A Priori L2-Error Estimates for Approximations of Functions on Compact Manifolds

Given a C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}^{2}}$$\end{document} -function f on a compact riemannian manifold (X,g) we give a set of frequencies L=Lf(ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L=L_{f}(\varepsilon)}$$\end{document} depending on a small parameter ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon > 0}$$\end{document} such that the relative L2-error ‖f-fL‖‖f‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{\|f-f^{L} \|}{\|f\|}}$$\end{document} is bounded above by ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon}$$\end{document}, where fL denotes the L-partial sum of the Fourier series f with respect to an orthonormal basis of L2(X) constituted by eigenfunctions of the Laplacian operator Δ associated to the metric g.


Introduction
The origin of this work was to give an answer to the following quite naive question: Given a 2π-periodic function f (θ) and a fixed ε > 0, is it possible to find an explicit subset of frequencies L = L f (ε) ⊂ Z for which we have the following a priori bound for the relative Here f L denotes the partial sum f L (θ) = ∈L f e i θ , f = 1 2π 2π 0 f (θ) e −i θ dθ are the Fourier coefficients and g = 1 2π 2π 0 |g(θ)| 2 dθ 1 2 is the L 2 -norm of a function g(θ). It turns out that such a bound can be explicitly constructed using only the quantities f , f , f and ε by an elementary application of the Chebyshev inequality in probability theory.
In fact, the context in which we was first interested was a little bit technically involved but heuristically analogous: we wanted to obtain a bound for the number of significant Fourier coefficients of an spherical function

Main Result
In order to fix the ideas, we fix an oriented compact riemannian manifold (X, g) and we consider the (scalar or hermitian) product in the space of (real or complex valued) square integrable functions on X defined by where dV is the volume element and we denote by A 0 (X) its L 2 -completion. The riemannian metric g over T X extends to every tensorial fiber bundle over X and in particular to the vector bundle Ω k (X) of differential k-form. On the other hand, we recall that every scalar product on a real vector space V extends in a natural way to a hermitian product on its complexification V ⊗ C. In this way, we can define a (scalar or hermitian) product in the space of (real or complex valued) differential k-forms on X by means of and we can consider its corresponding L 2 -completion, which will be denoted by A k (X).
We consider the exterior derivative operator d: A 0 (X) → A 1 (X) and its formal adjoint d * : A 1 (X) → A 0 (X) with respect to the (scalar o hermitian) products introduced below. It is well known that the Laplacian operator Δ := d * d + dd * = d * d over A 0 (X) is self-adjoint, positive definite and it has discrete spectrum. Consequently, there exists a countable orthonormal basis {ψ } ∈Λ of eigenfunctions of Δ. Thus, there exists a function λ: Λ → R + , → λ , such that Δψ = λ ψ for all ∈ Λ. For every f ∈ A 0 (X) we consider Vol. 12 (2015) Error Estimates for Approximations 53 its Fourier series ∈Λ f ψ with f = f, ψ ∈ C. For every subset L ⊂ Λ we define the partial sum of f over L as Our main result is the following.
In fact, L f (ε) can be chosen as the preimage by λ : where The proof is a direct application of the following two statements.

Lemma 2. Consider the (non-bounded) linear operators
Then for every f ∈ A 0 (X) for which D j (f ), j = 1, 2, are defined the following relations hold: Moreover, D 1 f 2 = X ∇ g f 2 g dV , where ∇ g and · g are the gradient operator and the norm with respect to the metric g. Furthermore the map λ: Λ → R + has discrete image and finite fibers.
Proof. Since D 2 ψ = λ ψ it follows that D 2 f = ∈Λ f λ ψ and consequently D 2 f 2 = ∈Λ |f | 2 λ 2 . On the other hand, for all , ∈ Λ we have dψ , dψ = ψ , d * dψ = ψ , Δψ = ψ , λ ψ = λ δ . Since The other expression for D 1 f follows from the well-known formula g(df, df ) = g(∇ g f, ∇ g f ). Finally, the last claim follows from the fact that the image of λ is the spectrum of the Laplacian Δ and every eigenvalue has finite multiplicity.
) satisfies the error estimates (2), where Moreover, if λ has discrete image and finite fibers then the partial sum (1) corresponding to L = L f (ε) has only a finite number of terms for every ε > 0.
Proof. Given f ∈ A 0 , consider a discrete random variable Z satisfying whose moments of order 1 and 2 are, respectively, thanks to Relations (4). The standard deviation of Z is given by On the other hand, given any pair of real numbers By definition of (5), it follows that L ± f (ε) = E(Z) ± ε −1 σ(Z). We conclude the proof by applying Chebyshev's inequality given by χ(ϕ) = e iϕ and the metric g = dϕ 2 4π 2 whose volume element is dV = dϕ 2π . The Laplacian can we written as Δ = −∂ 2 ϕ . An orthonormal eigenbasis of complex functions is given by ψ : given by χ(ϕ, θ) = (cos ϕ sin θ, sin ϕ sin θ, cos θ) and the metric g = 1 16π 2 sin 2 θ dϕ 2 + dθ 2 induced by that of R 3 , whose volume element is dV = sin θ 4π dϕ dθ. We consider the orthonormal basis given by the harmonic spherical functions ψ m : Error Estimates for Approximations 55 are the associated Legendre polynomials, see for instance [1].
and λ = ( + 1). Consequently, We can improve the choice of the set of frequencies L f (ε) for which the required estimate (2) is already fulfilled. One can proceed in the following way.
Theorem 6. Assume that we have already computed the Fourier coefficients of f for a given subset I ⊂ Λ. Then the inequality (2) also holds for the new set of frequencies Proof. Indeed, if we denote D 0 = Id, it follows from Parseval identity and Relations (4) that for each j = 0, 1, 2 the following equalities hold is a sum of n gaussian distributions with amplitudes a i , means μ i and standard deviations σ i , see Fig. 1.
We deal first with the unimodal case n = 1, a 1 = 1, μ 1 = 6, σ 1 = 1. We take I = λ −1 (J). Notice that the two first estimated intervals are approximately centered at μ 1 = 6. Now, we treat the bimodal case n = 2, a 1 = 1.5, a 2 = 1, μ 1 = 2, μ 2 = 13, σ 1 = σ 2 = 1 and I = λ −1 (J). To finish the theoretical part of the paper, we point out that the compactness assumption on X is necessary to state Theorem 1 in its present form, but there exists an alternative statement on the simplest non-compact manifold X = R n : belongs to L 2 (R n ) and it satisfies the following inequality The proof is completely analogous to the one of Theorem 1, using the Fourier transformf and its reconstruction formula instead of Fourier series. In this case, Λ = R n and all the summations ∈Λ a must be replaced by R n a(ξ)dξ. The analogues of Relations (4) follow from the well-known identity ∂f ∂xj (ξ) = 2iπξ jf (ξ) using the map λ: R n → R + given by λ(ξ) = 4π|ξ| 2 , where |ξ| denotes the euclidian norm of a vector ξ ∈ R n . In fact, as in the precedent version, L = λ −1 (I) for some compact interval I ⊂ R.
Remark 9. The uncertainty principle for f ∈ L 2 (R n ) asserts that D 0 (f )D 0 (f ) ≥ C n , see [3], where C n is some explicit positive constant depending only on the dimension n and The uncertainty principle applied to f can be interpreted as a lower bound for the midpoint μ of the interval I: Analogously, by applying the uncertainty principle to the partial derivatives

Application: Smooth Approximations of Polyhedral Objects
Theorem 1 can be applied to the problem of finding a smooth approximation of a geometric object, typically a curve or a surface, from which we only know a finite set of points.

Closed Curves
Let γ(s) = (x j (s)) n j=1 be a closed curve in R n of class C 2 parametrized by arc length s ∈ [0, L], where L is its total length. Assume we only know a finite number of consecutive points , with p 0 = p N , and we pretend to give an explicit parametrizationγ(s) approximating γ(s) with a relative error less than ε > 0, i.e. γ −γ ≤ ε γ , using for this the hermitian L 2 -product defined by f, g = 1 L L 0 fḡ ds. We consider the orthonormal basis given by the functions ψ (s) = e i2π s L , varying ∈ Z, and the corresponding Fourier series Theorem 1 gives us the bound . In order to obtain discrete counterparts of the continuous quantities involved in the precedent formula we proceed as follows. For each k = 1, . . . , N we define ds k = p k − p k−1 and s k = s k−1 + ds k taking also s 0 = 0. Then we can discretize the integrals involved above obtaining the following estimates for (a) the length L s N and the Fourier coefficients x j Besides the error ε given by considering only a finite number of Fourier terms, this procedure introduces two new sources of error, namely the approximations made in (a) and (b). Nevertheless, since the frequencies set Λ is discrete the method of choosing the subset L f (ε) is robust in the following sense. First, by perturbing slightly ε if necessary, we can assume that L ± f (ε) Vol. 12 (2015) Error Estimates for Approximations 59 are not close to integer numbers. Then, if the distances ds k between consecutive points are small enough then the set L f (ε), obtained by the discretization method described in (b), does not change.

Star-Shaped Surfaces
Let S ⊂ R 3 be a closed surface which is star-shaped with respect to the origin, i.e. for every u ∈ S 2 the half line {λ u, λ ∈ R + } cuts S in a unique point r(u)u determined by the radial function r: S 2 → R + which we can express as r = ( ,m)∈Λ r m ψ m according to the notations introduced in Example 2, where the coefficients where T i is the center of mass of the triangle T i , ψ m : R 3 → R is a degree polynomial extension of ψ m : S 2 → R and A(T i ) is the area of the spherical triangle obtained from T i by radial projection onto S 2 . In the same vein, the squared norm r 2 = 1 4π U r(χ(ϕ, θ)) 2 sin θ dθ dϕ can be approximated by In order to obtain a discrete counterpart of we need to consider the parametrization σ(ϕ, θ) = r(χ(ϕ, θ))χ(ϕ, θ) of S. A straightforward computation using that χ, ∂ θ χ and ∂ϕχ sin θ is a direct orthonormal basis, we obtain that the outer normal unitary vector N : S → S 2 of S satisfies the equality and consequently where N i is the outer normal unitary vector to the triangle T i , which can be easily computed from the given triangulation of S. Finally, to compute a discrete counterpart of the squared norm of the spherical laplacian of r, we apply the following formula given in [4] for the discrete version of the Laplacian of a function f defined in the vertex set of a triangulated surface M ⊂ R 3 : Here N (i) denotes the set indexing the vertex adjacent to p i . If j ∈ N (i) then α ij and β ij are the opposite angles to the edge p i p j . In our case we must take M = S 2 and f = r. Thus, is the set of vertex of the triangulation, T ij is the unique triangle containing the oriented edge p i p j and α ij and β ij are the opposite angles to the edge p i p j after projecting the triangulation radially onto S 2 .
Example 10. The triangulation of the surface of the left atrium of a human heart shown in Fig. 2 has N = 4,000 triangles and V = 2,002 vertex. From Vol. 12 (2015) Error Estimates for Approximations 61   Table 1, we list the L 2 -norm of the degree homogeneous part r = m=− r m ψ m of r for 0 ≤ ≤ 17. In Fig. 3 we represent graphically these norms in function of .