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51.
Khalfa Douak 《Journal of Computational and Applied Mathematics》1996,70(2):279-295
We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.
A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite. 相似文献
52.
《Discrete Mathematics》1996,150(1-3):303-313
Given a natural number n, an exact formula is derived for the minimal possible size MD(n) of a square grid, in which a digital convex n-gon can be inscribed. An exact construction of a digital convex n-gon which can be inscribed into a square grid of size MD(n) is also given. 相似文献
53.
We introduce and study matrix Christoffel functions for a matrix
weight W.
We find an explicit expression of the matrix Christoffel functions
in terms of any sequence of orthonormal matrix polynomials with
respect to W. An extremal property related to the matrix moment
problem defined by W is established for the matrix Christoffel
functions. We finally find the relative asymptotic behavior of
the matrix Christoffel functions associated to matrix weights in
the matrix Nevai class. 相似文献
54.
Let $\{P_n(x) \}_{n=0}^\infty$ be an orthogonal polynomial system
relative to a compactly supported measure. We find
characterizations for $\{P_n(x) \}_{n=0}^\infty$ to be a
Bochner--Krall orthogonal polynomial system, that is, $\{P_n(x)
\}_{n=0}^\infty$ are polynomial eigenfunctions of a linear
differential operator of finite order. In particular, we show that
$\{P_n(x) \}_{n=0}^\infty$ must be generalized Jacobi polynomials
which are orthogonal relative to a Jacobi weight plus two point
masses. 相似文献
55.
Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner productwhere the mass points ak belong to [−1,1], Mk,i0, i=0,…,Nk−1, and Mk,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants Mk,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in
. 相似文献
56.
57.
Fourier transform infrared (FTIR) spectroscopy has being emphasised as a widespread technique in the quick assess of food components. In this work, procyanidins were extracted with methanol and acetone/water from the seeds of white and red grape varieties. A fractionation by graded methanol/chloroform precipitations allowed to obtain 26 samples that were characterised using thiolysis as pre-treatment followed by HPLC-UV and MS detection. The average degree of polymerisation (DPn) of the procyanidins in the samples ranged from 2 to 11 flavan-3-ol residues. FTIR spectroscopy within the wavenumbers region of 1800-700 cm−1 allowed to build a partial least squares (PLS1) regression model with 8 latent variables (LVs) for the estimation of the DPn, giving a RMSECV of 11.7%, with a R2 of 0.91 and a RMSEP of 2.58. The application of orthogonal projection to latent structures (O-PLS1) clarifies the interpretation of the regression model vectors. Moreover, the O-PLS procedure has removed 88% of non-correlated variations with the DPn, allowing to relate the increase of the absorbance peaks at 1203 and 1099 cm−1 with the increase of the DPn due to the higher proportion of substitutions in the aromatic ring of the polymerised procyanidin molecules. 相似文献
58.
Hjalmar Rosengren 《Journal of Combinatorial Theory, Series A》2008,115(3):376-406
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions. 相似文献
59.
The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. 相似文献
60.
A result on pure shear provides the motivation for the determination of some new general results relating real second order
Cartesian tensors.
相似文献