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One-sided mean approximation

to the characteristic function of a spherical layer by algebraic polynomials

Anastasiya Torgashova Ural Federal University, Ekaterinburg, Russia

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Let Rm, m ≥ 2, be the Euclidean space with the inner product xy =

xkyk, x = (x1, x2, . . . , xm), y = (y1, y2, . . . , ym), and the norm |x| = √xx. For r > 0, let Sm−1(r) = {x ∈ Rm: |x| = r} be the sphere of radius r centered at the origin Sm−1 = Sm−1(1)

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For a pair of numbers η = (a, b), −1 ≤ a < b ≤ 1, consider the spherical layer G(η) = {x = (x1, x2, . . . , xm) ∈ Sm−1: a ≤ xm ≤ b} centered at the “north pole” em = (0, 0, . . . , 0, 1) of the sphere. In the case b = 1, a = h, −1 < h < 1, the set G(η) is the spherical cap C(h) = {x = (x1, x2, . . . , xm) ∈ Sm−1: xm ≥ h}.

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Let L(E) = L1(E) be the space of functions measurable and integrable on a set E with the norm ∥f∥L(E) =

Let L∞(E) be the space of measurable essentially bounded functions on E with the norm ∥f∥L∞(E) = ess sup {|f(x)|: x ∈ E}; this is the conjugate space for L(E). On the unit sphere Sm−1 of the space Rm, m ≥ 2, consider the classical (m − 1)-dimensional Lebesgue

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Denote by Pn,m the set of algebraic polynomials Pn(x) =

cαxα, xα = xα1

x = (x1, x2, . . . , xm) ∈ Rm,

degree (at most) n in m variables with real coefficients cα. For a pair of measurable functions f and g on the sphere Sm−1, the inequality f ≤ g means that f(x) ≤ g(x) for almost all x ∈ Sm−1.

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For a function f measurable and bounded on the sphere Sm−1, consider the sets P−

P+

In order that these sets were nonempty, assume that f is bounded from below and from above, respectively.

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Consider the values of the best approximation in the space L to a function f by the set Pn,m from below and from above: e−

e+

Polynomials at which the infima are attained are called the polynomials of best (integral) approximation to the function f from below and from above, respectively, or extremal polynomials.

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The main aim of this study is the best approximation from below in the space L(Sm−1) to the charateristic function

1G(η)(x) =

1, x ∈ G(η), 0, x ̸∈ G(η),

P−

e−

= inf{∥1G(η) − Pn∥L(Sm−1): Pn ∈ P−

(1) here, according to the notation introduced above, P−

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The crucial fact is that the function 1G(η)(x), x = (x1, x2, . . . , xm), defined on the sphere Sm−1, is zonal; i.e., this functions depends only on the coordinate xm

f(x1, x2, . . . , xm) = g(xm), x = (x1, x2, . . . , xm) ∈ Sm−1, where g is a univariate function defined

the interval[−1, 1]. For the function f = 1G(η), the function g in this relation is the characteristic function of the interval [a, b].

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Reduction to a one-dimensional problem The passage to polar coordinates on the sphere Sm−1 leads to the following representation of the integral of a function f ∈ L(Sm−1) over the unit sphere:

f(x)dx = |Sm−2|

g(t)(1 − t2)(m−3)/2 dt . where g(t) = 1

f

1 − t2 x′, t

dx′.

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For a real number t, denote by Λ(t) the hyperplane

points x = (x1, x2, . . . , xm−1, t) ∈ Λ(t) in the form x = (x1, x2, . . . , xm−1, t) = (x′, t), x′ = (x1, x2, . . . , xm−1) ∈ Rm−1. For t ∈ (−1, 1), the section of the sphere Sm−1 by the hyperplane Λ(t) is the (m − 2)-dimensional sphere Sm−2(a) of radius a = a(t) =

1 − t2 centered at the point tem and parallel to the coordinate space Rm−1 of points x′ = (x1, x2, . . . , xm−1). We identify this sphere with the sphere Sm−2(a) ⊂ Rm−1.

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The function g(t) = 1

f

1 − t2 x′, t

dx′ can be interpreted as the averaging g = Sf of the function f over sections of the sphere by hyperplanes. The averaging operator S defined by this formula is a bounded linear operator from the space L(Sm−1) to the space Lϕ

(−1, 1) with the ultraspherical weight ϕ(t) = (1 − t2)α, α = m − 3 2 . For the averaging operator, we have the inequality |Sm−2| · ∥Sf∥Lϕ

f ∈ L(Sm−1).

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For an algebraic polynomial Pn ∈ Pn,m of degree n in m variables, the function gn(t) = (SPn)(t) = 1

1 − t2 x′, t

dx′ is a univariate algebraic polynomial

the same degree n. Thus, SPn,m ⊂ Pn, Pn = Pn,1. Actually, it is not hard to understand that SPn,m = Pn.

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Lemma 1. Let m ≥ 3. If a function f is defined, integrable, bounded on the sphere Sm−1, and zonal, then S(P−

This fact is quite obvious.

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The function 1G(η) is zonal; more exactly,

1G(η)(x) = 1I(η)(xm),

x = (x1, x2, . . . , xm) ∈ S(m−1), where 1I(η) is the characteristic function of the interval I = I(η) = (a, b):

1I(η)(t) =

1, t ∈ (a, b), 0, t ∈ [−1, 1] \ (a, b). Consider the best approximation from below E−

= inf{∥1I(η) − pn∥Lϕ

(2) to the step function 1I(η) in the space Lϕ

the set P−

polynomials whose graphs lie under the graph of the function 1I(η).

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Lemma 2. For any m ≥ 3, n ≥ 0, and a, b ∈ (−1, 1), we have e−

and if a polynomial p∗

problem (2) (i.e., the infimum in (2) is attained at this polynomial), then the zonal polynomial P ∗

p∗

x = (x1, x2, . . . , xm) ∈ Rm, is extremal in problem (1) on the sphere.

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One-sided approximation on an interval Consider in more detail the problem

approximation from below to the characteristic function

1I(t) =

1, t ∈ (a, b), 0, t ̸∈ (a, b),

= (a, b) by the set of algebraic polynomials (in one variable) of given degree n ≥ 0 in the space Lψ = Lψ(−1, 1) with a more general nonnegative weight ψ (as compared to the ultraspherical weight ϕ) on (−1, 1). The problem is in calculating the value E−

(3)

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[BMQ] Bustamante J., Mart ´ ınez Cruz R., Quesada J.M. Quasi orthogonal Jacobi polynomials and best one-sided L1 approximation to step functions,

[BDR] Babenko A.G., Deikalova M.V., R´ ev´ esz Sz.G. Weighted

integral approximations to characteristic functions

intervals by polynomials

In these papers, problem (3) was solved in the case when one of the end-points of the interval I = (a, b) coincides with the corresponding end-point ±1 of the initial interval (−1, 1).

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These results and the method of their proving allow us to solve problem (3) under the assumption that the weight ψ is even and the interval I = (a, b) is symmetric about 0; i.e., a = −h and b = h, where 0 < h < 1. Indeed, consider the auxiliary problem of approximation

1h(t) =

1, t ∈ [0, h2), 0, t ∈ [h2, 1],

the results and method

[BDR] applied to the interval [0, 1].

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To obtain lower bounds, the authors of [BDR] followed the known scheme and used quadrature formulas with positive coefficients. Similarly, consider the positive quadrature formula

p(u)ψ(u) du =

λkp(uk), p ∈ Pℓ, (4) which is valid on the set Pℓ of polynomials of degree ℓ, h2 is its node, and all nodes {uk} lie inside the interval [0, 1]. The extremal polynomial p∗ in the approximation problem interpolates the function 1h at the nodes of quadrature formula (4). The method of constructing the polynomial goes back to A.Markov (1883) and Stieltjes (1884).

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Making the change of variable u = t2 on the left-hand side of (4) and considering the polynomial q(t) = p(t2), we obtain 2

q(t)ψ(t2)t dt =

λkq(√uk). (5) Making the change of variable v = −t in (5), we obtain 2

q(v)ψ(v2)|v| dv =

λkq(−√uk). (6)

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Combining (5) and (6), we obtain the new quadrature formula on the interval [−1, 1]:

q(t)ψ(t2)|t| dt =

λk 2 (q(−√uk) + q(√uk)). (7) This quadrature formula is valid for even polynomials q

formula (7) is also valid; moreover, its left- and right- hand sides are zero.

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Thus, quadrature formula (7) is valid for polynomials

points −h and h are its nodes. This quadrature formula is extremal in problem (2) with the weight ψ(t2)|t|. The extremal polynomial in this problem is q∗(t) = p∗(t2), which interpolates the function 1I(η) at the nodes of quadrature formula (7). The value of the best approximation is

ψ(t2)|t| dt −

λk, uS = h2.

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One-sided approximation of the characteristic function

1I of the interval I = (a, b) for an arbitrary pair of points

a and b such that −1 < a < b < 1 turned to be a difficult

(i) In the case when the points a and b are neighboring nodes of a positive quadrature formula valid on the set of polynomials of certain order n. In this case, the polynomial p∗

(ii) For polynomials of small degree n and specific points a and b; for example, for n = 7, the unit weight, and in the case when a and b are an arbitrary pair of the Gauss four-point quadrature formula (which is valid for polynomials of degree 7). In this case, the extremal polynomial interpolates the function 1I at the nodes of the Gauss quadrature formula.

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One-sided mean approximation to the characteristic function

For the present, we know a solution of the initial problem (1) of the one-sided approximation from below in the space L(Sm−1) to the characteristic function of the spherical layer G(η) = {x = (x1, x2, . . . , xm) ∈ Sm−1: a ≤ xm ≤ b}, η = (a, b), −1 ≤ a < b ≤ 1, by algebraic polynomials of arbitrary degree in m variables in the following cases:

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(1) for b = 1 and a = h, where −1 < h < 1, when the layer becomes the spherical cap C(h) = {x = (x1, x2, . . . , xm) ∈ Sm−1: xm ≥ h}; (2) for a layer symmetrical about the equator; more precisely, in the case a = −h and b = h, where 0 < h < 1. (3) The points a and b are neighboring nodes of a positive quadrature formula valid

the set

polynomials of certain order n. (4) In the three-dimensional space (m = 3) for polynomials of small degree n and specific points a and b; for example, for n = 7, in the case when a and b are an arbitrary pair of the Gauss four-point quadrature formula (which is valid for polynomials of degree 7).

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