Spherical Harmonics

Müller, Claus

doi:10.1007/bfb0094775

Cited by 646 publications

(381 citation statements)

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“…a ,k and x 0 = p. (14) We can now formulate the main result of this paper. It makes use of Corollary 1 in order to give estimates for the pointwise interpolation error.…”

Section: Lagrange Interpolation By Splinesmentioning

confidence: 99%

“…As our main analysis tool we use an orthonormal basis of L 2 (S n−1 ) which consists of spherical harmonics. For the reader's convenience, we describe some basic facts here, and refer to the book [14] for a more elaborate description.…”

Section: Prerequisitesmentioning

confidence: 99%

“…In our subsequent error estimates, which are simple consequences of Theorem 2, we will therefore use assumptions on the decay of the sequence ( a N (n, )) ≥0 , where a is defined in (14).…”

Section: This Implies Thatmentioning

confidence: 99%

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Error estimates for scattered data interpolation on spheres

Jetter¹,

Stöckler²,

1999

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Abstract. We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

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“…a ,k and x 0 = p. (14) We can now formulate the main result of this paper. It makes use of Corollary 1 in order to give estimates for the pointwise interpolation error.…”

Section: Lagrange Interpolation By Splinesmentioning

confidence: 99%

Section: Prerequisitesmentioning

confidence: 99%

See 1 more Smart Citation

Error estimates for scattered data interpolation on spheres

Jetter¹,

Stöckler²,

1999

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“…The following lemma, which is a variant of the Funk-Hecke formula (see [14]), facilitates the computation of each entry of the matrix B.…”

Section: The Elliptic Partial Differential Equation On Spheresmentioning

confidence: 99%

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

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Abstract. We present an overlapping domain decomposition technique for solving elliptic partial differential equations on the sphere. The approximate solution is constructed using shifts of a strictly positive definite kernel on the sphere. The condition number of the Schwarz operator depends on the way we decompose the scattered set into smaller subsets. The method is illustrated by numerical experiments on relatively large scattered point sets taken from MAGSAT satellite data.

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“…Dick Askey has informed us that this formula follows immediately from the Funk-Hecke theorem [1] and a familiar integral giving the Bessel functions as a Fourier transform. Bochner essentially has this in his book [17]; also see Claus Müller's lectures on spherical harmonics [43] and [44]; it is probably in some notes of Calderon published in Argentina, but they are not widely available. Also see [10], [11], and recent papers [15] and [16].…”

Section: Expansion Formula For a Plane Wavementioning

confidence: 99%

Solution of the Cauchy problem for a time-dependent Schrödinger equation

Meiler

Cordero-Soto

Suslov

2008

Journal of Mathematical Physics

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Abstract. We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.

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Spherical Harmonics

Cited by 646 publications

References 0 publications

Error estimates for scattered data interpolation on spheres

Error estimates for scattered data interpolation on spheres

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

Solution of the Cauchy problem for a time-dependent Schrödinger equation

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