Explicit families of functions on the sphere with exactly known Sobolev space smoothness

TY - CHAP

T1 - Explicit families of functions on the sphere with exactly known Sobolev space smoothness

AU - Brauchart, Johann S.

PY - 2018/5/23

Y1 - 2018/5/23

N2 - We analyze explicit trial functions defined on the unit sphere Sd in the Euclidean space ℝd+1, d ≥ 1, that are integrable in the Lp-sense, p ε [1,∞). These functions depend on two free parameters: one determines the support and one, a critical exponent, controls the behavior near the boundary of the support. Three noteworthy features are: (1) they are simple to implement and capture typical behavior of functions in applications, (2) their integrals with respect to the uniform measure on the sphere are given by explicit formulas and, thus, their numerical values can be computed to arbitrary precision, and (3) their smoothness can be defined a priori, that is to say, they belong to Sobolev spaces Hs(Sd) up to a specified index Ns determined by the parameters of the function.Considered are zonal functions g(x) = h(x · p), where p is some fixed pole on Sd. The function h(t) is of the type [max(t, T)]α or a variation of a truncated power function x (mapping) (x)α + C (which assumes 0 if x ≤ 0 and is the power xα if x > 0) that reduces to [max(t - T,0)]α, [max(t2 - T2, 0)]α, and [max(T2 - t2, 0)]α if α > 0. These types of trial functions have as support the whole sphere, a spherical cap centered at p, a bi-cap centered at the antipodes p, -p, or an equatorial belt. We give inclusion theorems that identify the critical smoothness s = s(T, α) and explicit formulas for the integral over the sphere. We obtain explicit formulas for the coefficients in the Laplace- Fourier expansion of these trial functions and provide the leading order term in the asymptotics for large index of the coefficients.

AB - We analyze explicit trial functions defined on the unit sphere Sd in the Euclidean space ℝd+1, d ≥ 1, that are integrable in the Lp-sense, p ε [1,∞). These functions depend on two free parameters: one determines the support and one, a critical exponent, controls the behavior near the boundary of the support. Three noteworthy features are: (1) they are simple to implement and capture typical behavior of functions in applications, (2) their integrals with respect to the uniform measure on the sphere are given by explicit formulas and, thus, their numerical values can be computed to arbitrary precision, and (3) their smoothness can be defined a priori, that is to say, they belong to Sobolev spaces Hs(Sd) up to a specified index Ns determined by the parameters of the function.Considered are zonal functions g(x) = h(x · p), where p is some fixed pole on Sd. The function h(t) is of the type [max(t, T)]α or a variation of a truncated power function x (mapping) (x)α + C (which assumes 0 if x ≤ 0 and is the power xα if x > 0) that reduces to [max(t - T,0)]α, [max(t2 - T2, 0)]α, and [max(T2 - t2, 0)]α if α > 0. These types of trial functions have as support the whole sphere, a spherical cap centered at p, a bi-cap centered at the antipodes p, -p, or an equatorial belt. We give inclusion theorems that identify the critical smoothness s = s(T, α) and explicit formulas for the integral over the sphere. We obtain explicit formulas for the coefficients in the Laplace- Fourier expansion of these trial functions and provide the leading order term in the asymptotics for large index of the coefficients.

UR - http://www.scopus.com/inward/record.url?scp=85053959204&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-72456-0_9

DO - 10.1007/978-3-319-72456-0_9

M3 - Chapter

SN - 9783319724553

SP - 153

EP - 177

BT - Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

PB - Springer International Publishing AG

ER -