Spaces of Functions on a Sphere

In this chapter, the theory of functions on a sphere is studied. The results obtained here are then systematically used in the book for proving the theorems on the existence and uniqueness of the BVE solution, for analyzing the asymptotic behavior of solutions, for deriving estimates in the norms of different functional spaces, for approximating the functions on the sphere by spherical polynomials (spectral method), for evaluating the rate of convergence of such approximations, etc. Noticeable activity in the study of various questions of the theory of Fourier-Laplace series is observed in the recent decades. This is connected mainly with the application of this series for the numerical solution of various problems on a sphere in such applied sciences as meteorology, weather forecast, climate theory, and so forth. In particular, questions such as convergence and summation of Fourier-Laplace series are of great importance when using the spectral method for discretizing partial differential equations on a sphere. Section 2.1 surveys briefly the main properties of Legendre polynomials, associated Legendre functions and spherical harmonics. In Sect. 2.2, geographical coordinates maps for the sphere are defined and the well-known theorem on the partition of unity is given which is an important tool in the theory of integration of functions on smooth compact manifolds. The orthogonal projections Y n ψ $$\mathrm{Y}_{n}\left (\psi \right )$$ of a function ψ $$\psi$$ onto the subspaces H n of the homogeneous spherical polynomials of degree n are defined in Sect. 2.3, and then they are used to introduce derivatives D s and Λ s $$\Lambda ^{s}$$ of real degree s of a function on the unit sphere. The orthogonal projections T N ψ $$\mathrm{T}_{N}\left (\psi \right )$$ of a function ψ onto the subspaces ℙ N $$\mathbb{P}^{N}$$ of spherical polynomials of degree n ≤ N are also defined in this section. Spaces of scalar functions having fractional derivatives on the sphere are mainly defined by two ways: either through the modulus of continuity of functions or using multiplier operators and geometrical properties of the sphere. In Sect. 2.4, we use the second approach, as more simple and constructive, and closely related to the spectral method of discretization of the partial differential equations on a sphere. The method is based on using the spherical Laplace operator, since it is the unique differential operator invariant with respect to any rotation of the sphere. The fractional derivatives are introduced through real degrees of the Laplace operator, and the smoothness of a function on the sphere is determined by the rate of convergence of its series of spherical harmonics. A family of Hilbert spaces ℍ s $$\mathbb{H}^{s}$$ of functions having fractional derivatives on the sphere up to real degree s is introduced, besides, ψ ∈ ℍ s $$\psi \in \mathbb{H}^{s}$$ if all its fractional derivatives up to order s belong to the Hilbert space 𝕃 2 ( S ) $$\mathbb{L}^{2}(S)$$ . Some structural properties of Hilbert spaces ℍ s $$\mathbb{H}^{s}$$ including various embedding theorems are given. According to the well-known Gibbs phenomenon and Parseval-Steklov identity, the convergence of Fourier series is deteriorated near to sharp changes of a continuous function. For the sphere, as a compact manifold, there is a ready instrument of approximation based on the expansion of space 𝕃 2 ( S ) $$\mathbb{L}^{2}(S)$$ in the orthogonal direct sum of the (2n + 1)-dimensional subspaces H n of homogeneous spherical polynomials of degree n which are invariant to the group SO(3) of sphere rotations. The rate of convergence of Fourier-Laplace series of functions from ℍ s $$\mathbb{H}^{s}$$ is estimated in Sect. 2.4 as well. The space ℂ ( S ) $$\mathbb{C}(S)$$ of continuous functions on a sphere is considered in Sect. 2.5. The chapter is concluded by Sect. 2.6 wheresome estimates are given in the norms of Banach spaces 𝕃 p ( S ) $$\mathbb{L}^{p}(S)$$ and 𝕃 p ( 0 , T ; 𝕏 ) $$\mathbb{L}^{p}(0,T; \mathbb{X})$$ .