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31.
Feng Dai 《Constructive Approximation》2005,22(3):417-436
Let $\sigma_k^\delta$ denote the Cesaro means of order $\delta > -1$ of the spherical harmonic expansions on the unit sphere $S^{d-1}$, and let $E_j(f, H^1)$ denote the best approximation of $f$ in the Hardy space $H^1(S^{d-1})$ by spherical polynomials of degree at most $j$. It is known that $\lambda:= (d-2)/2$ is the critical index for the summability of the Cesaro means on $H^1(S^{d-1})$. The main result of this paper states that, for $ f\in H^1(S^{d-1})$, $$\sum_{j=0}^N \f 1{j+1} \|\sigma_j^\lambda (f) -f\|_{H^1}\approx \sum_{j=0}^N \f 1{j+1} E_j(f, H^1),$$ where “$\approx$” means that the ratio of both sides lies between two positive constants independent of $f$ and $N$. 相似文献
32.
An elegant and fast method for the calculation of geometrical structure coefficients needed for an expansion of a few-body
wavefunction and interaction in hyperspherical harmonics has been proposed. A sum rule for the GSC has also been derived,
which is useful for an independent check of the coefficients. The proposed method of computation is many orders of magnitude
faster than conventional methods. 相似文献
33.
A complete potential harmonic scheme is presented,including the linked coupled hyperradial ordi nary differential equations and the secular equation of eigencnergy It has been used to directly solve the Scchrodinger equations of helium-like three-body systems (nuclear charge Z=1-9),and very accurate ground state eigonenergies as well as low-lying singlet excited state ones have been obtained 相似文献
34.
Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions. 相似文献
35.
Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) . This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space . The quality of the spectral approximation and the choice of the parameter c when approximating a function in by its truncated PSWFs series expansion, are the main issues. By considering a function as the restriction to of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples. 相似文献
36.
37.
A method is proposed to study the stress concentration around a shallow spheroidal crack in an infinite elastic body. The
stress concentration is due to the diffraction of a low-frequency plane longitudinal wave by the crack. The direction of wave
propagation is established in which the combined concentration of mode I and mode II stresses is maximum
__________
Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 70–77, January 2006. 相似文献
38.
39.
J. Haglund 《Advances in Mathematics》2011,227(5):2092
A special case of Haiman?s identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q,t. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman?s formula for the Hilbert series into an explicit polynomial in q,t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series. 相似文献
40.
Coulomb Sturmians are obtained in prolate spheroidal coordinates by separation of variables in the Schrödinger equation and direct solution of the appropriate one-dimensional equations. Molecular orbitals are expressed as linear combinations of the introduced Coulomb Sturmians and some low-lying energy terms and corresponding wave functions are calculated for one-electron diatomic molecules. It is shown that similarity of the one- and two-centre orbitals in spheroidal coordinates, combined with completeness and good convergence properties of Coulomb Sturmians, substantially speeds up convergence and makes the calculated results closer to the exact ones. Application of the elaborated calculating scheme for diatomic many-electron molecules is discussed. 相似文献