Error estimates for Hermite interpolation on spheres
Let \(S^{d-1}\) denote the unit sphere in the Euclidean space \(\mathbb{R}^d\) and \(C^r(S^{d-1})\) be the set of \(r\)-times differentiable functions on \(S^{d-1}\). Let \(xy\) denote the usual inner product of \(x,y\in\mathbb{R}^d\) and \(d(x, y)=\cos^{-1}xy\) be the geodesic distance between \(x,y\in S^{d-1}\). Let \(\mu\) be the normalised rotation invariant surface measure on \(S^{d-1}\). Furthermore, let \(P_m\) be the subspace of spherical polynomials of degree not greater than \(m\), \(H_m=P_m \cap(P_{m-1})^\perp\) be the set of spherical harmonics of degree \(m\), and \(d_m=\dim (H_m)\). Choose an orthonormal basis \(\{Y_{m,1}, \dots,Y_{m,d_m}\}\) for \(H_m\), \(m=0,1,\dots\). Then, every \(f\in L_1(S^{d-1})\) (the integrable functions on the sphere) has a Fourier series expansion \(f\sim\sum^\infty_{m=0} \sum^{d_m}_{j=1} \widehat f_{m,j}Y_{m,j}\), where \(\widehat f_{m,j}= \int_{S^{d-1}}fY_{m,j} d\mu\). Let \(h:[-\pi,\pi] \to\mathbb{R}\) be a continuous function and \(X=\{x_1, \dots, x_N\}\) be a set of distinct interpolation points. The usual interpolation problem is to find a function \(s(x)=\sum^N_{j=1}c_j h(d(x,x_j))+ p(x)\), \(p\in P_{m-1}\), such that \(s(x_i)=f(x_i)\), \(x_i\in X\), where \(c_j\), \(j=1, \dots,N\), satisfy \(\sum^N_{j=1} c_jq(x_i)=0\), \(\forall q\in P_{m-1}\). The node set \(x\) is assumed to be \(P_{m-1}\)-unisolvent, i.e. if \(p\in P_{m-1}\) and \(p(x)=0\) for all \(x\in X\) then \(p=0\). Furthermore \(N\geq\dim(P_{n-1})\). Let \(r\in \mathbb{N}\) and \(\wedge=\{L_1, \dots,L_N\}\subset (c^r(S^{d-1}))'\) (the dual of \(c^r (S^{d-1}))\). The authors consider the following Hermite interpolation problem: find a function \[ s(x)=\sum^N_{j=1} c_j\cdot L^y_j\biggl(h \bigl(d(x,y)\bigr) \biggr)+p(x),\;p\in P_{m-1}, \] such that \(L_i(s)=L_i(f)\), \(i=1, \dots,N\), where \(c_j\), \(j=1,\dots,N\), satisfy \(\sum^N_{j=1} c_jL_j(q)=0\), \(\forall q\in P_{m-1}\) and \(L^y\) denotes the action on the \(y\) variable. It is assumed that the function \(h\) is \(2r\)-times continuously differentiable and the functional set \(\wedge\) is \(P_{m-1}\)-unisolvent, i.e. if for some \(q\in P_{m-1}\), \(L(q)=0\) for all \(L\in \wedge\) then \(q=0\). Moreover, if \(h:[-\pi,\pi] \to\mathbb{R}\) has the absolutely convergent expansion \(h(t)=\sum^\infty_{k=0} h_k\cdot P_k^{(d-3)/2}(\cos t)\), then \(h\) is positive definite iff \(h_k\geq 0\) for every \(k\geq 0\) \((P_k^{(d-3)/2}\) are the Gegenbauer polynomials). The authors make the stronger assumption: \(h_k>0\) for all \(k\geq 0\). In this case the interpolation problem is to find a function \(s(x)=\sum^N_{j=1} c_jL_j^y(h(d(x,y)))\), such that \(L_i(s)=L_i(f)\), \(i=1,\dots, N\). The native space \(S_h\) is defined by \[ S_h=\left\{f: \sum^\infty_{k=0} {d_k \over h_k}\sum^{d_k}_{j=1} (\widehat f_{k,j})^2=: \|f\|^2_h <\infty \right\}. \] Furthermore, if \(A\) is an element of the Lie algebra of \(O^d\), the rotation group of \(\mathbb{R}^d\), and \(\sigma=\exp A\) then the differential operator \(A_\sigma\) is defined by \[ A_\sigma f(x)=\lim_{t\to 0}{f\bigl(\exp (tA)x\bigr) -f(x)\over t},x\in S^{d-1}. \] For \(\sigma=(\sigma_1, \dots, \sigma_j) \in(O^d)^j,\;A_\sigma= A_{\sigma_j}\circ \cdots\circ A_{\sigma_1}\) denotes the \(j\)th order rotational derivative. The main result is the following theorem. Let \(h\in C^{2r}[-\varepsilon,\varepsilon]\), for some \(\varepsilon>0\), with \(r\in\mathbb{N}\), be even, and positive definite on \(S^{d-1}\). Fix \(x\in S^{d-1}\) and let \(\Omega=\{y\in S^{d-1}: d(x,y)\leq \varepsilon/2 \}\). Assume that the interpolation distribution set \(\wedge= \{L_1,\dots,L_N\}\) contains a subset \(\wedge_1:=\{L_1= \delta_{x_i}\circ A_\sigma:i=1, \dots,Q\}\), for some fixed \(\sigma\in (O^d)^k\), with \(k\leq r\), such that the supporting set \(X:=\{x_1,\dots, x_Q\}\subset \Omega\) satisfies the following inequality: \(\max_{y \in\Omega} \min_{x_i\in X}\{d(y,x_i)\}\leq p\). Let \(f\) be from the native space \(S_h\), and let \(s_f(x):=\sum^N_{j=1} c_jL_j^u(h (d(x,y)))\), subject to the interpolation conditions \(L_i(s)= L_i(f)\), \(i=1, \dots,N\). Then for \(\gamma \in(O^d)^\ell\) with \(k+\ell\leq r\), we have \[ \biggl|\bigl[A_\gamma \circ A_\sigma(f-s_f) \bigr](x)\biggr |\leq cp^{r-k-\ell}. \]
scientific article; zbMATH DE number 1956843
scientific article