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Canonical Sets of Best -Approximation
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 -approximation with emphasis on multivariate interpolation and best -approximation by blending functions. The best -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 -approximation by sums of univariate functions. Explicit constructions of best one-sided -approximants give rise to well-known and new inequalities.
1. Canonical Interpolation Sets of Best -Approximation
We start with the notion of a canonical set of best -approximation. Let be a positive Borel measure defined on the compact set such that the quantity represents a norm in the linear space of functions that are continuous on . Let be a linear subspace of .
1.1. The Problem of Best -Approximation
Given a function , then we have the following.(A)Best -approximation of by elements of : find a function such that (B)Best one-sided from above -approximation by elements of : find a function , on , such that (C)Best one-sided from below -approximation by elements of : find a function , on , such that Any solution of the approximation problem (A) is called a best -approximant to from . Any solution of the approximation problem (B) is called a best one-sided from above -approximant to from ; respectively from below for the approximation problem (C). When is the usual Lebesgue measure, we use the notation -approximation, respectively, -approximant.For qualitative results on best -approximation from finite-dimensional subspaces see .
1.2. Lagrange Interpolation Problem
Given a function . We call a subset unisolvent for if the interpolation problem possesses a unique solution for every .
1.3. Canonical Sets of Best -Approximation
Let be a class of functions. We say that a set , which is unisolvent for , is a canonical set of best -approximation to , if for all the solution of the interpolation problem (1.4) is a best -approximant to from . Analogously, a unisolvent set for (resp., ) is called a canonical set of best one-sided from above (resp., from below) -approximation to , if for all the solution of the interpolation problem (1.4) with replaced by (resp., ) is a best one-sided from above (resp., from below) -approximant to from . In the case of one-sided -approximation the interpolation problem (1.4) is considered in a broader Lagrange-Hermite interpolation sense: for all points which are interior for we require matching not only the values of and but also the values of their first (partial) derivatives.
2. Canonical Sets of Univariate Best -Approximation
For details on the results in the present section see [1–6]. In the univariate case, when the set is an interval and the subspace 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  for details).
2.1. Haar-Chebyshev System of Order
A set of functions is a Haar-Chebyshev system (-system) of order on the interval if each nontrivial linear combination of has at most zeros in . In other words, a set of functions is a Haar-Chebyshev system of order on the interval if it is linearly independent with respect to an arbitrary chosen set of distinct points in . We say that an -dimensional subspace of is a Haar-Chebyshev space (T-space) of order on if every , , has no more than distinct zeros in . Equivalently, is a -space of order on , if for every basis of there exists an such that for all . It is also often said that the basis functions constitute a -space of order on .
2.2. Characterization of Best -Approximation by Canonical Interpolation Sets
Following , throughout this section we suppose that is a finite positive nonatomic measure. The problem of best -approximation by a Haar-Chebyshev space has an elegant solution via interpolation on a canonical set. Let be a Haar-Chebyshev space of order and let be a basis of . Consider the convex cone defined as for all .
Theorem 2.1. Let be a Haar-Chebyshev space of order on . Then the following holds true.(a)Uniqueness of the Canonical Set. There is a unique set of points such that (b)Best -Approximation via Interpolation. Let . Then the unique solution of the interpolation problem is the unique best -approximant to from .
Remark 2.2. Following Theorem 2.1, the set is a canonical set of best -approximation to the convex cone of functions from the -space . In many cases the approximating subspace can be a kernel of a certain differential operator , that is, and the convex cone of functions to be approximated can be defined by
Remark 2.3. Note that if a canonical set of best -approximation exists for a functional set, then the nonlinear problem of best -approximation becomes a linear interpolation problem in this functional set.
2.3. Best -Approximation by Algebraic Polynomials
Denote by the linear space of polynomials of degree . Taking into account that the polynomial basis of is a -system of order on any interval , the best polynomial -approximant to a given function from can be characterized as a polynomial interpolant to with respect to a canonical set if 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 and for . Then, the polynomial of degree is the unique polynomial of best -approximation to from on if and only if where the interpolation nodes , are the zeros of the -degree Chebyshev polynomial of second kind
Remark 2.5. By Theorem 2.4, the interpolation points form a canonical set of best -approximation from to the convex cone
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 -approximation depends on the degree of the polynomial approximant.
Corollary 2.6. The -degree polynomial is the unique monic (with leading coefficient 1) polynomial of degree which has a minimal -norm (minimal -deviation) on .
Theorem 2.7. Let be such that the set is a Haar-Chebyshev system of order on . Then the unique polynomial Lagrange interpolant to from with respect to the zeros of the -degree Chebyshev polynomial of second kind is the unique best -approximant to from .
Remark 2.8. In the particular case and of Theorem 2.4, the set of functions is a Haar-Chebyshev system of order in . Hence in this particular case, Theorem 2.4 is a corollary by Theorem 2.7. The notion of Haar-Chebyshev system implies that cannot have more than zeros on , where .
2.4. Best -Approximation by Trigonometric Polynomials
Let be the linear space of trigonometric polynomials of degree . Consider the normed linear space of -periodic functions whose absolute value has a finite integral on and equipped with the -norm The linear space is a finite dimensional subspace of . For a given , there exists a best -approximant (see  for details). Note that is a Haar-Chebyshev space of order on . Let denote the linear space of -periodic functions having continuous derivatives of order on . The next theorem  is a canonical set characterization of the best trigonometric -approximants from to functions from the convex cone where the differential operator is defined as
Theorem 2.9. Let . Then, the unique Lagrange trigonometric interpolant from to at the interpolation nodes is the unique best -approximant to from .
2.5. Characterization of Best One-Sided -Approximation
Best one-sided -approximation is related to the principal representations of the measure , 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 be a Haar-Chebyshev space of order .(a)If , then there exist unique sets of points and , and unique sets of positive numbers and such that the quadrature formulae are exact for all .(b)If , then there exist unique sets of points and , and unique sets of positive numbers and such that the quadrature formulae are exact for all .
Let be a basis of such that for . Hence, the set of functions is a Haar-Chebyshev system (-system) of order on or in other words is an -order -space. The following theorem holds true.
Theorem 2.10. Let be a -space of order . Then, for every there exist elements of best one-sided -approximation from below and from above . Moreover, the best one-sided approximants , to and the function satisfy (a) and , if , and(b) and , if .
In order to guarantee the uniqueness of the best one-sided -approximant to a given function we need the following restriction on .
2.5.2. Extended Haar-Chebyshev System of Multiplicity 2
A set of functions is an -order extended Haar-Chebyshev system of multiplicity 2 (-order ET-system of multiplicity 2) on the interval if each nontrivial linear combination of has at most zeros in , provided that the common zeros of and are counted twice (counting multiplicities 2). In other words, is an -order extended Haar-Chebyshev space of multiplicity 2 (-order ET-space of multiplicity 2) on the interval , if , and each nontrivial has at most zeros in , provided that the common zeros of and are counted twice (counting multiplicities 2).
2.5.3. Construction of Best One-Sided -Approximants via Interpolation on Canonical Sets
Theorem 2.11. Let be an -order ET-space of multiplicity 2 on the interval . Then, for every one has the following.(a)If , then the unique solutions and of the Lagrange-Hermite interpolation problems are the unique best one-sided from below, respectively, from above -approximants to from .(b)If , then the unique solutions and of the Lagrange-Hermite interpolation problems are the unique best one-sided from below, respectively, from above -approximants to from .
Theorem 2.12. Let be a differentiable, -periodic function and let denote the linear space of all trigonometric polynomials of degree at most . Then, the best one-sided -approximant from above, respectively, from below to from exists and is unique.
Remark 2.13. The set of trigonometric polynomials , where are best one-sided from below -approximants to the continuous -periodic function from . Hence, continuity is not enough for uniqueness of the best one-sided -approximant to be claimed.
The following theorem (see  for details) gives a characterization of the best one-sided -approximants via canonical sets. It can be considered as a refinement of Theorem 2.12 for functions from the convex cone .
Theorem 2.14. Let . Let be the unique interpolant from to at the interpolation nodes with multiplicities and . The interpolant to from is the unique best one-sided from below -approximant to from .
Let be the unique interpolant from to at the interpolation nodes with multiplicities and . The interpolant to from is the unique best one-sided from above -approximant to from .
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 where the functions are defined is an interval.(2)The approximating space in most of the cases is a kernel of a linear ordinary differential operator , that is, .(3)The set on which certain interpolation problem is unisolvent for consists of a finite number of points.(4)The convex cone of functions to be approximated is defined by (5)The best (one-sided) -approximants are characterized as Lagrange-Hermite interpolants on canonical sets.
Conditions – are due to the fact that is a -space or -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 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 should be nondenumerable point sets, and more precisely, -dimensional manifolds. Hence, an appropriate transfinite interpolation on lower dimensional manifolds can be a basis for a canonical set characterization of the best -approximants in the multivariate case. However, unlike the finite number of points in an interval 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 consisting of an approximation space and a set such that is unisolvent for . Such sets are candidates for canonical sets in the multivariate -approximation. There is no consistent general treatment of the canonical sets of best approximation in 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 -approximation by blending functions.
3. Algebraic Blending Functions
The linear space of univariate polynomials of degree can be defined as the general solution of the homogeneous, -order, linear differential equation . Hence, where by we denote the linear space of functions having continuous -order derivative on . The linear space is of finite dimension .
Let and where is the partial derivative of of order . denotes the linear space of all real valued functions having continuous partial derivatives up to order on .
The classical univariate polynomials have a natural multivariate extension by the so-called algebraic blending functions. Let be nonnegative integer such that . The space of algebraic blending functions of order on the unit square is defined as In other words, the linear space is the general solution of the homogeneous, -order, linear partial differential equation on or, saying it differently, the linear space is the kernel of the linear partial differential operator .
Each blending function of order can be represented in the form and obviously, each such function belongs to . Hence, Note that the above representation of a function from is not unique and the linear space 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 and be distinct points in . Then, there exists a unique polynomial interpolant to of degree such that with an error representation formula 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 univariate functions is a Haar-Chebyshev system (-system) of order in if each nontrivial linear combination of has at most zeros in . In other words, a set of functions is a Haar-Chebyshev system of order in if it is linearly independent with respect to an arbitrary chosen set of distinct points in . The concept of a Haar-Chebyshev system (-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 that contains an interior point, in particular, on . 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 , the linear space of algebraic blending functions of order 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 We define an interpolation grid associated with the 2 point sets of distinct in each set points and : The interpolation grid is a set of vertical and horizontal line-segments in .
4.3. Transfinite Lagrange Interpolation by Algebraic Blending Functions
Let . Let and be sets of distinct points (in each case) in . Then there exists a unique blending interpolant to , satisfying the following transfinite interpolation conditions: By using the univariate fundamental Lagrange interpolating polynomials and we give the following explicit construction of the unique transfinite Lagrange interpolant to on the grid : with the error representation formula From the above error representation formula, for each we have to conclude that for a fixed interpolation grid there exists a unique transfinite interpolant from to a given function .
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 satisfies , then on . In other words, an blending grid cannot be a zero set of , being nonidentically zero on .
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 of them are parallel to the one axis, and of them, to the other. The points where the lines intersect the coordinate axes are also naturally ordered. Each blending grid is unisolvent in transfinite Lagrange interpolation sense for . 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 and the set of blending grids where is the set of all blending grids of order in , consisting of vertical and horizontal line-segments.
4.5. Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions
Let be interpolation grid associated with the point sets and . The interpolation grid consists of vertical and horizontal line segments on . We associate with each point a multiplicity and with each point a multiplicity . Let and . Then for a given function , there exists a unique transfinite Lagrange-Hermite blending interpolant from to , satisfying the transfinite interpolation conditions with respect to the given blending grid , where as usual .
Explicit construction of the transfinite Lagrange-Hermite interpolant to from is where the fundamental Lagrange-Hermite interpolation polynomials and satisfy the interpolation conditions
4.6. Error Representation of Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions in Terms of B-Splines
Let where and . Then the following error representation formula holds where is the normalized -spline of degree with knots of multiplicities and is the normalized -spline of degree with knots of multiplicities . For details on B-splines see [11–13]. As a corollary we obtain Cauchy error representation for transfinite Lagrange-Hermite interpolation by blending functions Note that for a blending function . Therefore, each blending function from 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 -approximation by algebraic blending functions. For simplicity of the notations the results on transfinite interpolation and best -approximation by algebraic blending functions are stated in the bivariate case although they are entirely valid in higher dimensions.
5. Best -Approximation by Algebraic Blending Functions
Let be a normed linear space. Let be a subspace of . Then is called best approximant to from if for all Let satisfy on and let . Then an blending grid is a maximal set of zeros for in a sense that if on blending grid , then for . In other words, cannot vanish on , where .
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 -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  for details).
Theorem 5.1. Let satisfy for . Then possesses a unique best -approximant from that is the unique transfinite Lagrange interpolant to from with respect to the blending grid where and are the zeros of the Chebyshev polynomials of second kind and , respectively.
Remark 5.2. Following Theorem 5.1, we conclude that the blending grid is the canonical set of best -approximation from to the convex cone
Note that in the above convex cone, the non linear problem of best -approximation becomes a linear one.
Corollary by Theorem 5.1 is the interesting fact that the best -approximant to the polynomial from is also a polynomial, namely, and in particular the unique polynomial of minimal -norm (minimal -deviation) on from the class of monic polynomials is .
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 .
Let be the vector space of two-variable functions which are -periodic in each variable and is continuous on , where and is the partial derivative with respect to . The vector space of trigonometric blending functions of order is defined as
Each trigonometric blending function can be represented in the form where , and are sufficiently smooth -periodic functions.
Remark 6.1. Note that the above representation of is not unique in contrast to the uniqueness of the corresponding representation in .
Let be interpolation nodes with positive integer multiplicities and let be interpolation nodes with positive integer multiplicities .
6.1. Construction of Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions
Let . Given the interpolation nodes with corresponding multiplicities and , and. Find a trigonometric blending function of order , satisfying the following Lagrange-Hermite interpolation conditions ( denoting the partial derivative ):
Theorem 6.2. Let . Given the interpolation nodes with corresponding multiplicities and such that and , then there exists a unique trigonometric blending interpolant from (of order ) to satisfying the transfinite interpolation conditions (6.4). In addition, the following point-wise error representation holds: where , and is the first univariate Bernoulli function.
6.2. Uniqueness of the Lagrange-Hermite Trigonometric Blending Interpolant
Lemma 6.3. Let be the unique trigonometric blending interpolant from to satisfying the transfinite interpolation conditions (6.4) on the interpolating grid Then, for each point .
Corollary 6.4. Let be the unique trigonometric blending interpolant from to satisfying the transfinite interpolation conditions (6.4). If for , then for .
7. Best -Approximation by Trigonometric Blending Functions
Consider the normed vector space of functions that are -periodic in each variable, whose absolute value has a finite integral on and equipped with the norm where is the area measure. Denote for simplicity . We restrict to .
Let us consider the convex cone We construct best -approximants from to functions in the convex cone . Following the interpolation problem (6.4) let and , where all interpolation nodes are of multiplicity 1. By Theorem 6.2, there exists a unique trigonometric blending interpolant to on the blending grid consisting of vertical and horizontal line segments. The next theorem (see  for details) demonstrates that the interpolation grid is the canonical sets of best -approximation from to the convex cone .
Theorem 7.1. The unique trigonometric blending interpolant to on the blending grid is the unique best -approximant to from .
8. Canonical Set of Best -Approximation on a Triangle
Another canonical set result is presented in , where the domain is a triangle . The approximating space consists of all functions that are sums of functions of the barycentric coordinates with respect to the vertices of : is the kernel of the differential operator which acts on the linear space and is defined by
Let be the union of the medians in . The following interpolation theorem holds true.
Theorem 8.1. Let . Then there exists a unique transfinite interpolant to such that Moreover, if for some , then as well.
On the basis of Theorem 8.1 and an appropriate error representation formula, the corresponding best -approximation is characterized in terms of a canonical point set. The convex cone here is defined by another differential operator (see [18, page 456] for details) that is represented in terms of in a bit intricate way involving infinite series. The canonical result of best -approximation reads as follows.
Theorem 8.2. Let satisfy on . Then, the unique transfinite interpolant from Theorem 8.1 is the unique best -approximant to the function from .
Remark 8.3. By Theorem 8.2, the point set is a canonical set of best -approximation on a triangle.
9. Best -Approximation by Harmonic Functions
Here, we discuss results on best -approximation of subharmonic functions by harmonic functions. Let denote the open ball centered at of radius in the -dimensional space and let be the unit ball centered at in and . Denote by the linear space of harmonic functions on and let denote the convex cone of subharmonic functions on . Let be continuous on the closed unit ball and let . A function is called a best harmonic -approximant to on if for all . The next theorem gives a canonical set characterization of the best harmonic -approximants to the convex cone (see  for details).
Theorem 9.1. Let . Then the harmonic function is a best harmonic -approximant from