title: Necessary and sufficient conditions of nonnegativity for coordinate trigonometrical splines of the second order
metadata.name: dc.contributor.author
metadata.value: Dem’yanovich, Yuri K.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Makarov, Anton A.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2017-07-05T08:42:23Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2017-07-05T08:42:23Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2017-03
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Dem’yanovich Yu.K., Makarov A.A. Necessary and sufficient conditions of nonnegativity for coordinate trigonometrical splines of the second order. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2017, vol. 4 (62), issue 1, pp. 9–16.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu01.2017.102
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/6677
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: There are many different ways to define coordinate splines. The splines, which are defined with
  approximation relations, have the best approximation properties as to order of N-width for the standard
  compact sets. The suitable choice of the approximation relations gives the maximal smoothness under the
  condition of minimal support for coordinate splines. On the other hand, the choice of different generating
  function gives opportunity to receive splines of different types (polynomial, exponential, trigonometrical
  ones and so on).
  The paper is discussed the positivity of coordinate trigonometrical splines of the second order with
  maximal smoothness.
  Let us consider the real grid {xj}j∈Z,
  . . . < x−2 < x−1 < x0 < x1 < x2 < . . .
  By definition we put M def
  = ∪j∈Z (xj , xj+1), Sj
  def
  = [xj−1, xj+2], and consider three-component vectors:
  generating vector function ϕ(t) and family of vectors aj , defined with formulas
  ϕ(t) def
  = (1, sin t, cos t)T , aj
  def
  = det(ϕj ,ϕj+1,ϕ ′
  j+1)ϕ ′
  j − det(ϕ ′
  j , ϕj+1,ϕ ′
  j+1)ϕj ,
  where ϕj
  def
  = ϕ(xj), ϕ ′
  j
  def
  = ϕ′(xj).
  The coordinate trigonometric splines ωj ∈ C1 are defined by approximation relations
  ak−1ωk−1(t) + akωk(t) + ak+1ωk+1(t) ≡ ϕ(t) ∀t ∈ (xk, xk+1), ∀k ∈ Z,
  ωj(t) ≡ 0 ∀t ∈ M\Sj , ∀j ∈ Z.
  Theorem. The inequality xj+1 − xj < π ∀j ∈ Z is necessary and sufficient condition for
  nonnegativity of the functions ωj(t) ∀j ∈ Z.
  Furthermore, the sufficient conditions of convexity on the intervals (xj−1, xj), (xj+1, xj+2) and
  concavity on the interval (xj , xj+1) have been obtained in the paper. Refs 16. Figs 2.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: Работа частично выполнена при финансовой поддержке РФФИ (гранты №15-01-08847, №16-31-60060 мол_а_дк).
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 St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 4 (62); Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: nonnegativity
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: coordinate splines
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: trigonometrical splines
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Necessary and sufficient conditions of nonnegativity for coordinate trigonometrical splines of the second order
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 02-Demjanovich.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/6677/1/02-Demjanovich.pdf
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