title: Solution of rational interpolation problem via the Hankel polynomial construction
metadata.name: dc.contributor.author
metadata.value: Uteshev, Alexei Yu.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Baravy, Ivan I.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2017-02-09T09:06:05Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2017-02-09T09:06:05Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2016-12
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Uteshev A.Yu., Baravy I. I. Solution of rational interpolation problem via the Hankel polynomial construction. Vestnik of Saint Petersburg University. Series 10. Applied mathematics. Computer science. Control processes, 2016, issue 4, pp. 31–43.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu10.2016.403
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/5994
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The problem of rational interpolant construction is treated as
  r(x) = p(x)/q(x), {r(xj) = yj}Nj
  =1, {xj, yj}Nj
  =1
  ⊂ 􀀀, {p(x), q(x)} ⊂ 􀀀[x] .
  At the basis of one result by C. Jacobi, the interpolant is represented as a ratio of two
  Hankel polynomials, i. e. polynomials of the form HK(x) = det[ci+j−1 − ci+j−2x]K
  i,j=1. The
  generating sequence for these polynomials is selected as {
   N
  j=1 xkj
  yj/W (xj )}k∈􀀀 for q(x) and
  as {
   N
  j=1 xkj
  /(yjW (xj ))}k∈􀀀 for polynomial p(x); here W(x) =
   N
  j=1(x−xj). The conditions for the solubility of the problem and irreducibility of the obtained fraction are also presented.
  In addition to formal representation of the solution in determinantal form, the present paper is
  focused also at the effective computational algorithm for the Hankel polynomials. It is based on
  a little known identity by Jacobi and Joachimsthal connecting a triple of the Hankel polynomials
  of successive orders:
  αHK(x) − (x + β)HK−1(x) + 1/αHK−2(x) ≡ 0 ,
  here {α, β} ⊂ 􀀀 are some constants. The proof of this relation is also contained in the paper
  along with discussion of a degenerate case α = 0. With these results, a procedure for the
  Hankel polynomial computation can be developed which is recursive in its order. This gives an
  opportunity not only to compute a single interpolant with specialized degrees for p(x) and q(x)
  but also to compose the whole set of interpolants for an arbitrary combination for the degrees:
  deg p+deg q   N − 1. The results of the paper can be applied for problems of Approximation
  Theory, Control Theory (transfer function reconstruction from frequency responses) and for
  error-correcting coding (Berlekamp—Welch algorithm). Although the presented results are
  formulated for the case of infinite fields, they are applicable for finite fields as well. Refs 12.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes;Issue 4
metadata.lang: -
metadata.name: dc.subject
metadata.value: rational interpolation
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Hankel matrices and polynomials
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Berlekamp—Massey algorithm
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Solution of rational interpolation problem via the Hankel polynomial construction
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 03-Uteshev.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/5994/1/03-Uteshev.pdf
resources.format: Adobe PDF
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