One-sided mean approximation on the Euclidean sphere to the - PowerPoint PPT Presentation

One-sided mean approximation on the Euclidean sphere to the characteristic function of a spherical layer by algebraic polynomials Anastasiya Torgashova Ural Federal University, Ekaterinburg, Russia Joint research with Marina Deikalova

Notation. Statement of the problem Let R m , m ≥ 2 , be the Euclidean space with the inner product m ∑ xy = x k y k , k =1 x = ( x 1 , x 2 , . . . , x m ) , y = ( y 1 , y 2 , . . . , y m ) , and the norm | x | = √ xx. For r > 0 , let S m − 1 ( r ) = { x ∈ R m : | x | = r } be the sphere of radius r centered at the origin S m − 1 = S m − 1 (1)

For a pair of numbers η = ( a, b ) , − 1 ≤ a < b ≤ 1 , consider the spherical layer G ( η ) = { x = ( x 1 , x 2 , . . . , x m ) ∈ S m − 1 : a ≤ x m ≤ b } centered at the “north pole” e m = (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 = ( x 1 , x 2 , . . . , x m ) ∈ S m − 1 : x m ≥ h } .

Let L ( E ) = L 1 ( E ) be the space of functions measurable and integrable on a set E with the norm ∫ ∥ f ∥ L ( E ) = E | f ( x ) | dx. 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 S m − 1 of the space R m , m ≥ 2 , consider the classical ( m − 1) -dimensional Lebesgue measure. For a subset E ⊂ S m − 1 , denote by | E | m − 1 or | E | its measure.

Denote by P n,m the set of algebraic polynomials c α x α , ∑ P n ( x ) = | α | = α 1 + · · · + α m ≤ n, α = ( α 1 , . . . , α m ) ∈ Z m + x α = x α 1 1 x α 2 2 · · · x α m x = ( x 1 , x 2 , . . . , x m ) ∈ R m , m , of degree (at most) n in m variables with real coefficients c α . For a pair of measurable functions f and g on the sphere S m − 1 , the inequality f ≤ g means that f ( x ) ≤ g ( x ) for almost all x ∈ S m − 1 .

For a function f measurable and bounded on the sphere S m − 1 , consider the sets P − n,m ( f ) = { P ∈ P n,m : P ≤ f } , P + n,m ( f ) = { P ∈ P n,m : P ≥ f } . In order that these sets were nonempty, assume that f is bounded from below and from above, respectively.

Consider the values of the best approximation in the space L to a function f by the set P n,m from below and from above: e − n,m ( f ) = inf {∥ f − P ∥ : P ∈ P − n,m ( f ) } , e + n,m ( f ) = inf {∥ f − P ∥ : P ∈ P + n,m ( f ) } . 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 .

The main aim of this study is the best approximation from below in the space L ( S m − 1 ) to the charateristic function { 1 , x ∈ G ( η ) , 1 G ( η ) ( x ) = 0 , x ̸∈ G ( η ) , of the spherical layer G ( η ) by the set of polynomials P − n,m ( 1 G ( η ) ) . More exactly, we study the value e − n,m ( 1 G ( η ) ) = (1) = inf {∥ 1 G ( η ) − P n ∥ L ( S m − 1 ) : P n ∈ P − n,m ( 1 G ( η ) ) } ; here, according to the notation introduced above, P − n,m ( 1 G ( η ) ) = { P n ∈ P n,m : P n ≤ 1 G ( η ) } .

The crucial fact is that the function 1 G ( η ) ( x ) , x = ( x 1 , x 2 , . . . , x m ) , defined on the sphere S m − 1 , is zonal; i.e., this functions depends only on the coordinate x m of the point x = ( x 1 , x 2 , . . . , x m ) ∈ S m − 1 : f ( x 1 , x 2 , . . . , x m ) = g ( x m ) , x = ( x 1 , x 2 , . . . , x m ) ∈ S m − 1 , where g is a univariate function defined on the interval [ − 1 , 1] . For the function f = 1 G ( η ) , the function g in this relation is the characteristic function of the interval [ a, b ] .

Reduction to a one-dimensional problem The passage to polar coordinates on the sphere S m − 1 leads to the following representation of the integral of a function f ∈ L ( S m − 1 ) over the unit sphere: 1 ∫ ∫ g ( t )(1 − t 2 ) ( m − 3) / 2 dt . f ( x ) dx = | S m − 2 | S m − 1 − 1 where 1 (√ ) ∫ 1 − t 2 x ′ , t dx ′ . g ( t ) = f � � � S m − 2 � � S m − 2 �

For a real number t , denote by Λ( t ) the hyperplane of points x = ( x 1 , x 2 , . . . , x m − 1 , t ) ∈ R m . We will write points x = ( x 1 , x 2 , . . . , x m − 1 , t ) ∈ Λ( t ) in the form x = ( x 1 , x 2 , . . . , x m − 1 , t ) = ( x ′ , t ) , x ′ = ( x 1 , x 2 , . . . , x m − 1 ) ∈ R m − 1 . For t ∈ ( − 1 , 1) , the section of the sphere S m − 1 by the hyperplane Λ( t ) is the ( m − 2) -dimensional sphere √ 1 − t 2 centered at the S m − 2 ( a ) of radius a = a ( t ) = point te m and parallel to the coordinate space R m − 1 of points x ′ = ( x 1 , x 2 , . . . , x m − 1 ) . We identify this sphere with the sphere S m − 2 ( a ) ⊂ R m − 1 .

The function 1 (√ ) ∫ 1 − t 2 x ′ , t dx ′ g ( t ) = f � � � S m − 2 � � S m − 2 � 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 ( S m − 1 ) to the space L ϕ 1 ( − 1 , 1) of functions integrable over the interval ( − 1 , 1) with the ultraspherical weight α = m − 3 ϕ ( t ) = (1 − t 2 ) α , . 2 For the averaging operator, we have the inequality | S m − 2 | · ∥ Sf ∥ L ϕ f ∈ L ( S m − 1 ) . 1 ( − 1 , 1) ≤ ∥ f ∥ L ( S m − 1 ) ,

For an algebraic polynomial P n ∈ P n,m of degree n in m variables, the function 1 (√ ) ∫ 1 − t 2 x ′ , t dx ′ g n ( t ) = ( SP n )( t ) = S m − 2 P n � � � S m − 2 � � � is a univariate algebraic polynomial of the same degree n. Thus, S P n,m ⊂ P n , P n = P n, 1 . Actually, it is not hard to understand that S P n,m = P n .

Lemma 1. Let m ≥ 3 . If a function f is defined, integrable, bounded on the sphere S m − 1 , and zonal, then S ( P − n,m ( f )) = P − n ( Sf ) . This fact is quite obvious.

The function 1 G ( η ) is zonal; more exactly, x = ( x 1 , x 2 , . . . , x m ) ∈ S ( m − 1) , 1 G ( η ) ( x ) = 1 I ( η ) ( x m ) , where 1 I ( η ) is the characteristic function of the interval I = I ( η ) = ( a, b ) :  1 , t ∈ ( a, b ) ,  1 I ( η ) ( t ) = 0 , t ∈ [ − 1 , 1] \ ( a, b ) .  Consider the best approximation from below E − n,ϕ ( 1 I ( η ) ) = (2) 1 ( − 1 , 1) : p n ∈ P − = inf {∥ 1 I ( η ) − p n ∥ L ϕ n ( 1 I ( η ) ) } to the step function 1 I ( η ) in the space L ϕ 1 ( − 1 , 1) by n ( 1 I ( η ) ) = P − the set P − n, 1 ( 1 I ( η ) ) of (univariate) algebraic polynomials whose graphs lie under the graph of the function 1 I ( η ) .

Lemma 2. For any m ≥ 3 , n ≥ 0 , and a, b ∈ ( − 1 , 1) , we have n,m ( 1 G ( η ) ) = | S m − 2 | E − e − n,ϕ ( 1 I ( η ) ) and if a polynomial p ∗ n in one variable is extremal in problem (2) ( i.e., the infimum in (2) is attained at this polynomial ) , then the zonal polynomial P ∗ n ( x ) = p ∗ R m , is extremal in n ( x m ) , x = ( x 1 , x 2 , . . . , x m ) ∈ problem (1) on the sphere.

One-sided approximation on an interval Consider in more detail the problem of one-sided approximation from below to the characteristic function  1 , t ∈ ( a, b ) ,  1 I ( t ) = 0 , t ̸∈ ( a, b ) ,  of the interval I = ( a, b ) by the set of algebraic polynomials (in one variable) of given degree n ≥ 0 L ψ L ψ ( − 1 , 1) in the space = 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 − 1 ( − 1 , 1) : p n ∈ P − n,ψ ( 1 I ) = inf {∥ 1 I − p n ∥ L ψ n ( 1 I ) } (3)

[BMQ] Bustamante J., Mart ´ ınez Cruz R., Quesada J.M. Quasi orthogonal Jacobi polynomials and best one-sided L 1 approximation to step functions, J. Approx. Theory. 2015. Vol. 198. P. 10–23. [BDR] Babenko A.G., Deikalova M.V., R´ ev´ esz Sz.G. Weighted one-sided integral approximations to characteristic functions of intervals by polynomials on a closed interval, Proc. Steklov Inst. Math. 2017. Vol. 297, Suppl. 1. P. S11–S18 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) .

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 of the function  t ∈ [0 , h 2 ) , 1 ,  1 h ( t ) = t ∈ [ h 2 , 1] , 0 ,  on the interval [0 , 1] . Studying this problem, we use the results and method of [BDR] applied to the interval [0 , 1] .