that 𝑋 is unisolvent for 𝑈. Such sets 𝑋 are candidates for canonical sets in the multivariate 𝐿₁-approximation. There is no consistent general treatment of the canonical sets of best approximation in 𝐑^𝑑 and each known result is a solution of a particular problem. In the next sections we will discuss some of these results and we will focus on multivariate interpolation and best 𝐿₁-approximation by blending functions.

3. Algebraic Blending Functions

The linear space 𝜋_𝑚−1 of univariate polynomials of degree ≤𝑚−1 can be defined as the general solution of the homogeneous, 𝑚-order, linear differential equation 𝑝^(𝑚)(𝑥)=0,𝑥∈[−1,1]. Hence, 𝜋_𝑚−1=𝑝∈𝐶^𝑚(𝐼)∶𝑑^𝑚𝑝𝑑𝑥^𝑚=0,𝐼∶=[−1,1], (3.1) where by 𝐶^𝑚(𝐼) we denote the linear space of functions having continuous 𝑚-order derivative on [−1,1]. The linear space 𝜋_𝑚−1 is of finite dimension 𝑚.

Let 𝐼²=[−1,1]² and 𝐶^𝑚,𝑛∶=𝑓𝑥,𝑦∶𝐷^𝑖,𝑗𝑓𝑥,𝑦∈𝐶𝐼²,0≤𝑖≤𝑚,0≤𝑗≤𝑛, (3.2) where 𝐷^𝑖,𝑗𝑓(𝑥,𝑦)∶=(𝜕^𝑖+𝑗/𝜕𝑥^𝑖𝜕𝑦^𝑗)𝑓(𝑥,𝑦) is the partial derivative of 𝑓 of order (𝑖,𝑗). 𝐶^𝑚,𝑛 denotes the linear space of all real valued functions having continuous partial derivatives up to order (𝑚,𝑛) on 𝐼².

The classical univariate polynomials have a natural multivariate extension by the so-called algebraic blending functions. Let 𝑚,𝑛 be nonnegative integer such that 𝑚+𝑛≥1. The space 𝐵^𝑚,𝑛 of algebraic blending functions of order (𝑚,𝑛) on the unit square 𝐼² is defined as 𝐵^𝑚,𝑛∶=ℎ∈𝐶^𝑚,𝑛∶𝐷^𝑚,𝑛ℎ𝑥,𝑦=0,𝑥,𝑦∈𝐼². (3.3) In other words, the linear space 𝐵^𝑚,𝑛 is the general solution of the homogeneous, (𝑚,𝑛)-order, linear partial differential equation 𝐷^𝑚,𝑛ℎ(𝑥,𝑦)=0 on 𝐼² or, saying it differently, the linear space 𝐵^𝑚,𝑛 is the kernel of the linear partial differential operator 𝐷^𝑚,𝑛.

Each blending function ℎ∈𝐵^𝑚,𝑛 of order (𝑚,𝑛) can be represented in the form ℎ𝑥,𝑦=^𝑚−1_𝑘=0𝑎_𝑘𝑦𝑥^𝑘+^𝑛−1_𝑙=0𝑏_𝑙(𝑥)𝑦^𝑙,𝑎_𝑘𝑦∈𝐶^𝑛(𝐼),𝑏_𝑙(𝑥)∈𝐶^𝑚(𝐼) (3.4) and obviously, each such function belongs to 𝐵^𝑚,𝑛. Hence, 𝐵^𝑚,𝑛=ℎ∶ℎ𝑥,𝑦=^𝑚−1_𝑘=0𝑎_𝑘𝑦𝑥^𝑘+^𝑛−1_𝑙=0𝑏_𝑙(𝑥)𝑦^𝑙,𝑎_𝑘∈𝐶^𝑛(𝐼),𝑏_𝑙∈𝐶^𝑚(𝐼). (3.5) Note that the above representation of a function from 𝐵^𝑚,𝑛 is not unique and the linear space 𝐵^𝑚,𝑛 is of infinite dimension.

4. Transfinite Interpolation by Algebraic Blending Functions

In two-dimensional case, transfinite interpolation (beyond the finite interpolation) is to construct a simple interpolation function (blending interpolation function for example) over a planar domain in such a way that it matches (interpolates) a given function and its partial derivatives on curves. Transfinite interpolation is an approximate recovery of functions via interpolation with variety of applications, for example, in geometric modeling and finite element methods. The notion can be extended in a natural way for higher dimensions. The Dirichlet problem is an example of a transfinite interpolation scheme in the linear space of harmonic functions.

In contrast to the transfinite interpolation schemes, the classical interpolation schemes are restricted to a finite or denumerable number of interpolation points. For example, approximate recovery of a univariate function can be obtained by a univariate Lagrange interpolation polynomial: let 𝑓∈𝐶[−1,1] and 𝑥₁,𝑥₂,…,𝑥_𝑚 be 𝑚 distinct points in [−1,1]. Then, there exists a unique polynomial interpolant 𝑝_𝑓∈𝜋_𝑚−1 to 𝑓 of degree ≤𝑚−1 such that 𝑝_𝑓𝑥_𝑘=𝑓𝑥_𝑘,𝑘=1,2,…,𝑚 (4.1) with an error representation formula 𝑓(𝑥)−𝑝_𝑓(𝑥)=𝑓^(𝑚)(𝜉)𝑚!𝑥−𝑥₁⋯𝑥−𝑥_𝑚,𝑥∈[−1,1],𝜉∈(−1,1). (4.2) Existence and uniqueness of univariate Lagrange interpolants are closely connected with the notion of Haar-Chebyshev system in the univariate case.

4.1. Haar-Chebyshev Systems in the Multivariate Case

Let us remind that a set of 𝑚 univariate functions {𝑢₁,𝑢₂,…,𝑢_𝑚}⊂𝐶[−1,1] is a Haar-Chebyshev system (𝑇-system) of order 𝑚 in [−1,1] if each nontrivial linear combination of 𝑢₁,𝑢₂,…,𝑢_𝑚 has at most 𝑚−1 zeros in [−1,1]. In other words, a set of 𝑚 functions is a Haar-Chebyshev system of order 𝑚 in [−1,1] if it is linearly independent with respect to an arbitrary chosen set of 𝑚 distinct points in [−1,1]. The concept of a Haar-Chebyshev system (𝑇-system) is based on counting the zeros and it is essentially restricted to the univariate case. It is known [9, 10] that there is no universal Haar-Chebyshev system on any set in 𝐑^𝑑 that contains an interior point, in particular, on 𝐼². Hence, the point-wise Lagrange univariate interpolation scheme cannot be extended to the point-wise interpolation scheme in the multivariate case.

However, we can extend the notion of a Haar-Chebyshev system in the multivariate case with respect to transfinite interpolation by blending functions on grids (curves) with a prescribed geometry. Being defined as the kernel of the differential operator 𝐷^𝑚,𝑛, the linear space 𝐵^𝑚,𝑛 of algebraic blending functions of order (𝑚,𝑛) shares interpolation properties of the algebraic polynomials. In particular, Lagrange interpolation by algebraic polynomials has a multivariate extension to transfinite Lagrange interpolation by blending functions.

4.2. Transfinite Interpolation Grid

Let 𝜔_{𝐱𝐦,𝐲𝐧}𝑥,𝑦∶=^𝑚_𝑠=1𝑥−𝑥_𝑠^𝑛_𝑘=1𝑦−𝑦_𝑘. (4.3) We define an (𝑚,𝑛) interpolation grid 𝐆_𝐦,𝐧 associated with the 2 point sets of distinct in each set points 𝐱_𝐦∶={𝑥_𝑠,1≤𝑠≤𝑚} and 𝐲_𝐧∶={𝑦_𝑘,1≤𝑘≤𝑛}: 𝐆_𝐦,𝐧∶=𝑥,𝑦∈𝐼²∶𝜔_{𝐱𝐦,𝐲𝐧}𝑥,𝑦=0. (4.4) The interpolation grid 𝐆_𝐦,𝐧 is a set of 𝑚 vertical and 𝑛 horizontal line-segments in 𝐼².

4.3. Transfinite Lagrange Interpolation by Algebraic Blending Functions

Let 𝑓∈𝐶^𝑚,𝑛. Let 𝐱_𝐦∶={𝑥_𝑠,1≤𝑠≤𝑚} and 𝐲_𝐧∶={𝑦_𝑘,1≤𝑘≤𝑛} be sets of distinct points (in each case) in 𝐼. Then there exists a unique blending interpolant ℎ_𝑓∈𝐵^𝑚,𝑛 to 𝑓, satisfying the following transfinite interpolation conditions: ℎ_{𝑓∣𝐆𝑚,𝑛}=𝑓_{∣𝐆𝑚,𝑛}. (4.5) By using the univariate fundamental Lagrange interpolating polynomials 𝑙_𝑠,𝑚(𝑥),𝑠=1,…,𝑚 and 𝑙_𝑘,𝑛(𝑦),𝑘=1,…,𝑛 we give the following explicit construction of the unique transfinite Lagrange interpolant to 𝑓 on the grid 𝐆_𝑚,𝑛: ℎ_𝑓𝑥,𝑦=^𝑚_𝑠=1𝑓𝑥_𝑠,𝑦𝑙_𝑠,𝑚(𝑥)+^𝑛_𝑘=1𝑓𝑥,𝑦_𝑘𝑙_𝑘,𝑛𝑦−^𝑚𝑛_𝑠=1𝑘=1𝑓𝑥_𝑠,𝑦_𝑘𝑙_𝑠,𝑚(𝑥)𝑙_𝑘,𝑛𝑦 (4.6) with the error representation formula 𝑓𝑥,𝑦−ℎ_𝑓=𝐷𝑥,𝑦^𝑚,𝑛𝑓𝜉,𝜂𝜔𝑚!𝑛!^{𝐱𝐦,𝐲𝐧}𝑥,𝑦,𝑥,𝑦∈𝐼²,𝜉,𝜂∈𝐼². (4.7) From the above error representation formula, for each 𝑔∈𝐵^𝑚,𝑛 we have ℎ_𝑔=𝑔 to conclude that for a fixed (𝑚,𝑛) interpolation grid 𝐆_𝐦,𝐧 there exists a unique transfinite interpolant ℎ_𝑓 from 𝐵^𝑚,𝑛 to a given function 𝑓∈𝐶^𝑚,𝑛.

In addition, the univariate concept of a Haar-Chebyshev system can be extended in the infinite dimensional linear space of two-variable blending functions and interpolation grids consisting of vertical and horizontal line-segments as follows: if ℎ∈𝐵^𝑚,𝑛 satisfies ℎ_{∣𝐆𝐦,𝐧}=0, then ℎ=0 on 𝐼². In other words, an (𝑚,𝑛) blending grid cannot be a zero set of ℎ∈𝐵^𝑚,𝑛, ℎ being nonidentically zero on 𝐼².

4.4. Haar-Chebyshev Pair of a Linear Space and a Set of Grids in the Multivariate Case

In the one-dimensional Lagrange interpolation problem we have a finite set of points which are similar in the sense they obey the natural ordering on the real line. In the bivariate case we have set of grid-lines which are also similar to each other in the sense that 𝑚 of them are parallel to the one axis, and 𝑛 of them, to the other. The points where the lines intersect the coordinate axes are also naturally ordered. Each blending grid 𝐺_𝑚,𝑛 is unisolvent in transfinite Lagrange interpolation sense for 𝐵^𝑚,𝑛. On the other hand, the interpolation sets in the bivariate (multivariate) case may have diverse geometry and we can not talk about Haar-Chebyshev systems in general. However, we can argue that the pair of the linear space 𝐵^𝑚,𝑛 and the set of blending grids 𝒢_𝑚,𝑛𝐵^𝑚,𝑛,𝒢_𝑚,𝑛isaHaar-Chebyshevpairon𝐼², (4.8) where 𝒢_𝑚,𝑛 is the set of all blending grids of order (𝑚,𝑛) in 𝐼², consisting of 𝑚 vertical and 𝑛 horizontal line-segments.

4.5. Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions

Let 𝐆_𝐦,𝐧∶=𝑥,𝑦∈𝐼²∶𝜔_{𝐱𝐦,𝐲𝐧}𝑥,𝑦=0 (4.9) be (𝑚,𝑛) interpolation grid associated with the point sets 𝐱_𝐦∶={𝑥_𝜇,1≤𝜇≤𝑚} and 𝐲_𝐧∶={𝑦_𝜈,1≤𝜈≤𝑛}. The interpolation grid 𝐆_𝑚,𝑛 consists of 𝑚 vertical and 𝑛 horizontal line segments on 𝐼². We associate with each point 𝑥_𝜇 a multiplicity 𝑠_𝜇 and with each point 𝑦_𝜈 a multiplicity 𝑗_𝜈. Let ∑^𝑚_𝜇=1𝑠_𝜇=𝑀 and ∑^𝑛_𝜈=1𝑗_𝜈=𝑁. Then for a given function 𝑓∈𝐶^𝑀,𝑁, there exists a unique transfinite Lagrange-Hermite blending interpolant ℎ_𝑓 from 𝐵^𝑀,𝑁 to 𝑓, satisfying the transfinite interpolation conditions 𝐷^𝑠,0ℎ_𝑓𝑥_𝜇,𝑦=𝐷^𝑠,0𝑓𝑥_𝜇,𝑦,𝑠=0,…,𝑠_𝜇[],𝐷−1,𝜇=1,…,𝑚,𝑦∈−1,1_0,𝑗ℎ_𝑓𝑥,𝑦_𝜈=𝐷^0,𝑗𝑓𝑥,𝑦_𝜈,𝑗=0,…,𝑗_𝜈−1,𝜈=1,…,𝑛,𝑥∈[−1,1] (4.10) with respect to the given (𝑚,𝑛) blending grid 𝐆_𝐦,𝐧, where as usual 𝐷^0,0𝑓=𝑓.

Explicit construction of the transfinite Lagrange-Hermite interpolant ℎ_𝑓 to 𝑓 from 𝐵^𝑀,𝑁 is ℎ_𝑓𝑥,𝑦=^𝑚𝑠_𝜇=1^𝜇−1_𝑠=0𝐷^𝑠,0𝑓𝑥_𝜇,𝑦𝑙^𝑠_𝜇,𝑀(𝑥)+^𝑛𝑗_𝜈=1^𝜈−1_𝑗=0𝐷^0,𝑗𝑓𝑥,𝑦_𝜈𝑙^𝑗_𝜈,𝑁𝑦−^𝑚𝑠_𝜇=1^𝜇−1𝑛_𝑠=0^𝑗_𝜈=1^𝜈−1_𝑗=0𝐷^𝑠,𝑗𝑓𝑥_𝜇,𝑦_𝜈𝑙^𝑠_𝜇,𝑀(𝑥)𝑙^𝑗_𝜈,𝑁𝑦, (4.11) where the fundamental Lagrange-Hermite interpolation polynomials 𝑙^𝑠_𝜇,𝑀(𝑥),𝜇=1,…,𝑚 and 𝑙^𝑗_𝜈,𝑁(𝑦),𝜈=1,…,𝑛 satisfy the interpolation conditions 𝐷^𝑟,0𝑙^𝑠_𝜇,𝑀𝑥_𝜅=𝛿_𝜇,𝜅𝛿_𝑠,𝑟,𝜅=1,…,𝑚,𝑟=0,…,𝑠_𝜇𝐷−1,_0,𝑟𝑙^𝑗_𝜈,𝑁𝑦_𝜅=𝛿_𝜈,𝜅𝛿_𝑗,𝑟,𝜅=1,…,𝑛,𝑟=0,…,𝑗_𝜈−1. (4.12)

4.6. Error Representation of Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions in Terms of B-Splines

Let Ω_{(𝐱𝐦,𝐬𝐦);(𝐲𝐧,𝐣𝐧)}𝑥,𝑦∶=^𝑚_𝜇=1𝑥−𝑥_𝜇^𝑠𝜇𝑛_𝜈=1𝑦−𝑦_𝜈^𝑗𝜈, (4.13) where 𝐬_𝐦∶=(𝑠₁,…,𝑠_𝑚) and 𝐣_𝐧∶=(𝑗₁,…,𝑗_𝑛). Then the following error representation formula holds 𝑓𝑥,𝑦−ℎ_𝑓𝑥,𝑦=Ω_{(𝐱𝐦,𝐬𝐦);(𝐲𝐧,𝐣𝐧)}×𝑥,𝑦₁₋₁𝐵_{𝑀−1;[(𝐱𝐦,𝐬𝐦),𝑥]}(𝑣)𝐵_{𝑁−1;[(𝐲𝐧,𝐣𝐧),𝑥]}(𝑤)𝐷^𝑀,𝑁𝑓(𝑣,𝑤)𝑑𝑣𝑑𝑤, (4.14) where 𝐵_{𝑀−1;[(𝐱𝐦,𝐬𝐦),𝑥]} is the normalized 𝐵-spline of degree 𝑀−1 with knots (𝐱_𝑚,𝑥) of multiplicities (𝐬_𝐦,1) and 𝐵_{𝑁−1;[(𝐲𝐧,𝐣𝐧),𝑦]} is the normalized 𝐵-spline of degree 𝑁−1 with knots (𝐲_𝑛,𝑦) of multiplicities (𝐣_𝐧,1). For details on B-splines see [11–13]. As a corollary we obtain Cauchy error representation for transfinite Lagrange-Hermite interpolation by blending functions 𝑓𝑥,𝑦−ℎ_𝑓=𝐷𝑥,𝑦^𝑀,𝑁𝑓𝜉,𝜂Ω𝑀!𝑁!^{(𝐱𝐦,𝐬𝐦);(𝐲𝐧,𝐣𝐧)}𝑥,𝑦,𝑥,𝑦∈𝐼²,𝜉,𝜂∈𝐼². (4.15) Note that ℎ_𝑔=𝑔 for a blending function 𝑔∈𝐵^𝑀,𝑁. Therefore, each blending function from 𝐵^𝑀,𝑁 can be represented as transfinite Lagrange-Hermite interpolant on a grid consisting of horizontal and vertical line-segments.

Remark 4.1. The multivariate results which resemble the broadest extent the univariate theory concern transfinite interpolation and best 𝐿₁-approximation by algebraic blending functions. For simplicity of the notations the results on transfinite interpolation and best 𝐿₁-approximation by algebraic blending functions are stated in the bivariate case 𝑑=2 although they are entirely valid in higher dimensions.

5. Best 𝐿₁-Approximation by Algebraic Blending Functions

Let (ℱ,‖∘‖) be a normed linear space. Let 𝑈 be a subspace of ℱ. Then 𝑢_𝑓∈𝑈 is called best approximant to 𝑓∈ℱ from 𝑈 if for all 𝑢∈𝑈𝑓−𝑢_𝑓≤𝑓−𝑢. (5.1) Let 𝑓 satisfy 𝐷^𝑚,𝑛𝑓(𝑥,𝑦)>0 on 𝐼² and let ℎ∈𝐵^𝑚,𝑛. Then an (𝑚,𝑛) blending grid is a maximal set of zeros for 𝑓−ℎ in a sense that if 𝑓(𝑥,𝑦)−ℎ(𝑥,𝑦)=0 on (𝑚,𝑛) blending grid 𝐆_𝐦,𝐧, then 𝑓(𝑥,𝑦)−ℎ(𝑥,𝑦)≠0 for (𝑥,𝑦)∈𝐼²⧵𝐆_𝐦,𝐧. In other words, 𝑓−ℎ cannot vanish on 𝐆_𝐦,𝐧∪(𝑥₀,𝑦₀), where (𝑥₀,𝑦₀)∈𝐼²⧵𝐆_𝐦,𝐧.

Approximation by blending functions is useful in problems of improving the efficiency of data transfer systems, image processing, reducing the size of the table of a function of many variables, cubature formulae, and numerical solution of differential and integral equations. For results on existence of best algebraic blending 𝐿₁-approximants see [14, 15]. The above notion of a Haar-Chebyshev system in the multivariate case is essential in the proof of the next theorem (see [16] for details).

Theorem 5.1. Let 𝑓∈𝐶^𝑚,𝑛 satisfy 𝐷^𝑚,𝑛𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼². Then 𝑓 possesses a unique best 𝐿₁-approximant from 𝐵^𝑚,𝑛 that is the unique transfinite Lagrange interpolant ℎ_𝑓 to 𝑓 from 𝐵^𝑚,𝑛 with respect to the blending grid 𝐆_𝐦,𝐧=𝑥,𝑦∶^𝑚𝑛_𝑖=1𝑗=1𝑥−𝑥_𝑖𝑦−𝑦_𝑗=0, (5.2) where 𝑥_𝑘,1≤𝑘≤𝑚 and 𝑦_𝑗,1≤𝑗≤𝑛 are the zeros of the Chebyshev polynomials of second kind 𝑈_𝑚(𝑥) and 𝑈_𝑛(𝑦), respectively.

Remark 5.2. Following Theorem 5.1, we conclude that the blending grid 𝐆_𝐦,𝐧 is the canonical set of best 𝐿₁-approximation from 𝐵^𝑚,𝑛 to the convex cone 𝑓∈𝐶^𝑚,𝑛𝐼²∶𝐷^𝑚,𝑛𝑓𝑥,𝑦≥0for𝑥,𝑦∈𝐼². (5.3) Note that in the above convex cone, the non linear problem of best 𝐿₁-approximation becomes a linear one.
Corollary by Theorem 5.1 is the interesting fact that the best 𝐿₁-approximant to the polynomial 𝑥^𝑚𝑦^𝑛 from 𝐵^𝑚,𝑛 is also a polynomial, namely, 𝑈_𝑚(𝑥)𝑈_𝑛𝑦2^𝑚+𝑛_𝐿1(𝐼2)≤𝑥^𝑚𝑦^𝑛−ℎ𝑥,𝑦_𝐿1(𝐼2),ℎ∈𝐵^𝑚,𝑛 (5.4) and in particular the unique polynomial of minimal 𝐿₁-norm (minimal 𝐿₁-deviation) on 𝐼² from the class of monic polynomials {∑^𝑚_𝑖=0∑^𝑛_𝑗=0𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗∶𝑎_𝑚𝑛=1} is 2^{−𝑚−𝑛}𝑈_𝑚(𝑥)𝑈_𝑛(𝑦).

Remark 5.3. Note that in contrast to the result in Theorem 5.1, the best uniform (Chebyshev) approximant from 𝐵^1,1 to 𝑓∈𝐶^1,1 satisfying 𝐷^1,1𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼², is never unique unless 𝑓∈𝐵^1,1 (see [17] for details).

6. Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions

In bivariate case, transfinite interpolation by trigonometric blending functions is to construct a blending trigonometric interpolant over a planar domain in such a way that matches (interpolates) a given function and its partial derivatives on curves that constitute the interpolation set. For details on the results in the present section see [7].

Let 𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)² be the vector space of two-variable functions 𝑓(𝜃,𝜂) which are 2𝜋-periodic in each variable and 𝔇^{(2𝑚+1,2𝑛+1)}𝑓𝜃,𝜂∶=𝔇_𝜃^(2𝑚+1)𝔇_𝜂^(2𝑛+1)𝑓𝜃,𝜂 (6.1) is continuous on [0,2𝜋)², where 𝔇_𝜃^(2𝑚+1)=𝔇_𝜃∏^𝑚_𝑘=1(𝔇²_𝜃+𝑘²) and 𝔇_𝜃=𝜕/𝜕𝜃 is the partial derivative with respect to 𝜃. The vector space 𝐵^𝑡_𝑚,𝑛 of trigonometric blending functions of order (𝑚,𝑛) is defined as 𝐵^𝑡_𝑚,𝑛∶=ℎ_𝑡𝜃,𝜂∈𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)²∶𝔇^{(2𝑚+1,2𝑛+1)}ℎ_𝑡𝜃,𝜂=0. (6.2)

Each trigonometric blending function ℎ_𝑡∈𝐵^𝑡_𝑚,𝑛 can be represented in the form ℎ_𝑡𝜃,𝜂=^𝑚_𝑘=0𝑎_𝑘𝜂cos(𝑘𝜃)+𝑏_𝑘𝜂sin(𝑘𝜃)+^𝑛_𝑝=0𝑐_𝑝(𝜃)cos𝑝𝜂+𝑑_𝑝(𝜃)sin𝑝𝜂, (6.3) where 𝑎_𝑘,𝑏_𝑘,𝑐_𝑝, and 𝑑_𝑝 are sufficiently smooth 2𝜋-periodic functions.

Remark 6.1. Note that the above representation of ℎ_𝑡∈𝐵^𝑡_𝑚,𝑛 is not unique in contrast to the uniqueness of the corresponding representation in 𝑇_𝑛.
Let 𝜃_𝑙1∶=(𝜃₀,𝜃₁,…,𝜃_𝑙1),0≤𝜃₀<𝜃₁<⋯<𝜃_𝑙1<2𝜋 be 𝑙₁+1 interpolation nodes with positive integer multiplicities 𝜆_𝑙1∶=(𝜆₀,𝜆₁,…,𝜆_𝑙1) and let 𝜂_𝑙2∶=(𝜂₀,𝜂₁,…,𝜂_𝑙2),0≤𝜂₀<𝜂₁<⋯<𝜂_𝑙2<2𝜋 be 𝑙₂+1 interpolation nodes with positive integer multiplicities 𝛽_𝑙2∶=(𝛽₀,𝛽₁,…,𝛽_𝑙2).

6.1. Construction of Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions

Let 𝑓(𝜃,𝜂)∈𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)². Given the interpolation nodes with corresponding multiplicities (𝜃_𝑙1,𝜆_𝑙1) and (𝜂_𝑙2,𝛽_𝑙2), ∑^𝑙1_𝑖=0𝜆_𝑖=2𝑚+1 and∑^𝑙2_𝑗=0𝛽_𝑗=2𝑛+1. Find a trigonometric blending function ℎ_𝑡,𝑓∈𝐵^𝑡_𝑚,𝑛 of order (𝑚,𝑛), satisfying the following Lagrange-Hermite interpolation conditions (𝑓^(𝑖,𝑗) denoting the partial derivative 𝜕^𝑖+𝑗𝑓/𝜕𝜃^𝑖𝜕𝜂^𝑗): ℎ^(𝑖,0)_𝑡,𝑓𝜃_𝑠1,𝜂=𝑓^(𝑖,0)𝜃_𝑠1,𝜂,𝑖=0,1,…,𝜆_𝑠1−1;𝑠₁=0,1,…,𝑙₁;ℎ^(0,𝑗)_𝑡,𝑓𝜃,𝜂_𝑠2=𝑓^(0,𝑗)𝜃,𝜂_𝑠2,𝑗=0,1,…,𝛽_𝑠2−1;𝑠₂=0,1,…,𝑙₂. (6.4)

Theorem 6.2. Let 𝑓(𝜃,𝜂)∈𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)². Given the interpolation nodes with corresponding multiplicities (𝜃_𝑙1,𝜆_𝑙1) and (𝜂_𝑙2,𝛽_𝑙2) such that ∑^𝑙1_𝑖=0𝜆_𝑖=2𝑚+1 and ∑^𝑙2_𝑗=0𝛽_𝑗=2𝑛+1, then there exists a unique trigonometric blending interpolant ℎ_𝑡,𝑓 from 𝐵^𝑡_𝑚,𝑛 (of order (𝑚,𝑛)) to 𝑓 satisfying the transfinite interpolation conditions (6.4). In addition, the following point-wise error representation holds: 𝑓𝜃,𝜂−ℎ_𝑡,𝑓=𝔇𝜃,𝜂^{(2𝑚+1,2𝑛+1)}𝑓(𝜉,𝜁)[𝑚!]²[𝑛!]²𝐵₁𝜃,𝜂−ℎ_𝑡,𝐵1𝜃,𝜂, (6.5) where (𝜉(𝜃,𝜂),𝜁(𝜃,𝜂))∈(0,2𝜋)², 𝐵₁(𝜃,𝜂)∶=𝐵₁(𝜃)𝐵₁(𝜂) and 𝐵₁(𝜃)=𝜃−𝜋,𝜃∈[0,2𝜋) is the first univariate Bernoulli function.

6.2. Uniqueness of the Lagrange-Hermite Trigonometric Blending Interpolant

Suppose that there is another trigonometric blending interpolant ℎ_𝑡,𝑓∈𝐵^𝑡_𝑚,𝑛 to 𝑓 satisfying the interpolation conditions (6.4). Evidently, ℎ_𝑡,ℎ_𝑡,𝑓=ℎ_𝑡,𝑓. Therefore, by using (6.5) ℎ_𝑡,𝑓−ℎ_𝑡,ℎ_𝑡,𝑓=𝔇^{(2𝑚+1,2𝑛+1)}ℎ_𝑡,𝑓(𝜉,𝜁)[𝑚!]²[𝑛!]²𝐵₁𝜃,𝜂−ℎ_𝑡,𝐵1⇒ℎ𝜃,𝜂=0_𝑡,𝑓=ℎ_𝑡,ℎ_𝑡,𝑓=ℎ_𝑡,𝑓. (6.6)

Lemma 6.3. Let ℎ_𝑡,𝐵1(𝜃,𝜂) be the unique trigonometric blending interpolant from 𝐵^𝑡_𝑚,𝑛 to 𝐵₁(𝜃,𝜂)=𝐵₁(𝜃)𝐵₁(𝜂) satisfying the transfinite interpolation conditions (6.4) on the interpolating grid𝐆_{𝐥𝟏,𝐥𝟐}∶=𝜃,𝜂∈[0,2𝜋]²∶^𝑙1,𝑙2_{𝑠1=0,𝑠2=0}𝜃−𝜃_𝑠1𝜂−𝜂_𝑠2=0. (6.7) Then, [𝐵₁(𝜃,𝜂)−ℎ_𝑡,𝐵1(𝜃,𝜂)]≠0 for each point (𝜃,𝜂)∉𝐆_{𝐥𝟏,𝐥𝟐}.

Corollary 6.4. Let ℎ_𝑡,𝑓 be the unique trigonometric blending interpolant from 𝐵^𝑡_𝑚,𝑛 to 𝑓∈𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)² satisfying the transfinite interpolation conditions (6.4). If 𝔇^{(2𝑚+1,2𝑛+1)}𝑓(𝜉,𝜁)≠0 for (𝜉,𝜁)∈[0,2𝜋)², then 𝑓(𝜃,𝜂)−ℎ_𝑡,𝑓(𝜃,𝜂)≠0 for (𝜃,𝜂)∉𝐆_{𝐥𝟏,𝐥𝟐}.

7. Best 𝐿₁-Approximation by Trigonometric Blending Functions

Consider the normed vector space 𝐿₁[0,2𝜋]² of functions that are 2𝜋-periodic in each variable, whose absolute value has a finite integral on [0,2𝜋]² and equipped with the norm 𝑓_{𝐿1[0,2𝜋]}²∶=_[]0,2𝜋2𝑓𝜃,𝜂𝑑𝐴=Fubini^′sTheorem₀^2𝜋𝑓𝜃,𝜂𝑑𝜃𝑑𝜂, (7.1) where 𝐴 is the area measure. Denote for simplicity ‖𝑓‖₁∶=‖𝑓‖_{𝐿1[0,2𝜋]}². We restrict 𝐵^𝑡_𝑚,𝑛 to 𝐵^𝑡_𝑚,𝑛⊂𝐿₁[0,2𝜋]².

Let us consider the convex cone 𝐶₊^{2𝑚+1,2𝑛+1}∶=𝑓∈𝐶^{2𝑚+1,2𝑛+1}[0,2𝜋)²∶𝔇^{(2𝑚+1,2𝑛+1)}𝑓𝜃,𝜂≥0for𝜃,𝜂∈[0,2𝜋)². (7.2) We construct best 𝐿₁-approximants from 𝐵^𝑡_𝑚,𝑛 to functions in the convex cone 𝐶₊^{2𝑚+1,2𝑛+1}. Following the interpolation problem (6.4) let 𝜃={𝑖𝜋/(𝑚+1),𝑖=1,…,2𝑚+1} and 𝜂={𝑗𝜋/(𝑛+1),𝑗=1,…,2𝑛+1}, where all interpolation nodes are of multiplicity 1. By Theorem 6.2, there exists a unique trigonometric blending interpolant ℎ_𝑡,𝑓∈𝐵^𝑡_𝑚,𝑛 to 𝑓∈𝐶₊^{2𝑚+1,2𝑛+1} on the blending grid 𝐆_𝐦,𝐧∶=𝜃,𝜂∈[0,2𝜋]²∶^{2𝑚+1,2𝑛+1}_{𝑖=1,𝑗=1}𝜃−𝑖𝜋(𝑚+1)𝜂−𝑗𝜋(),𝑛+1=0 (7.3) consisting of 2𝑚+1 vertical and 2𝑛+1 horizontal line segments. The next theorem (see [7] for details) demonstrates that the interpolation grid 𝐆_𝐦,𝐧 is the canonical sets of best 𝐿₁-approximation from 𝐵^𝑡_𝑚,𝑛 to the convex cone 𝐶₊^{2𝑚+1,2𝑛+1}.

Theorem 7.1. The unique trigonometric blending interpolant ℎ_𝑡,𝑓∈𝐵^𝑡_𝑚,𝑛 to 𝑓∈𝐶₊^{2𝑚+1,2𝑛+1} on the blending grid 𝐆_𝐦,𝐧 is the unique best 𝐿₁-approximant to 𝑓 from 𝐵^𝑡_𝑚,𝑛.

8. Canonical Set of Best 𝐿₁-Approximation on a Triangle

Another canonical set result is presented in [18], where the domain is a triangle Δ∈𝐑². The approximating space consists of all functions that are sums of functions of the barycentric coordinates {𝜆_𝑖}³_𝑖=1 with respect to the vertices of Δ: 𝐵(Δ)∶=ℎ∈𝐶³(Δ)∶ℎ𝜆₁,𝜆₂=ℎ₁𝜆₁+ℎ₂𝜆₂+ℎ₃𝜆₃,𝜆₁+𝜆₂+𝜆₃=1,𝜆_𝑖≥0. (8.1)𝐵(Δ) is the kernel of the differential operator 𝜕³ which acts on the linear space 𝐶³(Δ) and is defined by 𝜕³𝑓𝜆₁,𝜆₂𝜕∶=𝜕𝜆₁𝜕𝜕𝜆₂𝜕𝜕𝜆₁−𝜕𝜕𝜆₂𝑓𝜆₁,𝜆₂. (8.2)

Let ℳ∶=ℳ₁∪ℳ₂∪ℳ₃ be the union of the medians {ℳ_𝑖}³_𝑖=1 in Δ. The following interpolation theorem holds true.

Theorem 8.1. Let 𝑓∈𝐶³(Δ). Then there exists a unique transfinite interpolant ℎ_𝑓∈𝐵(Δ) to 𝑓 such that ℎ_𝑓∣ℳ=𝑓_∣ℳ. (8.3) Moreover, if 𝑓∈𝐶^𝑚(Δ) for some 𝑚>3, then ℎ_𝑓∈𝐶^𝑚(Δ) as well.

On the basis of Theorem 8.1 and an appropriate error representation formula, the corresponding best 𝐿₁(Δ)-approximation is characterized in terms of a canonical point set. The convex cone here is defined by another differential operator 𝒟 (see [18, page 456] for details) that is represented in terms of 𝜕³ in a bit intricate way involving infinite series. The canonical result of best 𝐿₁-approximation reads as follows.

Theorem 8.2. Let 𝑓∈𝐶³(Δ) satisfy 𝒟𝑓≥0 on Δ. Then, the unique transfinite interpolant ℎ_𝑓 from Theorem 8.1 is the unique best 𝐿₁-approximant to the function 𝑓 from 𝐵(Δ).

Remark 8.3. By Theorem 8.2, the point set ℳ is a canonical set of best 𝐿₁-approximation on a triangle.

9. Best 𝐿₁-Approximation by Harmonic Functions

Here, we discuss results on best 𝐿₁-approximation of subharmonic functions by harmonic functions. Let 𝐵(𝑟) denote the open ball centered at 𝟎 of radius 𝑟>0 in the 𝑑-dimensional space 𝐑^𝑑 and let 𝐵∶=𝐵(1) be the unit ball centered at 𝟎 in 𝐑^𝑑 and 𝐵₀∶=𝐵(2^−1/𝑑). Denote by ℋ(𝐵) the linear space of harmonic functions on 𝐵 and let 𝒮(𝐵) denote the convex cone of subharmonic functions on 𝐵. Let 𝑓∈𝐶(𝐵) be continuous on the closed unit ball and let 𝑓∈𝒮(𝐵). A function ℎ_𝑓∈𝐶(𝐵)∩ℋ(𝐵) is called a best harmonic 𝐿₁-approximant to 𝑓 on 𝐵 if 𝑓−ℎ_{𝑓𝐿1(𝐵)}≤𝑓−ℎ_𝐿1(𝐵) (9.1) for all ℎ∈𝐶(𝐵)∩ℋ(𝐵). The next theorem gives a canonical set characterization of the best harmonic 𝐿₁-approximants to the convex cone 𝐶(𝐵)∩𝒮(𝐵) (see [19] for details).

Theorem 9.1. Let 𝑓∈𝐶(𝐵)∩𝒮(𝐵). Then the harmonic function ℎ_𝑓 is a best harmonic 𝐿₁-approximant from 𝐶(𝐵)∩ℋ(𝐵) to 𝑓 if and only if (i)ℎ_𝑓=𝑓 on 𝜕𝐵₀ (that is, ℎ_𝑓 is a transfinite harmonic interpolant to 𝑓 on 𝜕𝐵₀);(ii)ℎ_𝑓≤𝑓on𝐵⧵𝐵₀.

Remark 9.2. If a best harmonic 𝐿₁-approximant to 𝑓∈𝐶(𝐵)∩𝒮(𝐵) exists, then it is unique. However, a best harmonic 𝐿₁-approximant to an arbitrary function 𝑓∈𝐶(𝐵)∩𝒮(𝐵) need not exist. For example, 𝑔(𝑥,𝑦)∶=𝑥⁴𝑦⁴ is a subharmonic polynomial on the unit disk in 𝐑² which does not possess a best 𝐿₁-approximant from 𝐶(𝐵)∩ℋ(𝐵). However obviously, 𝐷^2,2𝑔(𝑥,𝑦)≥0 and according to Theorem 5.1, a unique best 𝐿₁-approximant on [−1,1]² to 𝑔 from 𝐵^2,2 exists. For other results on best harmonic 𝐿₁-approximation see [20, 21].

Remark 9.3. The constructive characterization of Theorem 9.1 is in terms of the Dirichlet problem on 𝜕𝐵₀. It can be considered as a transfinite interpolation problem by harmonic functions on the spherical interpolation grid 𝜕𝐵₀. The canonical set of best harmonic 𝐿₁-approximation to the convex cone 𝐶(𝐵)∩𝒮(𝐵) on the unit ball is 𝜕𝐵₀.

10. Best One-Sided 𝐿₁-Approximation by Algebraic Blending Functions

Let 𝑓∈𝐶(𝐼²). A function ℎ^∗∈𝐵^𝑚,𝑛 is called best one-sided from above 𝐿₁-approximant to 𝑓 from the linear space 𝐵^𝑚,𝑛 if ℎ^∗𝑥,𝑦≥𝑓𝑥,𝑦,𝑥,𝑦∈𝐼²,_𝐼2ℎ^∗−𝑓≤_𝐼2ℎ−𝑓 (10.1) for all ℎ∈𝐵^𝑚,𝑛(𝐼²) such that ℎ(𝑥,𝑦)≥𝑓(𝑥,𝑦),(𝑥,𝑦)∈𝐼². Analogously, we define best one-sided from below 𝐿₁-approximant to 𝑓 from 𝐵^𝑚,𝑛.

This type of one-sided 𝐿₁-approximation (with respect to a convex set rather than a subspace) has been a subject of much research activity (see [1] for details). The results in this area have mainly dealt with the case when the best approximant is from a finite-dimensional linear space. However, as we have mentioned above, the linear space 𝐵^𝑚,𝑛 of algebraic blending functions of order (𝑚,𝑛) is of infinite dimension.

First we reformulate a general result in the particular case of 𝐵^𝑚,𝑛 approximating linear space (see [22, 23] for details). It shows that the canonical sets must satisfy certain conditions.

Theorem 10.1. Let 𝑓∈𝐶(𝐼²). Let 𝒰𝑓∶=ℎ∈𝐵^𝑚,𝑛∶ℎ𝑥,𝑦≥𝑓𝑥,𝑦on𝐼². (10.2) Let ℎ^∗∈𝒰(𝑓) and let 𝑍∶=𝑍(ℎ^∗−𝑓) be the zero set of (ℎ^∗−𝑓) in 𝐼². Then, the following are equivalent.(a)The blending function ℎ^∗ is a best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^𝑚,𝑛 on 𝐼².(b)Gaussian property for the zero set 𝑍 holds _𝐼2ℎ≥0wheneverℎ∈𝐵^𝑚,𝑛,ℎ≥0on𝑍. (10.3)(c)The domain 𝐼² is a quadrature domain with respect to the point set Z and the linear space 𝐵^𝑚,𝑛; that is, there is a positive measure 𝜇 with support in 𝑍 such that _𝐼2ℎ=_𝑍ℎ𝑑𝜇 (10.4) for all ℎ∈𝐵^𝑚,𝑛.

By analogy with the canonical sets in the best 𝐿₁-approximation by blending functions one can expect that the best one-sided 𝐿₁-approximants by algebraic blending functions are Lagrange-Hermite transfinite interpolants on grids consisting of vertical and horizontal lines. However, Theorem 10.1 and the next result (see [24] for details) show that this is not the case: transfinite interpolation grids consisting of vertical and horizontal line-segments in 𝐼² cannot be canonical sets of best one-sided 𝐿₁-approximation by blending functions.

Theorem 10.2. Let 𝑚≥1 and 𝑛≥1. Then there exists an algebraic blending function ℎ∈𝐵^2𝑚,2𝑛 such that it is positive on the Legendre (𝑚,𝑛) grid 𝐆_{𝐦,𝐧,𝐋}∶=𝑥,𝑦∈𝐼²∶𝐿_𝑚(𝑥)𝐿_𝑛𝑦=0, (10.5) where 𝐿_𝑚 and 𝐿_𝑛 are the Legendre polynomials of degree 𝑚, respectively𝑛, that is, ℎ_{∣𝐆𝐦,𝐧,𝐋}>0, (10.6) but its integral on 𝐼² is negative: _𝐼2ℎ<0. (10.7)

Analogous result concerning best approximation by trigonometric blending functions is published in [7].

10.1. Best One-Sided 𝐿₁-Approximation by Algebraic 𝐵^1,1-Blending Functions

We give a constructive characterization in terms of canonical sets for the best one-sided 𝐿₁-approximant to a function 𝑓∈𝐶^1,1 satisfying 𝐷^1,1𝑓≥0 on 𝐼² from the infinite-dimensional linear space 𝐵^1,1. The best 𝐿₁-approximants are transfinite Lagrange-Hermite interpolants on the diagonals of 𝐼² as canonical sets. The occurrence of the diagonals as canonical sets of best one-sided 𝐿₁-approximation (not interpolation grids consisting of vertical and horizontal line-segments in 𝐼² !) is a fact which, to the best of our knowledge, has been first observed in [25].

Note that 𝐵^1,1=ℎ𝑥,𝑦∶ℎ𝑥,𝑦=𝑎(𝑥)+𝑏𝑦,𝑎∈𝐶¹(𝐼),𝑏∈𝐶¹(𝐼). (10.8) In other words, we approximate two-variable functions by sums of univariate functions on 𝐼².

The proof of existence, uniqueness and explicit construction of the best one-sided 𝐿₁-approximant from above and from below consists of three main steps (see [25, 26] for details).

(A) Transfinite Lagrange-Hermite Interpolation Formula on an Appropriate Grid with an Error Remainder Term

Theorem 10.3. Let 𝑓∈𝐶^1,1.
(a) The 𝐵^1,1 blending function ℎ^∗_𝑓𝑥,𝑦∶=𝑓(−1,−1)+^𝑥₋₁𝐷^1,0𝑓(𝑡,𝑡)𝑑𝑡+^𝑦₋₁𝐷^0,1𝑓(𝑡,𝑡)𝑑𝑡 (10.9) is the unique Lagrange-Hermite transfinite interpolant to 𝑓 from 𝐵^1,1 satisfying the transfinite interpolation conditions ℎ^∗_𝑓∣Δ∗=𝑓_∣Δ∗,gradℎ^∗_𝑓∣Δ∗=grad𝑓_∣Δ∗ (10.10) on the main diagonal Δ^∗ of 𝐼². Moreover, the error representation formula holds 𝑓𝑥,𝑦−ℎ^∗_𝑓𝐷𝑥,𝑦=−^1,1𝑓𝜉,𝜂2𝑥−𝑦²,𝑥,𝑦∈𝐼²,𝜉,𝜂∈𝐼². (10.11)
(b) The 𝐵^1,1 blending function ℎ_∗𝑓𝑥,𝑦∶=𝑓(−1,1)+^𝑥₋₁𝐷^1,0𝑓(𝑡,−𝑡)𝑑𝑡−^−𝑦₋₁𝐷^0,1𝑓(𝑡,−𝑡)𝑑𝑡 (10.12) is the unique Lagrange-Hermite transfinite interpolant to 𝑓 from 𝐵^1,1 satisfying the transfinite interpolation conditions ℎ_{∗𝑓∣Δ∗}=𝑓_∣Δ∗,gradℎ_{∗𝑓∣Δ∗}=grad𝑓_∣Δ∗ (10.13) on the antidiagonal Δ_∗ of I². Moreover, the following error representation formula holds: 𝑓𝑥,𝑦−ℎ_∗𝑓=𝐷𝑥,𝑦^1,1𝑓𝜌,𝜎2𝑥+𝑦²,𝑥,𝑦∈𝐼²,𝜌,𝜎∈𝐼². (10.14)

The following observation is an essential fact in the proof of the next theorem. Let 𝑓 satisfy 𝐷^1,1f(𝑥,𝑦)>0 on 𝐼² and let ℎ∈𝐵^1,1(𝐼²). If (𝑓−ℎ)=0 on Δ^∗ and (grad(𝑓−ℎ))_|Δ^∗=(0,0), then (𝑓−ℎ)≠0 on 𝐼²⧵Δ^∗. In other words, (𝑓−ℎ) can not vanish on Δ^∗∪(𝑥₀,𝑦₀), where (𝑥₀,𝑦₀)∈𝐼²⧵Δ^∗.

(B) Transfinite Cubature Formulae on Δ^∗ and Δ_∗ with 𝐵^1,1 Blending Degree of Precision

Let 𝑓∈𝐶^1,1. Then _𝐼2𝑓−2_𝐼𝑓()4𝑥,𝑥𝑑𝑥=−3𝐷^1,1𝑓𝛼,𝛽,𝛼,𝛽∈𝐼². (10.15) In particular _𝐼2ℎ=2_𝐼ℎ(𝑥,𝑥)𝑑𝑥,ℎ∈𝐵^1,1. (10.16) Analogously, _𝐼2𝑓−2_𝐼()4𝑥,−𝑥𝑑𝑥=3𝐷^1,1𝑓𝛾,𝛿,𝛾,𝛿∈𝐼². (10.17) In particular _𝐼2ℎ=2_𝐼ℎ(𝑥,−𝑥)𝑑𝑥,ℎ∈𝐵^1,1. (10.18)

(C) Constructive Characterization of the Best One-Sided 𝐿₁-Approximants from 𝐵^1,1.

Theorem 10.4. Let 𝑓∈𝐶^1,1 satisfy 𝐷^1,1𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼². Then we have the following.(a)The function 𝑓 has a unique best one-sided from above 𝐿₁-approximantfrom 𝐵^1,1. The unique best one-sided from above 𝐿₁-approximant ℎ^∗_𝑓 is characterized by the Lagrange-Hermite type transfinite interpolation conditions on the main diagonal Δ^∗ of 𝐼²: ℎ^∗_𝑓|Δ∗=𝑓_|Δ∗,gradℎ^∗_𝑓∣Δ∗=grad𝑓_∣Δ∗. (10.19)(b)The function 𝑓 has a unique best one-sided from below 𝐿₁-approximant from 𝐵^1,1. Theunique best one-sided from below 𝐿₁-approximant ℎ_∗𝑓 is characterized by the Lagrange-Hermite type transfinite interpolation conditions on the antidiagonal Δ_∗ of 𝐼²: ℎ_{∗𝑓∣Δ∗}=𝑓_∣Δ∗,gradℎ_{∗𝑓∣Δ∗}=grad𝑓_∣Δ∗. (10.20)

Proof of Theorem 10.4. We sketch the proof of (a). The proof of (b) follows the same steps. Given 𝑓∈𝐶^1,1 such that 𝐷^1,1𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼². By Theorem 10.3, ℎ^∗_𝑓≥𝑓 on 𝐼². Consider an arbitrary blending function ℎ∈𝐵^1,1 such that ℎ≥𝑓 on 𝐼². By using the transfinite cubature formula from (B), we have ℎ−𝑓=_𝐼2𝐼2ℎ−ℎ^∗_𝑓+_𝐼2ℎ^∗_𝑓−𝑓=2¹₋₁ℎ−ℎ^∗_𝑓(𝑥,𝑥)𝑑𝑥+_𝐼2ℎ^∗_𝑓≥−𝑓_𝐼2ℎ^∗_𝑓−𝑓, (10.21) taking into account that ∫¹₋₁(ℎ−ℎ^∗)(𝑥,𝑥)𝑑𝑥≥0 which follows from ℎ≥𝑓 on 𝐼² and (ℎ^∗_𝑓)_∣Δ∗=𝑓_∣Δ∗. Hence, ℎ−𝑓≥_𝐼2𝐼2ℎ^∗_𝑓−𝑓, (10.22) for each function ℎ∈𝐵^1,1 satisfying ℎ≥𝑓 on 𝐼².

Uniqueness of the Best One-Sided from above 𝐿₁-Approximant from 𝐵^1,1 to 𝑓 in the Convex Cone {𝑓∈𝐶^1,1∶𝐷^1,1𝑓(𝑥,𝑦)≥0,(𝑥,𝑦)∈𝐼²}.

Suppose that ℎ∈𝐵^1,1 is another best one-sided 𝐿₁-approximant to 𝑓 from 𝐵^1,1. It follows from (10.21) that 𝑓_∣Δ∗=(ℎ^∗_𝑓)_∣Δ∗=ℎ_∣Δ^∗. From ℎ≥𝑓 on 𝐼² we conclude that (gradℎ)_∣Δ∗=(grad𝑓)_∣Δ∗. By Theorem 10.3, ℎ=ℎ^∗_𝑓. The proof is completed.

Example 10.5. Consider 𝑓(𝑥,𝑦)=𝑥𝑦,𝐷^1,1𝑓(𝑥,𝑦)=1>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to 𝑓 from 𝐵^1,1ℎ^∗_𝑓𝑥,𝑦=𝑓(−1,−1)+^𝑥₋₁𝐷^1,0𝑓(𝑡,𝑡)𝑑𝑡+^𝑦₋₁𝐷^0,1𝑓(𝑡,𝑡)𝑑𝑡=1+^𝑥₋₁𝑡𝑑𝑡+^𝑦₋₁𝑥𝑡𝑑𝑡=²+𝑦²2 (10.23) is the unique best one-sided from above 𝐿₁-approximant from 𝐵^1,1 to 𝑓(𝑥,𝑦)=𝑥𝑦 (see Figure 1).

Figure 1 

𝑓(𝑥,𝑦)=𝑥𝑦 and ℎ∗𝑓(𝑥,𝑦).

Example 10.6. Let 𝑓(𝑥,𝑦)=𝑥^𝑝𝑦^𝑞, 𝑥≥0,𝑝>0,𝑦≥0,𝑞>0, 𝑝+𝑞=1. We compute 𝐷^1,0𝑓(𝑥,𝑦)=𝑝𝑥^𝑝−1𝑦^𝑞, 𝐷^0,1𝑓(𝑥,𝑦)=𝑞𝑥^𝑝𝑦^𝑞−1, 𝐷^1,1𝑓(𝑥,𝑦)=𝑝𝑞𝑥^𝑝−1𝑦^𝑞−1>0 for 𝑥,𝑦>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to 𝑓 from 𝐵^1,1ℎ^∗_𝑓𝑥,𝑦=𝑓(0,0)+^𝑥₀𝐷^1,0𝑓(𝑡,𝑡)𝑑𝑡+^𝑦₀𝐷^0,1𝑓()=𝑡,𝑡𝑑𝑡_𝑥0𝑝𝑑𝑡+^𝑦₀𝑞𝑑𝑡=𝑝𝑥+𝑞𝑦 (10.24) is the unique best one-sided from above 𝐿₁-approximant from 𝐵^1,1 to 𝑓(𝑥,𝑦)=𝑥^𝑝𝑦^𝑞 on [0,1]².

10.2. Best One-Sided 𝐿₁-Approximation by Algebraic 𝐵^2,1-Blending Functions

Next theorem (see [27] for details) gives characterization of the best one-sided 𝐿₁-approximants by algebraic 𝐵^2,1-blending functions in terms of transfinite Lagrange-Hermite interpolation on canonical sets.

Theorem 10.7. Let 𝑓∈𝐶^2,1 and 𝐷^2,1𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼².
(a) Let 𝐯^∗∶={(𝑥,2|𝑥|−1)∶𝑥∈[−1,1]}. (10.25) The function 𝑓 possesses a unique best one-sided from above 𝐿₁-approximant ℎ^∗ from 𝐵^2,1. The unique best one-sided from above 𝐿₁-approximant ℎ^∗_𝑓 to 𝑓 is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation conditions: ℎ^∗_{𝑓∣𝐯∗}=𝑓_∣𝐯∗,gradℎ^∗_{𝑓∣𝐯∗}=grad𝑓_∣𝐯∗. (10.26)
(b) Let 𝐯_∗∶={(𝑥,1−2|𝑥|)∶𝑥∈[−1,1]}. (10.27) The function 𝑓 possesses a unique best one-sided from below 𝐿₁-approximant ℎ_∗𝑓 from 𝐵^2,1. The unique best one-sided from below 𝐿₁-approximant ℎ_∗𝑓 to 𝑓 is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation condition: ℎ_{∗𝑓∣𝐯∗}=𝑓_∣𝐯∗,gradℎ_{∗𝑓∣𝐯∗}=grad𝑓_∣𝐯∗. (10.28)

The following two steps (see [27] for details) are essential in the proof of Theorem 10.7.

(A) Transfinite Interpolation Formulas with Remainder Term
Let 𝑓∈𝐶^2,1 and let ℎ^∗_𝑓 and ℎ_∗𝑓 be the transfinite interpolants to 𝑓 from Theorem 5.1. Then 𝑓𝑥,𝑦−ℎ^∗_𝑓𝐷𝑥,𝑦=−^2,1𝑓𝜉₁,𝜂₁|𝑥|24𝑦−2+1²𝑦+4|𝑥|+1, (10.29) where (𝑥,𝑦)∈𝐼² and (𝜉₁,𝜂₁)∈𝐼² and 𝑓𝑥,𝑦−ℎ_∗𝑓=𝐷𝑥,𝑦^2,1𝑓𝜉₂,𝜂₂|𝑥|24𝑦−2+1²−𝑦+4|𝑥|+1, (10.30) where (𝑥,𝑦)∈𝐼² and (𝜉₂,𝜂₂)∈𝐼².

(B) Transfinite Cubature Formulae on 𝐯^∗ and 𝐯_∗ with 𝐵^2,1 Blending Degree of Precision
Let 𝑓∈𝐶^2,1. Then _𝐼2𝑓=2¹₋₁𝑓(|𝑥|)1𝑥,2−1𝑑𝑥−3𝐷^2,1𝑓𝜌₁,𝜎₁,𝜌₁,𝜎₁∈𝐼²,_𝐼2𝑓=2¹₋₁𝑓(|𝑥1𝑥,1−2|)𝑑𝑥+3𝐷^2,1𝑓𝜌₂,𝜎₂,𝜌₂,𝜎₂∈𝐼². (10.31)

Example 10.8. Consider 𝑓(𝑥,𝑦)=𝑥²𝑦,   𝐷^2,1𝑓(𝑥,𝑦)=2>0 on 𝐼². Then, by Theorem 10.7, the 𝐵^2,1 transfinite blending interpolant ℎ^∗_𝑓4𝑥,𝑦∶=3𝑥²|𝑥|−𝑥²+1121+𝑦³ (10.32) to 𝑓 is the unique best one-sided from above 𝐿₁-approximant from 𝐵^2,1 to 𝑓(𝑥,𝑦)=𝑥²𝑦. Analogously, the 𝐵^2,1 transfinite interpolant ℎ_∗𝑓4𝑥,𝑦∶=−3𝑥²|𝑥|+𝑥²−1121+𝑦³ (10.33) to 𝑓 is the unique best one-sided from below 𝐿₁-approximant from 𝐵^2,1 to 𝑓(𝑥,𝑦)=𝑥²𝑦 (see Figure 2). Note that the best one-sided approximants to the polynomial 𝑓(𝑥,𝑦)=𝑥²𝑦 are not polynomials. Moreover, they have a limited 𝐶^2,1-smoothness, contrary to the unconstrained best 𝐿₁-approximation by blending functions (see Theorem 5.1).

Figure 2 

𝑓(𝑥,𝑦)=𝑥2𝑦, ℎ∗𝑓(𝑥,𝑦) and ℎ∗𝑓(𝑥,𝑦).

10.3. Best One-Sided 𝐿₁-Approximation by 𝐵^2,2-Blending Functions

Let 𝑓∈𝐶^2,2 and 𝐷^2,2𝑓(𝑥,𝑦)≥0 for (𝑥,𝑦)∈𝐼². Then (see [28] for details), the unique best one-sided from above 𝐿₁-approximant ℎ^∗_𝑓 to 𝑓 from 𝐵^2,2 is the unique transfinite Lagrange-Hermite interpolant to 𝑓 on the canonical grid 𝐱∶=𝑥,𝑦∈𝐼²∶|𝑥|=𝑦 (10.34) satisfying the transfinite interpolation conditions ℎ^∗_𝑓|𝐱=𝑓_∣𝐱,gradℎ^∗_𝑓∣𝐱=grad𝑓_∣𝐱. (10.35) For 𝑓(𝑥,𝑦) even with respect to one of the variables 𝑥 or 𝑦, the unique best one-sided from below 𝐿₁-approximant ℎ_∗𝑓 to 𝑓 from 𝐵^2,2 is the unique transfinite Lagrange-Hermite interpolant to 𝑓 on the canonical grid ⋄∶=𝑥,𝑦∈𝐼²∶|𝑥|+𝑦=1 (10.36) satisfying the transfinite interpolation conditions ℎ_{∗𝑓∣⋄}=𝑓_∣⋄,gradℎ_{∗𝑓∣⋄}=grad𝑓_∣⋄. (10.37) Surprisingly, there is no universal canonical grid for the entire convex cone 𝑓∈𝐶^2,2,𝐷^2,2𝑓𝑥,𝑦≥0,𝑥,𝑦∈𝐼² (10.38) concerning the best one-sided from below 𝐿₁-approximation from 𝐵^2,2 (see [28] for details).

The best one-sided from above 𝐿₁-approximant to 𝑓 has the smoothness of 𝑓. The best one-sided from below 𝐿₁-approximant to 𝑓 (if it exists) is a blending transfinite spline function with two line-segment knots {(𝑥,0)∶𝑥∈𝐼} and {(0,𝑦)∶𝑦∈𝐼}.

Example 10.9. Consider 𝑓(𝑥,𝑦)=𝑥²𝑦², 𝐷^2,2𝑓(𝑥,𝑦)=4>0. Then ℎ^∗_𝑓(𝑥,𝑦)∶=(𝑥⁴+𝑦⁴)/2 is the unique best one-sided from above 𝐿₁-approximant to 𝑓(𝑥,𝑦)=𝑥²𝑦² from 𝐵^2,2 and ℎ_∗𝑓1𝑥,𝑦∶=2𝑥⁴+𝑦⁴−43|𝑥|³+𝑦³+𝑥²+𝑦²−16 (10.39) is the unique best one-sided from below 𝐿₁-approximant to 𝑓(𝑥,𝑦)=𝑥²𝑦² from 𝐵^2,2 (see Figure 3). Note that ℎ_∗∉𝐶^3,0∪𝐶^0,3. Hence, the best one-sided from below 𝐿₁-approximant from 𝐵^2,2 to 𝑓 does not inherit the smoothness of 𝑓.

Figure 3 

𝑓(𝑥,𝑦)=𝑥2𝑦2, ℎ∗𝑓(𝑥,𝑦) and ℎ∗𝑓(𝑥,𝑦).

11. Best One-Sided 𝐿₁-Approximation by Quasi-Blending Functions

The characterization result of best one-sided 𝐿₁-approximation by algebraic 𝐵^2,1-blending functions (see Theorem 10.7) has a natural extension to approximation by quasi-blending functions (see [29] for details). Define the space 𝑄𝐵^𝑚,1 of all quasi-blending functions of order (𝑚,1) by 𝑄𝐵^𝑚,1∶=ℎ∈𝐶^0,1𝐼²∶𝐷^0,1ℎ⋅,𝑦∈𝜋_𝑚−1forafixed𝑦∈𝐼, (11.1) where 𝜋_𝑚−1 is the space of all univariate algebraic polynomials of degree not exceeding 𝑚−1. Any ℎ∈𝑄𝐵^𝑚,1 can be represented in the form ℎ𝑥,𝑦=𝑏(𝑥)+^𝑚−1_𝜇=0𝑎_𝜇𝑦𝑥^𝜇,𝑥,𝑦∈𝐼², (11.2) with 𝑎₀,𝑎₁,…,𝑎_𝑚−1∈𝐶¹(𝐼) and 𝑏∈𝐶⁰(𝐼). Obviously, this representation is not unique. The canonical sets 𝐯_∗ and 𝐯^∗ for the best 𝐿₁-approximation by 𝐵^2,1-blending functions (see Theorem 10.7) are examples of the so-called 𝑚-oscillating point sets.

Given 𝑚+1 points −1=𝑥₀<𝑥₁<⋯<𝑥_𝑚=1. Let 𝜂∈{−1,1}. A continuous function 𝜑∶𝐼→𝐼 is called 𝑚-oscillating if (i)𝜑(𝑥_𝜇)=𝜂(−1)^𝜇,  0≤𝜇≤𝑚;(ii)The restriction 𝜑_𝜇∶=𝜑_{|[𝑥𝜇−1,𝑥𝜇]} is a homeomorphism between [𝑥_𝜇−1,𝑥_𝜇] and 𝐼.

A point set Γ∈𝐼² is called 𝑚-oscillating if it is the graph of an 𝑚-oscillating function. The 𝑚-oscillating point sets are unisolvent for the following transfinite interpolation by quasi-blending functions.

Let 𝑚∈𝐍, 𝑓∈𝐶^𝑚,1(𝐼²), and Γ be an 𝑚-oscillating point set associated with a given 𝑚-oscillating function 𝜑. Then, there exists a unique quasi-blending function ℎ∈𝑄𝐵^𝑚,1 satisfying the following transfinite interpolation conditions: ℎ_∣Γ=𝑓_∣Γ,𝐷^0,1ℎ_∣Γ=𝐷^0,1𝑓_∣Γ. (11.3)

We define two 𝑚-oscillating point sets which turn out to be canonical sets of best 𝐿₁-approximation by quasi-blending functions from 𝑄𝐵^𝑚,1. Given 𝑚∈𝐍 and 𝑥,𝑠∈(−1,1), consider the functions 𝑞_𝑚()𝑃𝑥,𝑠∶=^{𝑘((1+𝑠)/2,−(1+𝑠)/2)}(𝑥)𝑃_𝑘^{((1+𝑠)/2,−(1+𝑠)/2)}(−𝑥),if𝑃𝑚=2𝑘,_{𝑘(−(1−𝑠)/2,−(1+𝑠)/2)}(𝑥)𝑃_𝑘^{((1−𝑠)/2,(1+𝑠)/2)}(𝑥),if𝑚=2𝑘+1, (11.4) where 𝑃_𝑛^(𝛼,𝛽) denote the corresponding Jacobi polynomials of degree 𝑛. Each polynomial 𝑞_𝑚(⋅,𝑠) has 𝑚 distinct zeros −1<𝜓₁(𝑠)<⋯<𝜓_𝑚(𝑠)<1. Moreover, it can be shown (see [29] for details) that the inverse functions 𝜑_𝜇∶=𝜓_𝜇⁻¹∶[𝑥_𝜇−1,𝑥_𝜇]→𝐼, 1≤𝜇≤𝑚 exist, are continuously differentiable, and join together to an 𝑚-oscillating function 𝜑∶𝐼→𝐼 such that the quadrature formulae _𝐼2𝑓𝑥,𝑦𝑑𝑥𝑑𝑦≈2¹₋₁𝑓𝑠,𝜑(𝑠)𝑑𝑠;_𝐼2𝑓𝑥,𝑦𝑑𝑥𝑑𝑦≈2_𝐼2𝑓𝑠,−𝜑(𝑠)𝑑𝑠 (11.5) are exact in the space 𝑄𝐵^𝑚,1. Let the 𝑚-oscillating point sets Γ_𝑚 and 𝐿_𝑚 be given by Γ_𝑚∶=𝑥,(−1)^𝑚𝜑(𝑥)∈𝐼²,𝐿∶𝑥∈𝐼_𝑚∶=𝑥,(−1)^𝑚+1𝜑(𝑥)∈𝐼²∶𝑥∈𝐼. (11.6) The next theorem shows that the sets Γ_𝑚 and 𝐿_𝑚 are canonical sets of best one-sided 𝐿₁-approximation from 𝑄𝐵^𝑚,1.

Theorem 11.1. Let 𝑓∈𝐶^𝑚,1(𝐼²) satisfy 𝐷^𝑚,1𝑓≥0 on 𝐼². Then the following holds. (a)The function f possesses a unique best one-sided 𝐿₁-approximant from above from 𝑄𝐵^𝑚,1. The best one-sided from above approximant ℎ^∗_𝑓 is characterized by the following Lagrange-Hermite transfinite interpolation conditions: (i)ℎ^∗_{𝑓∣Γ𝑚}=𝑓_∣Γ𝑚,(ii)𝐷^0,1ℎ^∗_{𝑓∣Γ𝑚}=𝐷^0,1𝑓_∣Γ𝑚. (11.7)(b)The unique best one-sided from below 𝐿₁-approximant ℎ_∗𝑓 to 𝑓 from 𝑄𝐵^𝑚,1 is characterized by the following Lagrange-Hermite transfinite interpolation conditions: (i)ℎ_{∗𝑓∣𝐿𝑚}=𝑓_∣𝐿𝑚,(ii)𝐷^0,1ℎ^∗_{𝑓∣𝐿𝑚}=𝐷^0,1𝑓_∣𝐿𝑚. (11.8)

12. Best One-Sided 𝐿₁-Approximation by Sums of Univariate Functions

Here we present a multivariate extension of Theorem 10.4 (see [30] for details). Let 𝑑≥2 and 𝐶^1,𝑑 be the linear space of all differentiable functions 𝑓, defined on the 𝑑-dimensional cube 𝐼^𝑑∶=[−1,1]^𝑑 and having continuous mixed derivatives 𝐷^1,1_𝑖,𝑗𝜕𝑓∶=²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗,1≤𝑖<𝑗≤𝑑. (12.1) We denote by 𝐵^1,𝑑 the subspace of all 𝑑-variable functions ℎ which are sums of univariate ones, that is, 𝐵^1,𝑑∶=ℎ∈𝐶^1,𝑑∶ℎ(𝐱)=ℎ𝑥₁,…,𝑥_𝑑∶=^𝑑_𝑖=1ℎ_𝑖𝑥_𝑖. (12.2) or equivalently 𝐵^1,𝑑=ℎ∈𝐶^1,𝑑∶𝐷^1,1_𝑖,𝑗ℎ=0,1≤𝑖<𝑗≤𝑑. (12.3) Let Δ^∗∶={(𝑡,…,𝑡)∈𝐼^𝑑∶𝑡∈[−1,1]} be the main diagonal of the 𝑑-dimensional cube 𝐼^𝑑. Let grad𝑓∶=𝜕𝑓𝜕𝑥₁,…,𝜕𝑓𝜕𝑥_𝑑 (12.4) denote the gradient of the 𝑑-variable function 𝑓.

12.1. Transfinite Lagrange-Hermite Interpolation to a Function 𝑓∈𝐶^1,𝑑 from 𝐵^1,𝑑 on the Diagonal Δ^∗ of 𝐼^𝑑

Theorem 12.1. Let 𝑓∈𝐶^1,𝑑. Then we have the following(a)The function ℎ^∗_𝑓∈𝐵^1,𝑑, where ℎ^∗_𝑓𝑥₁,…,𝑥_𝑑∶=𝑓(−1,…,−1)+^𝑑_𝑖=1^𝑥𝑖₋₁𝜕𝑓𝜕𝑥_𝑖(𝑡,…,𝑡)𝑑𝑡 (12.5)is the unique transfinite interpolant to 𝑓 from 𝐵^1,𝑑 satisfying the following Lagrange-Hermite transfinite interpolation conditions: ℎ^∗_𝑓∣Δ∗=𝑓_|Δ∗,gradℎ^∗_𝑓|Δ∗=grad𝑓_|Δ∗. (12.6) (b)The following error representation formula holds 𝑓(𝐱)−ℎ^∗_𝑓(𝐱)=−_{1≤𝑖<𝑗≤𝑑}𝜕²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗𝝃_𝑖,𝑗𝑥_𝑖−𝑥_𝑗²2, (12.7)

where 𝐱=(𝑥₁,…,𝑥_𝑑)∈𝐼^𝑑 and 𝝃_𝑖,𝑗∈𝐼^𝑑.

Proof. Proof of (a). Let 𝐱=(𝑥,…,𝑥)∈Δ^∗. Then (𝜕ℎ^∗_𝑓/𝜕𝑥_𝑖)(𝐱)=(𝜕𝑓/𝜕𝑥_𝑖)(𝑥,…,𝑥)=(𝜕𝑓/𝜕𝑥_𝑖)(𝑥) and, from here, (gradℎ^∗_𝑓)_|Δ∗=(grad𝑓)_|Δ∗.
Proof of (b). Let 𝐱=(𝑥₁,…,𝑥_𝑑)∈𝐼^𝑑. Without any restriction (with a permutation of the variables if necessary) we suppose 𝑥₁≤𝑥₂⋯≤𝑥_𝑑. Consider the auxiliary function 𝑔(𝐱)=𝑓(𝐱)−ℎ^∗_𝑓(𝐱). Then by (12.5), 𝑔_|Δ∗=0 and (grad𝑔)_|Δ∗=0, and in view of this, for each 𝑖,1≤𝑖≤𝑑𝜕𝑔𝜕𝑥_𝑖(𝐱)=𝜕𝑔𝜕𝑥_𝑖𝑥₁,…,𝑥_𝑑−𝜕𝑔𝜕𝑥_𝑖𝑥_𝑖,…,𝑥_𝑖=−^𝑑_{𝑗=1,𝑗≠𝑖}𝑥_𝑖−𝑥_𝑗¹₀𝜕²𝑔𝜕𝑥_𝑖𝜕𝑥_𝑗𝑥₁+𝑠𝑥_𝑖−𝑥₁,…,𝑥_𝑑+𝑠𝑥_𝑖−𝑥_𝑑𝑑𝑠. (12.8) Denote 𝐱_𝑘∶=(𝑥_𝑘,…,𝑥_𝑘,𝑥_𝑘+1,…,𝑥_𝑑),𝑘=1,…,𝑑, 𝐱₁=𝐱=(𝑥₁,…,𝑥_𝑑), 𝐱_𝑑=(𝑥_𝑑,…,𝑥_𝑑). By using the representation (12.8) for 𝜕𝑔/𝜕𝑥_𝑖 we obtain 𝑔(𝐱)=^𝑑−1_𝑘=1𝑔𝐱_𝑘−𝑔𝐱_𝑘+1=−^𝑑−1_𝑘=1^𝑥𝑘+1_𝑥𝑘𝑑𝑔𝑑𝑡𝑡,…,𝑡,𝑥_𝑘+1,…,𝑥_𝑑𝑑𝑡=−^𝑑−1_𝑘=1^𝑥𝑘+1_𝑥𝑘^𝑘_𝑖=1𝜕𝑔𝜕𝑥_𝑖𝑡,…,𝑡,𝑥_𝑘+1,…,𝑥_𝑑=𝑑𝑡_{𝑑−1𝑘𝑘=1𝑑𝑖=1𝑗=𝑘+1𝑥𝑘+1𝑥𝑘}𝑡−𝑥_𝑗¹₀𝜕²𝑔𝜕𝑥_𝑖𝜕𝑥_𝑗𝐱_{𝑠,𝑡,𝑘}𝑑𝑠𝑑𝑡, (12.9) where 𝐱_{𝑠,𝑡,𝑘}∶=(𝑡,…,𝑡,𝑥_𝑘+1+𝑠(𝑡−𝑥_𝑘+1),…,𝑥_𝑑+𝑠(𝑡−𝑥_𝑑)). Changing twice the order of summation in (12.9) and applying the integral mean value theorem we obtain 𝑔(𝐱)=^𝑑−1_𝑖=1^𝑑−1𝑑_{𝑘=𝑖𝑗=𝑘+1}^𝑥𝑘+1_𝑥𝑘𝑡−𝑥_𝑗¹₀𝜕²𝑔𝜕𝑥_𝑖𝜕𝑥_𝑗𝐱_{𝑠,𝑡,𝑘}=𝑑𝑠𝑑𝑡_{𝑑−1𝑑𝑖=1𝑗=𝑖+1𝑗−1𝑘=𝑖𝑥𝑘+1𝑥𝑘}𝑡−𝑥_𝑗¹₀𝜕²𝑔𝜕𝑥_𝑖𝜕𝑥_𝑗𝐱_{𝑠,𝑡,𝑘}=𝑑𝑠𝑑𝑡_{𝑑−1𝑑𝑖=1𝑗=𝑖+1𝑗−1𝑘=𝑖}𝑥_𝑘+1−𝑥_𝑗²2−𝑥_𝑘−𝑥_𝑗²2𝐷^1,1_𝑖,𝑗𝑔𝐱_𝑘,𝑗=−^𝑑−1𝑑_{𝑖=1𝑗=𝑖+1}𝐷^1,1_𝑖,𝑗𝑔𝝃_𝑖,𝑗𝑥_𝜉−𝑥_𝑗²2,𝜉_𝑖,𝑗∈𝐼^𝑑,1≤𝑖<𝑗≤𝑑. (12.10) This completes the proof.

12.2. Transfinite Cubature Formula, Exact in the Linear Space 𝐵^1,𝑑

Let ℎ∈𝐵^1,𝑑. Then the following transfinite cubature formula holds: ⋯_𝐼𝑑ℎ𝑥₁,…,𝑥_𝑑𝑑𝑥₁⋯𝑑𝑥_𝑑=2^𝑑−1_Δ∗ℎ(𝑡,…,𝑡)𝑑𝑡. (12.11)

Proof. Integrate (12.7) on 𝐼^𝑑 for 𝑓=ℎ∈𝐵^1,𝑑, taking into account that ℎ^∗_ℎ=ℎ if ℎ∈𝐵^1,𝑑 and the representation (12.5) of ℎ^∗_ℎ.

The next theorem gives a canonical set characterization of the best one-sided from above 𝐿₁-approximant from 𝐵^1,𝑑 to a function, belonging to the convex cone 𝒞^+,𝑑∶=𝑓∈𝐶^1,𝑑∶𝐷^1,1_𝑖,𝑗𝑓≥0on𝐼²;1≤𝑖<𝑗≤𝑑. (12.12)

Theorem 12.2. Let 𝑓∈𝒞^+,𝑑. Then the unique transfinite Lagrange-Hermite interpolant ℎ^∗_𝑓 from 𝐵^1,𝑑 to 𝑓 satisfying the transfinite interpolation conditions ℎ^∗_𝑓∣Δ∗=𝑓_|Δ∗,gradℎ^∗_𝑓|Δ∗=grad𝑓_|Δ∗ (12.13) is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^1,𝑑.

Proof. By (12.7) we conclude that 𝑓(𝐱)≤ℎ^∗_𝑓(𝐱),𝐱∈𝐼^𝑑. Taking into account the cubature (12.11) we conclude by Theorem 10.1 that the transfinite interpolant ℎ^∗_𝑓 to 𝑓 is the best one-sided from above 𝐿₁-approximant to 𝑓∈𝒞^+,𝑑 from 𝐵^1,𝑑.
Uniqueness of the Best One-Sided from above 𝐋_𝟏-Approximant from 𝐁^𝟏,𝐝 to 𝓒^+,𝐝.
Suppose ℎ∈𝐵^1,𝑑, ℎ≥𝑓 on 𝐼^𝑑 is another best one-sided from above 𝐿₁-approximant to 𝑓∈𝒞^+,𝑑 from 𝐵^1,𝑑. Then, by using (12.6) and (12.11) we obtain 0=_𝐼𝑑ℎ^∗_𝑓−𝑓−_𝐼𝑑ℎ−𝑓=_𝐼𝑑ℎ^∗_𝑓−ℎ=2^𝑑−1_Δ∗ℎ^∗_𝑓−ℎ=2^𝑑−1_Δ∗𝑓−ℎ≤0. (12.14) Hence, ℎ_|Δ^∗=𝑓_|Δ∗. However, ℎ≥𝑓 on 𝐼^𝑑 and from here, (𝑔𝑟𝑎𝑑ℎ)_|Δ^∗=(𝑔𝑟𝑎𝑑𝑓)_|Δ^∗. By Theorem 12.1 we conclude that ℎ=ℎ^∗_𝑓. The proof is completed.

Remark 12.3. According to Theorem 12.2, the set Δ^∗ is the canonical point set of best one-sided from above 𝐿₁-approximation from 𝐵^1,𝑑 to the convex cone 𝒞^+,𝑑.
Denote by 𝐸^𝑑 the set 𝐸^𝑑∶=𝜺=𝜀₁,…,𝜀_𝑑,𝜀_𝑖=±1,𝑖=1,…,𝑑 (12.15) of all 𝑑-dimensional vectors with entries ±1. For a fixed 𝜀∈𝐸^𝑑 consider the 𝜀-diagonal of 𝐼^𝑑: Δ_𝜺∶={(𝜀₁𝑡,…,𝜀_𝑑𝑡)∶𝑡∈[−1,1]}. By using the linear transformation 𝑢_𝑖=𝜀_𝑖𝑥_𝑖,𝑖=1,…,𝑑, the following corollary by Theorems 12.1 and 12.2 holds.

Corollary 12.4. Let, for a fixed vector 𝜺=(𝜀₁,…,𝜀_𝑑)∈𝐸^𝑑, the function 𝑓∈𝐶^1,𝑑 satisfy 𝜀_𝑖𝜀_𝑗𝐷^1,1_𝑖,𝑗𝑓≥0on𝐼^𝑑,1≤𝑖<𝑗≤𝑑. (12.16) Then the unique Lagrange-Hermite interpolant ℎ^∗_𝜺,𝑓𝑥₁,…,𝑥_𝑑=𝑓−𝜀₁,…,−𝜀_𝑑+^𝑑_𝑖=1𝜀_𝑖^{𝜀𝑖𝑥𝑖}₋₁𝜕𝑓𝜕𝑥_𝑖𝜀₁𝑡,…,𝜀_𝑑𝑡𝑑𝑡 (12.17) from 𝐵^1,𝑑 to 𝑓, satisfying the transfinite interpolation conditions ℎ^∗_{𝜺,𝑓|Δ𝜺}=𝑓_|Δ𝜺,gradℎ^∗_{𝜺,𝑓|Δ𝜺}=grad𝑓_|Δ𝜺, (12.18) is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^1,𝑑.

Remark 12.5. Theorems 12.1 and 12.2 are common basis for well-known classical and new inequalities. For example, the inequalities 9, 13, 16, 25, and 61 published in [31] are corollaries from the explicit constructions (12.5) and (12.7) of the best one-sided from above 𝐿₁-approximants to appropriately chosen functions (see [30] for details). More precisely, the right-hand side expressions of these inequalities are best one-sided from above 𝐿₁-approximants to the left-hand side ones in the sense of Theorems 12.1 and 12.2. We give some examples.
For a set of 𝑑 numbers {𝑥₁,…,𝑥_𝑑} denote 𝑥_∗∶=min{𝑥₁,…,𝑥_𝑑}, 𝑥^∗∶=max{𝑥₁,…,𝑥_𝑑}.

Example 12.6. Consider 𝑓(𝑥₁,…,𝑥_𝑑)=𝑥^𝑝1₁⋯𝑥^𝑝𝑑_𝑑, where 𝑥_𝑖>0,𝑝_𝑖>0,𝑖=1,…,𝑑 and ∑^𝑑_𝑖=1𝑝_𝑖=1. We compute 𝜕𝑓𝜕𝑥_𝑖(𝐱)=𝑝_𝑖𝑥^𝑝𝑖_𝑖^𝑑−1_{𝑗=1,𝑗≠𝑖}𝑥^𝑝𝑗_𝑗;𝜕²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗(𝐱)=𝑝_𝑖𝑥^𝑝𝑖_𝑖⁻¹𝑝_𝑗𝑥^𝑝𝑗_𝑗^𝑑−1_{𝑘=1,𝑘≠𝑖,𝑗}𝑥^𝑝𝑘_𝑘>0 (12.19) for 𝑥_𝑖>0,𝑖=1,…,d. Then ℎ^∗_𝑓𝑥₁,…,𝑥_𝑑=𝑓𝑥_∗,…,𝑥_∗+^𝑑_𝑖=1^𝑥𝑖_𝑥∗𝜕𝑓𝜕𝑥_𝑖()=𝑡,…,𝑡𝑑𝑡_{𝑑𝑖=1𝑥𝑖0}𝑝_𝑖𝑡^{𝑝𝑖𝑑−1}_{𝑗=1,𝑗≠𝑖}𝑡^𝑝𝑗=𝑑𝑡_{𝑑𝑖=1𝑥𝑖0}𝑝_𝑖𝑑𝑡=^𝑑_𝑖=1𝑝_𝑖𝑥_𝑖 (12.20) is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^1,𝑑 on [𝑥_∗,𝑥^∗]^𝑑. Hence, 𝑥^𝑝1₁⋯𝑥^𝑝𝑑_𝑑≤𝑝₁𝑥₁+⋯+𝑝_𝑑𝑥_𝑑 (12.21) with the case of equality only for 𝑥₁=⋯=𝑥_𝑑. The inequality is easily extended to 𝑥_𝑖≥0,𝑖=1,…,𝑑. This is the well-known inequality between the geometric mean and the arithmetic mean of a set of nonnegative numbers.

Example 12.7. Let 𝑓(𝑡) be a univariate function satisfying 𝑓^′(𝑡)+𝑡𝑓^′′(𝑡)>0 for 𝑡>0. Then, for the 𝑑-variable function 𝑓(𝑥₁⋯x_𝑑),𝑥_𝑖>0,𝑖=1,…,𝑑 we compute 𝜕𝑓𝜕𝑥_𝑖(𝐱)=^𝑑_{𝑗=1,𝑗≠𝑖}𝑥_𝑗𝑓^′𝑥₁⋯𝑥_𝑑𝜕²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗(𝐱)=^𝑑_{𝑘=1,𝑘≠𝑖,𝑗}𝑥_𝑘𝑓^′𝑥₁⋯𝑥_𝑑+^𝑑_{𝑘=1,𝑘≠𝑖}𝑥_𝑘^𝑑_{𝑠=1,𝑠≠𝑗}𝑥_𝑠𝑓^′′𝑥₁⋯𝑥_𝑑=^𝑑_{𝑘=1,𝑘≠𝑖,𝑗}𝑥_𝑘𝑓^′𝑥₁⋯𝑥_𝑑+𝑥₁⋯𝑥_𝑑𝑓^′′𝑥₁⋯𝑥_𝑑>0 (12.22) for 𝑥_𝑖>0,𝑖=1,…,𝑑. In view of Theorems 12.1 and 12.2 the unique transfinite interpolant to 𝑓 from 𝐵^1,𝑑 on [𝑥_∗,𝑥^∗]²ℎ^∗_𝑓𝑥₁,…,𝑥_𝑑=𝑓𝑥_∗+^𝑑_𝑖=1^𝑥𝑖_𝑥∗𝜕𝑓𝜕𝑥_i()𝑥𝑡,…,𝑡𝑑𝑡=𝑓_∗+^𝑑_𝑖=1^𝑥𝑖_𝑥∗𝑡^𝑑−1𝑓^′𝑡^𝑑1𝑑𝑡=𝑑^𝑑_𝑖=1𝑓𝑥^𝑑_𝑖 (12.23) is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^1,𝑑 on [𝑥_∗,𝑥^∗]^𝑑. Hence, 𝑓𝑥₁⋯𝑥_𝑑≤1𝑑^𝑑_𝑖=1𝑓𝑥^𝑑_𝑖 (12.24) with the case of equality only for 𝑥₁=⋯=𝑥_𝑑.

Example 12.8. Let 𝑓(𝑡)=(𝛽+𝑡)^𝛼,𝛽≥0,𝛼>0,𝑡>0. We compute 𝑓^′(𝑡)+𝑡𝑓^′′(𝑡)=𝛽+𝑡^𝛼−2𝛼𝛽+𝛼𝑡>0. (12.25) Then as a corollary by Example 12.7, we obtain the inequality 𝛽+𝑥₁⋯𝑥_𝑑^𝑎≤1𝑑^𝑑_𝑖=1𝛽+𝑥^𝑑_𝑖^𝛼,𝑥_𝑖>0,𝑖=1,…,𝑑 (12.26) with the case of equality only for 𝑥₁=⋯=𝑥_𝑑. Obviously, the inequality holds for 𝑥_𝑖≥0,𝑖=1,…,𝑑.

Example 12.9. Let 𝑓(𝑡)=sin(𝑡),𝑡≥0. We compute 𝑓^′(𝑡)+𝑡𝑓^′′(𝑡)=cos(𝑡)−𝑡sin(𝑡)>0 for 𝑡∈(0,𝑡_∗), where 𝑡_∗≈0.8603336 is the unique solution of the nonlinear equation cot(𝑡)=𝑡,𝑡∈[0,𝜋/2]. Then, following Example 12.7, we obtain the inequality sin𝑥₁⋯𝑥_𝑑≤1𝑑^𝑑_𝑖=1sin𝑥^𝑑_𝑖,0<𝑥_𝑖<𝑡_∗^1/𝑑,1≤𝑖≤𝑑 (12.27) with the case of equality only for 𝑥₁=⋯=𝑥_𝑑. Obviously the inequality holds for 0≤𝑥_𝑖≤𝑡_∗^1/𝑑,1≤𝑖≤𝑑.

Example 12.10. Let the univariate function 𝑓(𝑡) satisfy 2𝑓^′(𝑡)+𝑡𝑓^′′(𝑡)>0 for 𝑡>0. Consider the 𝑑-variable function 𝑓𝑥₁,…,𝑥_𝑑∶=_{1≤𝑖<𝑗≤𝑑}𝑓1𝑥_𝑖+𝑥_𝑗,𝑥_𝑖>0,1≤𝑖≤𝑑. (12.28) Computing partial derivatives of 𝑓 we obtain 𝜕𝑓𝜕𝑥_𝑖=−^𝑑_{𝑗=1,𝑗≠𝑖}1𝑥_𝑖+𝑥_𝑗²𝑓^′1𝑥_𝑖+𝑥_𝑗𝜕²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗=2𝑥_𝑖+𝑥_𝑗³𝑓^′1𝑥_𝑖+𝑥_𝑗+1𝑥_𝑖+𝑥_𝑗⁴𝑓^′′1𝑥_𝑖+𝑥_𝑗=1𝑥_𝑖+𝑥_𝑗³2𝑓^′1𝑥_𝑖+𝑥_𝑗+1𝑥_𝑖+𝑥_𝑗𝑓^′′1𝑥_𝑖+𝑥_𝑗>0, (12.29) where 1≤𝑖<𝑗≤𝑑. In view of Theorems 12.1 and 12.2, the transfinite interpolant to 𝑓 from 𝐵^1,𝑑 on [𝑥_∗,𝑥^∗]^𝑑ℎ^∗_𝑓𝑥₁,…,𝑥_𝑑=𝑓𝑥_∗,…,𝑥_∗+^𝑑_𝑖=1𝜕𝑓𝜕𝑥_𝑖()=𝑑()𝑡,…,𝑡𝑑𝑡𝑑−12𝑓12𝑥_∗−(𝑑−1)^𝑑_𝑖=1^𝑥𝑖_𝑥∗14𝑡²𝑓^′1=2𝑡𝑑𝑡𝑑−12^𝑑_𝑖=1𝑓12𝑥_𝑖 (12.30) is the unique best one-sided from above 𝐿₁-approximant to 𝑓 from 𝐵^1,𝑑. Hence, the following inequality holds: _{1≤𝑖<𝑗≤𝑑}𝑓1𝑥_𝑖+𝑥_𝑗≤𝑑−12^𝑑_𝑖=1𝑓12𝑥_𝑖 (12.31) with the case of equality only for 𝑥₁=⋯=𝑥_𝑑. In the particular cases 𝑓(𝑡)=𝑡 and 𝑓(𝑡)=ln(𝑡) we obtain the inequalities _{1≤𝑖<𝑗≤𝑑}1𝑥_𝑖+𝑥_𝑗≤𝑑−14^𝑑_𝑖=11𝑥_𝑖;_{1≤𝑖<𝑗≤𝑑}1ln𝑥_𝑖+𝑥_𝑗≤𝑑−12^𝑑_𝑖=11ln2𝑥_𝑖 (12.32) with the case of equalities only for 𝑥₁=⋯=𝑥_𝑑.

Acknowledgment

Thanks are due to the anonymous referees for valuable comments and suggestions which improved the quality of the manuscript.