W. Freeden, M. Schreiner, and R. Franke, A survey on spherical spline approximation, Surveys Math. Indust., 7:29--85, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....so called variational methods have their counterparts on the sphere. Strictly speaking, variational methods belong to the previous subsection since the kernels 20 associated with the extremal problem also happen to be radially symmetric. The methods have been frequently used in geosciences, see [29 31,33,75 77]. Another popular method, often applied in meteorology, is the classical spectral method, which uses spherical harmonics of high degrees to approximate functions on the sphere (which corresponds to using ordinary polynomials in the plane) see [44] 3.4. Distance Weighting Methods This class of ....

....processing to CAGD and computer animation. That said, there do not seem to exist many methods based on subdivision and or wavelets, specifically designed to deal with the problem of functions on surfaces. Notable exceptions are the various existing constructions of wavelets on the sphere (see [16,32,33,43,51,67]) and the paper [72] where wavelets are constructed on general surfaces. 3.9. Visualization of Surfaces on Surfaces Although visualization is not addressed in this chapter, it is important to stress that a good visualization of the reconstructed modeled surfaces and functions is essential in ....

W. Freeden, M. Schreiner, and R. Franke, A survey on spherical spline approximation, Surveys Math. Indust., 7:29--85, 1997.

....by a function of the form # ## 2q # l # =1 c f(##, # #) can be performed. When the points # belong to a real sphere contained in# 2q then the restriction of the above interpolant to that sphere becomes a spherical radial basis function or spherical spline; the reader is referred to [3,6] and references therein for acquaintance with this recent and important field of research in approximation theory. The main purpose of this paper is to present a complete characterization of continuous bizonal positive definite kernels on# 2q thus generalizing a famous result of I. J. Schoenberg ....

W. Freeden, M. Schreiner and R. A. Franke, A survey on spherical spline approximation, Surveys Math. Indust. 7 (1997), 29--85.

....the corresponding error bounds based on power functions. There are different ways to define native spaces (see [10] for comparisons) but in the first part [20] of the survey we wanted to provide just one technique that is general enough to unify different constructions (e.g. on the sphere [3] or on Riemannian manifolds [2,13] But we avoided advanced tools like Fourier transforms or expansions into series of spherical harmonics or eigenfunctions of the Laplace Beltrami operator. In this continuation of [20] we start with embedding of native spaces into L 2 ( Omega Gamma4 This ....

Freeden, W., R. Franke and M. Schreiner, A survey on spherical spline approximation, Surveys on Mathematics for Industry 7 (1997), 29--85.

....motions. ffl Invariance on the sphere S d Gamma1 under all orthogonal transformations leads to zonal functions Phi(x; y) OE(x T y) for OE : Gamma1; 1] IR. ffl Spaces of periodic functions induce periodic reproducing kernels. See [5] for basis functions on topological groups, and see [3] for a review of results on the sphere. The paper [13] introduces the theory of basis functions on general manifolds, and corresponding error bounds are in [2] x4. Native Spaces of Positive Definite Functions Instead of a single point x 2 Omega with a single evaluation functional ffi x 2 H ....

Freeden, W., R. Franke, and Schreiner, M., A survey on spherical spline approximation, Surveys on Mathematics for Industry 7 (1997) 29--85

....W for our present investigation of cubature formulas. The degree of exactness of such formulas is usually defined in terms of graded spaces of spherical harmonics. Let W = e V denote the space of spherical harmonics up to order . The fundamental properties of these functions are described in [2,7,9]. Furthermore, it is known that W satisfies a Markov inequality with constant c W = 4, Section 2.3] Note that the related inequality in the survey paper by W. Freeden et al. 2, Lemma 2.1] does not reproduce the best possible result as far as the power of the polynomial degree is ....

W. Freeden, M. Schreiner and R. Franke, A survey on spherical spline approximation, Surv. Math. Ind. 7 (1997), 29--85.

....n Gamma1 ) i.e. in the norm induced by the (normalized) inner product hf; gi = 1 n Gamma1 Z S n Gamma1 f(x)g(x) dx; f; g 2 L 2 (S n Gamma1 ) 4) with n Gamma1 the surface area of the unit sphere S n Gamma1 ae IR n . For more details we refer to the books [4, 6] the recent survey [1], or [2] Classical Hilbert spaces, such as Sobolev spaces H s (S n Gamma1 ) of real order s 0, can be defined by introducing a weight in the summability condition on the Fourier coefficients, e.g. see [5] For two points x; y 2 S n Gamma1 and a set X ae S n Gamma1 we denote the ....

Freeden, W., M. Schreiner, and R. Franke, A survey on spherical spline approximation, Surv. Math. Ind. 7 (1997), 29--85.

.... kGk H(fA Gamma1 n g; Omega Gamma : Hence, Delta; Delta) H(f1g; Omega Gamma defines a duality between H(fA n g; Omega Gamma and H(fA Gamma1 n g; Omega Gamma4 Of importance for our considerations are Sobolev spaces equipped with a reproducing kernel structure (cf. 8] [10] for more details) Theorem 2.1. Let fA n g be summable (i.e. A n 6= 0 for all n 0 and P 1 n=0 2n 1 4 1 A 2 n 1) Then the space H = H(fA n g; Omega Gamma (furtheron we often write H instead of H(fA n g; Omega Gamma if no confusion is likely to arise) is a functional Hilbert subspace ....

....just a graphical impression in Figure 4.2. 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 3 2 1 0 1 2 3 Figure 4.2: Modified rational (P )wavelets Psi j (cos #) # 2 [ Gamma ; j = 0; 2, 0 (x) 1 x 2 ) Gamma4 . 4.2. 3 Exponential Wavelets From the Sobolev Lemma (cf. e.g. [10]) we know that the order of differentiability of the rational scaling function and its modification is finite. A scaling function of class C (1) can be obtained via the generator 0 (x) e Gammah(x) x 2 [0; 1) where h : 0; 1) R is assumed to satisfy the following conditions: i) h 2 ....

[Article contains additional citation context not shown here]

Freeden, W., Schreiner, M., Franke, R. (1995): A Survey on Spherical Spline Approximation. Surveys on Mathematics for Industry (accepted for publication)

No context found.

Freeden, W., Schreiner, M., Franke, R. (1996b) A Survey on Spherical Spline Approximation. Surv. Math. Ind., 7, 29-85.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC