Spline Interpolation and Smoothing on the Sphere

We extend the notion of periodic polynomial splines on the circle and thin plate splines on Euclidean d-space to splines on the sphere which are invariant under arbitrary rotations of the coordinate system. We solve the following problem: Find u Hₘ (S), a suitably defined reproducing kernel (Sobolev) space on the sphere S to, A) minimize Jₘ (u) subject to u (Pᵢ) = zᵢ, i = 1, 2, , n, and B) minimize \ 1 n₉ = ₁ⁿ ({u ({Pᵢ) - zᵢ }) ² } + Jₘ (u), \ where \ gathered Jₘ (u) = ₀^2 ₀^ { ({ ^{{m / 2} u (, ) }) } } ² d d, m even \\ = ₀^2 ₀^ {\{ {{ ({ ^{{{ (m-1) / 2} u}) _ ² }} { ² } + (^{{{ (m-1) / 2}} u}) _ ² } \}} } d d, m\, odd. \\ gathered \ Here is the Laplace–Beltrami operator on the sphere and Jₘ (u) is the natural analogue on the sphere, of the quadratic functional ₀^2 ({u^{ (m) () }) } ² d on the circle, which appears in the definition of periodic polynomial splines. Jₘ (u) may also be considered to be the analogue of \ ₉ = ₀ᵐ ({array{*{20c} m \\ j \\ array }) } - ^ - ^ { ({{ ᵐ u { xʲ y^{m - j }}}) } } ² dxdy \ appearing in the definition of thin plate splines on the plane. The solution splines are obtained in the form of infinite series, which do not appear to be convenient for certain kinds of computation. We then replace Jₘ in A) and B) by a quadratic functional Qₘ which is topologically equivalent to Jₘ on Hₘ (S) and obtain closed form solutions to the modified problems which are suitable for numerical calculation, thus providing practical pseudo-spline solutions to interpolation and smoothing problems on the sphere. Convergence rates of the splines and pseudo-splines will be the same. A number of results established or conjectured for polynomial and thin plate splines can be extended to the splines and pseudo-splines constructed here.