A note on local polynomial approximation on compact manifolds

J. Levesley and D. L. Ragozin

Version 1.4 Last Changed 00/07/06 July 21, 2000

Abstract

In [1] we give a local error estimate for approximation of smooth functions by spherical harmonics. For a k–times differentiable function f we show that there is a spherical polynomial p such that |f(x) − p(x)| ≤ Cρ^{k}, for x ∈ D_{ρ}, where D_{ρ} is a disk of radius ρ on the sphere. We generalise this result to all compact manifolds in a very elementary way, retaining the essence of the argument used on the sphere.

Discussion and Result

Let M be a compact d–dimensional C^{∞}-manifold embedded in IR^{d+k} for some k > 0. Let the distance, d(x, y), between x, y ∈ M be measured as the minimum arc length from x to y. Fix x ∈ M and, for ρ > 0, let D_{ρ}(x) = {y ∈ M : d(x, y) ≤ ρ}. Let T_{x} = IR^{d} be the tangent space to M at x considered as a d-dimensional hyperplane in IR^{d+k}. Let N_{ρ} = {y ∈ IR^{d} : ∥y∥ ≤ ρ}. Since M is an embedded C^{∞} manifold the orthogonal projection, π_{x}, from IR^{d+k} to T_{x} has a derivative at x which is the identity map when restricted to T_{x}. Hence by the inverse function theorem there exists ρ_{0} > 0 and a C^{∞} map ϕ : N_{ρ_{0}} → M which is the inverse of the orthogonal projection restricted to a closed neighborhood of x in M. Since ϕ : N_{ρ_{0}} ⊂ IR^{d} → M is an invertible C^{∞} map on a compact neighborhood, whose inverse is orthogonal projection restricted to M, there exists a constant C_{1} such that

C_{1}d(ϕ(y), ϕ(z)) ≤ ∥y − z∥ ≤ ||ϕ(y) − ϕ(z)||
≤ d(ϕ(y), ϕ(z)), y, z ∈ N_{ρ_{0}}. (1)

The polynomials of degree n on M, denoted P_{n}, are the restrictions of the polynomials of degree n in IR^{d+k} to M.

Fix f ∈ C^{k}(M). By (1) for ρ ≤ ρ_{0}, f ∘ ϕ ∈ C^{k}(ϕ^{−1}D_{ρ}). We can now perform a multivariate Taylor expansion,

f ∘ ϕ(y) = ∑_{|α|<k} c_{α}y^{α} + R_{k}(f, y),  (2)

where the remainder

|R_{k}(f, y)| ≤ C_{k}(f) max_{t∈D_{ρ}}(||ϕ^{−1}(t)||)^{k}.  (3)

Let t = ϕ(y) and define p_{k−1} ∈ P_{k−1} by

p_{k−1}(t) = ∑_{|α|<k} c_{α}y^{α},

a polynomial since ϕ is the inverse of orthogonal projection. Then we see, using Equations (2) and (3) , that, for ρ ≤ ρ_{0},

max_{y∈D_{ρ}} |f(t) − p_{k−1}(t)| ≤ C_{k}(f) max_{t∈D_{ρ}}(||ϕ^{−1}(t)||)^{k}.

Since

max_{t∈D_{ρ}} ||ϕ^{−1}(t)|| ≤ ρ,

we arrive, using the compactness of the manifold, at the following result:

Theorem 1 Let f ∈ C^{k}(M). Then there exists a positive ρ_{0} and a constant C_{k}(f) such that for each x ∈ M there exists a degree k − 1 polynomial p_{k−1} such that, for every 0 < ρ ≤ ρ_{0},

max_{t∈D_{η}} |f(t) − p_{k−1}(t)| ≤ C_{k}(f)ρ^{k},

where the constant C_{k}(f) does not depend on ρ.

Remarks 1 Such a local polynomial approximation theorem is often referred to as a Whitney–type theorem.

References

[1] J. Levesley and D.L. Ragozin, Local approximation on manifolds using radial functions and polynomials, in Curve and Surface Fitting: Saint Malo 1999, A. Cohen, C. Rabut and L. Schumaker(eds.), Vanderbilt University Press, Nashville, 2000, 291– 300.