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1.
Summary. In this paper, after recalling the two definitions of the generalizations of the Padé approximants to orthogonal series, we will define the Padé–Legendre approximants of a Legendre series. We will propose two algorithms for the recursive computation of some sequences of these approximants. We will also estimate the speed of convergence of the columns of the Padé–Legendre table from the asymptotic behaviour of the coefficients of the Legendre series. Finally we will illustrate these results with some numerical examples. Received June 20, 1998 / Published online March 20, 2001  相似文献   

2.
In this paper we study Padé-type and Padé approximants for rectangular matrix formal power series, as well as the formal orthogonal polynomials which are a consequence of the definition of these matrix Padé approximants. Recurrence relations are given along a diagonal or two adjacent diagonals of the table of orthogonal polynomials and their adjacent ones. A matrix qd-algorithm is deduced from these relations. Recurrence relations are also proved for the associated polynomials. Finally a short presentation of right matrix Padé approximants gives a link between the degrees of orthogonal polynomials in right and left matrix Padé approximants in order to show that the latter are identical. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

3.
Laurent Padé-Chebyshev rational approximants,A m (z,z −1)/B n (z, z −1), whose Laurent series expansions match that of a given functionf(z,z −1) up to as high a degree inz, z −1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z −1)B n (z, z −1)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.  相似文献   

4.
When constructing multivariate Padé approximants, highly structured linear systems arise in almost all existing definitions [10]. Until now little or no attention has been paid to fast algorithms for the computation of multivariate Padé approximants, with the exception of [17]. In this paper we show that a suitable arrangement of the unknowns and equations, for the multivariate definitions of Padé approximant under consideration, leads to a Toeplitz-block linear system with coefficient matrix of low displacement rank. Moreover, the matrix is very sparse, especially in higher dimensions. In Section 2 we discuss this for the so-called equation lattice definition and in Section 3 for the homogeneous definition of the multivariate Padé approximant. We do not discuss definitions based on multivariate generalizations of continued fractions [12, 25], or approaches that require some symbolic computations [6, 18]. In Section 4 we present an explicit formula for the factorization of the matrix that results from applying the displacement operator to the Toeplitz-block coefficient matrix. We then generalize the well-known fast Gaussian elimination procedure with partial pivoting developed in [14, 19], to deal with a rectangular block structure where the number and size of the blocks vary. We do not aim for a superfast solver because of the higher risk for instability. Instead we show how the developed technique can be combined with an easy interval arithmetic verification step. Numerical results illustrate the technique in Section 5.Research partly funded by FWO-Vlaanderen.  相似文献   

5.
Properties of Padé approximants to the Gauss hypergeometric function 2F1(a,b;c;z) have been studied in several papers and some of these properties have been generalized to several variables in [6]. In this paper we derive explicit formulae for the general multivariate Padé approximants to the Appell function F1(a,1,1;a+1;x,y)=i,j=0(axiyj/(i+j+a)), where a is a positive integer. In particular, we prove that the denominator of the constructed approximant of partial degree n in x and y is given by , where the integer m, which defines the degree of the numerator, satisfies mn+1 and m+a2n. This formula generalizes the univariate explicit form for the Padé denominator of 2F1(a,1;c;z), which holds for c>a>0 and only in half of the Padé table. From the explicit formulae for the general multivariate Padé approximants, we can deduce the normality of a particular multivariate Padé table. AMS subject classification 41A63, 41A21  相似文献   

6.
Let p n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g Ω (z) . We prove corresponding results when we restrict the freedom of p n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.  相似文献   

7.
The Baker-Gammel-Wills Conjecture states that if a functionf is meromorphic in a unit diskD, then there should, at least, exist an infinite subsequenceNN such that the subsequence of diagonal Padé approximants tof developed at the origin with degrees contained inN converges tof locally uniformly inD/{poles off}. Despite the fact that this conjecture may well be false in the general Padé approximation in several respects. In the present paper, six new conjectures about the convergence of diagonal Padé approximants are formulated that lead in the same direction as the Baker-Gammel-Wills Conjecture. However, they are more specific and they are based on partial results and theoretical considerations that make it rather probable that these new conjectures hold true.  相似文献   

8.
This paper gives the definition and some properties of a new family of Padé-type approximants (PTA) for k-variate formal power series (FPS). These PTA have the form P(t)/Q(t) where Q(t) = Πri = 0(1 ? x(it), {x(i), 0 ? i ? r} being a given set of points in
, and x·t is the scalar product of x and t in
. Some results about the approximation order, the unicity and some invariance properties of these PTA are proved together with a convergence result when the FPS is defined by a Stieltjes integral.  相似文献   

9.
The connection between orthogonal polynomials, Padé approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Padé and Padé-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

10.
Summary. For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function being rational, is well-known. In [Lubi88] Lubinsky proved that if is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of is such that it generates a non-normal Padé table. This implies that the Padé operator is an almost always continuous operator because it is continuous when computing a normal Padé approximant [Wuyt81]. In this paper we generalize the above results to the case of multivariate Padé approximation. We distinguish between two different approaches for the definition of multivariate Padé approximants: the general order one introduced in [Levi76, CuVe84] and the so-called homogeneous one discussed in [Cuyt84]. Received December 19, 1994  相似文献   

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