Needlet approximation for isotropic random fields on the sphere

Abstract

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets -a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2. For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2-weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion. polynomial of degree 2 j − 1, associated with a positive weight quadrature rule with degree of precision 2 j+1 − 1. Main results.Needlets form a multiscale tight frame for square-integrable functions on S d , see [26,27]. The resulting tight frame has good approximation performance for random fields on the sphere.Needlet decomposition for random fields. For a random field T on S d , the (semidiscrete) needlet approximation of order J for J ∈ N 0 is defined as( 1.1) Let ⌈·⌉ be the ceiling function. Given 1 ≤ p < ∞ and d ≥ 2, the needlet approximation given by (1.1) of a ⌈p⌉-weakly isotropic random field T on S d converges to T in L p Ω × S d . (See Theorem 3.4, and see Section 2 for the definition of n-weakly isotropic, n ∈ N.)In Theorem 3.9, we also prove that when p is a positive integer the needlet approximation on the set of p-weakly isotropic random fields is bounded in the L p Ω × S d sense.Let d ≥ 2. Let T be a 2-weakly isotropic random field satisfying ∞ ℓ=1 A ℓ ℓ 2s+d−1 < ∞. Here A ℓ is the angular power spectrum of T , see Section 4.1. Then T (ω) is in the Sobolev space W s 2 (S d ) P-almost surely (or P-a. s.). (This fact was proved in [19, Section 4] and [3]; we restate this result in Corollary 4.4.)Semidiscrete needlet approximation. In Theorems 4.6 and 4.7, we prove that the semidiscrete needlet approximation V need J (T ) has the following mean and pointwise approximation errors. Let s > 0.( 1.2)The needlet approximation V need J (T ; ω) has the following L 2 (S d )-error bound: P-a. s.,where the constants c in (1.2) and (1.3) depend only on d, s, h and κ. Fully discrete needlet approximation. For implementation, we discretise the needlet coefficient (T (ω), ψ jk ) L 2 (S d ) = S d T (ω, x)ψ jk (x) dσ d (x) by a positive weight numerical integration rule,This then gives the fully discrete needlet approximation for the random field T :In Theorems 4.10 and 4.11, we prove that for s > d/2, the fully discrete needlet approximation V need J,N (T ) has the same mean and pointwise error orders as those for the semidiscrete needlet approximation V need J (T ), cf. (1.2)-(1.3). Let Q N be a discretisation quadrature exact for polynomials of degree 3 · 2 J−1 − 1. Then 4) and P-a. s.5) (ii) for p = ∞, lim J→∞ E T − V need J (T ) L∞(S d ) = 0.Remark. By the Fubini theorem, (3.7) implies that the nee...

Asymptotic Behaviour of Level Sets of Needlet Random Fields

“…Moreover, since it is known that the derivative of an operator kernel related to the Laplace-Beltrami operator on the sphere is a kernel itself (see equation (12) in [24]) we can establish the same property on the kernels ∂ ∂ x,y Ψ…”

“…They are used to extract information from spherically observed signals in these fields, since the presence of a masked region in the domain of observation does lower the efficiency of the analysis. Spherical wavelet systems, so called needlets, have been introduced by [24], [25] and investigated in the last years by many authors, see for instance [4], [12], [31], [11], [3], [5], [10]. In particular in [3], [5], [10] the properties of needlets applied to random fields are studied.…”

Approximation and localized polynomial frame on double hyperbolic and conic domains

“…The kernel is highly localized if it decays at rates faster than any inverse polynomial rate away from the main diagonal y = x in Ω × Ω with respect to the distance d on Ω; see the definition in the next section. These kernels provide important tools for analysis on regular domains, such as the unit sphere and the unit ball, and are essential ingredient in recent study of approximation and localized polynomial frames; see, for example, [3,8,11,14,17,18,19] for some of the results on the spheres and balls and [1,2,12,13,15,25] for various applications. The reason that highly localized kernels are known only on a few regular domains lies in the addition formula for orthogonal polynomials, which are closed form formulas for the reproducing kernels of orthogonal polynomials.…”