Moduli of smoothness and approximation on the unit sphere and the unit ball

Abstract

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a K-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding K-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.

Best Polynomial Approximation on the Unit Sphere and the Unit Ball

“…These moduli of smoothness and K-functionals were also defined in [8], and e Theorem 3.6 was proved there. For d = 1, they agree with the Ditzian-Totik moduli of smoothness and K-functionals.…”

“…. and 0 < t < 1, 2 , but they should hold for all µ ≥ 0 and perhaps even µ > −1/2, which, however, requires a different proof from that of [8].…”

“…An extensive theory of harmonic analysis for orthogonal expansions with respect to h 2 κ (x)dσ has been developed (cf. [8,16]), in parallel with the classical theory for spherical harmonic expansions. The weighted best approximation in L p (h 2 κ ; S d−1 ) norm was studied in [29], where analogues of the modulus of smoothness ω * r (f, t) p and K-functional K * r (f, t) p are defined with · p replaced by the norm of L p (h 2 κ ; S d−1 ) for h κ invariant under a reflection group, and a complete analogue of Theorem 2.3 was established.…”

“…Since the K-functional K * r (f, t) p is defined in terms of ∆ 0 and the K-function K r (f, t) is defined in terms of D i,j , it indicates that K r (f, t) p may be stronger than K * r (f, t) p if we believe that the parts encode more information than the whole. The main advantage of the modulus of smoothness ω r (f, t) p lies in the fact that it is defined in terms of moduli of smoothness of one variable, which allows us to tap into the well established theory of trigonometric approximation of one variable, and it also means that ω r (f, t) p can be computed relatively easily (see [8] for examples).…”

“…One advantage of the moduli of smoothness is that they can be relatively easily computed. Indeed, they can be computed just like the second modulus of smoothness on the sphere; see [8] for several examples.…”