Showing papers on "Biorthogonal system published in 1989"

Multivariable Meixner, Krawtchouk, and Meixner–Pollaczek polynomials

Abstract: A multivariable biorthogonal generalization of the Meixner, Krawtchouk, and Meixner–Pollaczek polynomials is presented. It is shown that these are orthogonal with respect to subspaces of lower degree and biorthogonal within a given subspace. The weight function associated with the Krawtchouk polynomials is the multivariate binomial distribution.

Some properties of biorthogonal polynomials

Abstract: BRIEF COMMUNICATIONS SOME PROPERTIES OF BIORTHOGONAL POLYNOMIALS A. P. Golub UDC 517.53 In studying Pade approximations of functions with the aid of generalized moment repre- sentations [i] one is obliged to construct and investigate various systems of biorthogonal polynomials. In the present paper we establish some general results concerning biorthogonal po lynomials. The starting point for obtaining these results is the following proposition, applied here in a somewhat more general form than in the primary source. THEOREM 1 (Dzyadyk [i])

Axioms of determinacy and biorthogonal systems

Abstract: If allΠn1 games are determined, every non-norm-separable subspaceX ofl∞(N) which is W* —Σn+1/1 contains a biorthogonal system of cardinality 2ℵ0. In Levy’s model of Set Theory, the same is true of every non-norm-separable subspace ofl∞(N) which is definable from reals and ordinals. Under any of the above assumptions,X has a quotient space which does not linearly embed into 1∞(N).

A new modal formalism in analysis of radiation and scattering problems

Abstract: A biorthogonal eigenmode formalism is presented for the analysis of radiation and scattering problems. The proposed formalism is shown to be computationally simpler, faster in convergence, and easier in application than other mode formalisms previously proposed. Mode computation examples are presented. >

On biorthogonality of Hermitian and skew-Hermitian Szego/Levinson polynomials

Abstract: A system of polynomials, biorthogonal with respect to a given weight function which is not necessarily even, is constructed from a corresponding system of Szego or Levinson polynomials. Two-term coupled recurrences are presented for the biorthogonal polynomial system, and it is shown that these recurrences reduce to a pair of three-term uncoupled recurrences in the case that the unit circle weight function is even. Several observations relative to recent linear prediction algorithms utilizing Hermitian and skew-Hermitian Levinson polynomials are discussed. >

A system of biorthogonal polynomials and its applications

Abstract: A. P. Go lub UDC 517.53 In [i] there were constructed generalized moment representations for basic hypergeo-metric series. There arose the problem of the biorthogonalization of the sequences of func-\"-q~'-1 q~+~ 1 , i = O, r ~v (q) = tions {~i(~)= t~162 and {[~1(1)-/v't-xi(q~}~=o, where L~+I (q)= q-1 (q-l) q l q-p (1) i=0 N BN (t)-S d~'v'[5~ (t), d~ Iv) ::/: 0, N = 0, or l=0 I AM (t) BN (t) dt o We will prove the following result. THEOREM i. The polynomials AM(t), M = 0, =, and (2) can be represented in the form M ra{ra-I) m 2 qM-~+l __ 1 AM(t) = E (-l)~q H qr_ 1 ra=O r~l N n(n-l) B~v (t) = E (-l)\"q ~ n~O r=l (1') = O, M=/=N. (4) (5) (q-1/ t ~(~-l) ql~r+l 1 bl(t)-') (-l)'~q\" ~ [~m(t), 1=0~. Proof. It is obvious that if we define an operator A:C[0, i] ~ C[0, i] by the formula (A~) (t) = t o ~ ep (tqu) uVdu o and a polynomial AM(t) by (3), then the relations (AkAM)(1) = 0, k = i, M, M = i, ~, will hold. Therefore if we take into account the equality I I ((A(p) (t) ~p (t) dt = ~ q) (t) (B~p) (t) dr, ~ o where the operator B:LI[O, 1] ~ Li[O, 1) has the form 1 p-{-I-?q-itl