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81.
J. Arvesú L. L. Littlejohn F. Marcellán 《Journal of Computational Analysis and Applications》2002,4(4):363-387
In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L2(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L2(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A1, first observed by Everitt, and the self-adjoint operator A1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A). 相似文献
82.
This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier--Legendre sums of order n with 2r terms of the form
k=1
2r
akPn+k(x) added; here P
m(x) denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval [-1,1], which, in fact, for r= = 1 allows us to significantly improve the approximation properties of partial Fourier--Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions
and A
q
(B). With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties. 相似文献
83.
G. Min 《Journal of Approximation Theory》1999,98(2):197
This note characterizes the denseness of rational systems
in C[−1, 1], where the nonreal poles in {ak}∞k=1
\[−1, 1] are paired by complex conjugation. This extends an Achiezer's result. 相似文献
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84.
It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfLp, 1p<∞, the former only has rateω(f, n−1)p, and the latter cannot even be bounded byC fp. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔn(yj) of each sign changeyj. 相似文献
85.
In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance 2. The value of the average density for the case of =0 and 2=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula. 相似文献
86.
Shuhong Gao 《Proceedings of the American Mathematical Society》1999,127(6):1615-1623
A method is given for constructing elements in whose orders are larger than any polynomial in when becomes large. As a by-product a theorem on multiplicative independence of compositions of polynomials is proved.
87.
H. M. Srivastava 《Proceedings of the American Mathematical Society》1999,127(2):385-396
For a natural number , the author derives several families of series representations for the Riemann Zeta function . Each of these series representing converges remarkably rapidly with its general term having the order estimate:
Relevant connections of the results presented here with many other known series representations for are also pointed out.
88.
Daniel Carando Ignacio Zalduendo 《Proceedings of the American Mathematical Society》1999,127(1):241-250
We study the problem of extendibility of polynomials over Banach spaces: when can a polynomial defined over a Banach space be extended to a polynomial over any larger Banach space? To this end, we identify all spaces of polynomials as the topological duals of a space spanned by evaluations, with Hausdorff locally convex topologies. We prove that all integral polynomials over a Banach space are extendible. Finally, we study the Aron-Berner extension of integral polynomials, and give an equivalence for non-containment of .
89.
Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type
In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.
90.
We study limit distribution of partial sums SN,k(t) =
s = 1
[N t]
Ak(Xs) of Appell polynomials of the long-range dependent moving average process Xt> = i t bt - i i, where {i} is a strictly stationary and weakly dependent martingale difference sequence, and bi id - 1 (0 < d < 1/2). We show that if k(1-2 d)<1, then suitably normalized partial sums SN,k(t) converge in distribution to the kth order Hermite process. This result generalizes the corresponding results of Surgailis, and Avram and Taqqu obtained in the case of the i.i.d. sequence { i}. 相似文献