Convergence of polyharmonic splines on semi-regular grids $\mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}}$ for ${\boldsymbol{a}\rightarrow\mathbf {0}}$
Tóm tắt
Let p, n ∈ ℕ with 2p ≥ n + 2, and let I
a
be a polyharmonic spline of order p on the grid ℤ × aℤ
n
which satisfies the interpolating conditions
$I_{a}\left( j,am\right) =d_{j}\left( am\right) $
for j ∈ ℤ, m ∈ ℤ
n
where the functions d
j
: ℝ
n
→ ℝ and the parameter a > 0 are given. Let
$B_{s}\left( \mathbb{R}^{n}\right) $
be the set of all integrable functions f : ℝ
n
→ ℂ such that the integral
$$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$
is finite. The main result states that for given
$\mathbb{\sigma}\geq0$
there exists a constant c>0 such that whenever
$d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$
j ∈ ℤ, satisfy
$\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $
for all j ∈ ℤ there exists a polyspline S : ℝ
n+1 → ℂ of order p on strips such that
$$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$
for all y ∈ ℝ
n
, t ∈ ℝ and all 0 < a ≤ 1.
Tài liệu tham khảo
Aronszajn, N., Creese, T., Lipkin, L.: Polyharmonic Functions. Clarendon, Oxford (1983)
Bejancu, A., Kounchev, O., Render, H.: Cardinal interpolation with biharmonic polysplines on strips, curve and surface fitting (Saint Malo 2002), pp. 41–58. Mod. Methods Math., Nashboro, Brentwood, TN (2003)
Bejancu, A., Kounchev, O., Render, H.: Cardinal interpolation with periodic polysplines on strips. (to appear in Calcolo)
Buhmann, M.: Radial Basis Functions : Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)
Dyn, N.: Interpolation and approximation by radial and related functions. Approximation theory VI, vol. I (College Station, TX, 1989), pp. 211–234. Academic, Boston, MA (1989)
Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Pseudo-Differential Operators. Springer-Verlag, Berlin–Heidelberg–New York–Tokyo (1983)
Kounchev, O.: Optimal recovery of linear functionals of Peano type through data on manifolds. Comput. Math. Appl. 30(3–6), 335–351 (1995)
Kounchev, O.I.: Multivariate Polysplines. Applications to Numerical and Wavelet Analysis. Academic, London–San Diego (2001)
Kounchev, O., Render, H.: Wavelet analysis of cardinal L-splines and construction of multivariate prewavelets. In: Chui, C.K., et al. (eds.) Approximation Theory X. Wavelets, Splines, and Applications, pp. 333–353. Vanderbilt University Press, Nashville, TN. Innovations in Applied Mathematics (2002)
Kounchev, O., Render, H.: The approximation order of polysplines. Proc. Am. Math. Soc. 132, 455–461 (2004)
Kounchev, O., Render, H.: Polyharmonic splines on grids ℤ×aℤn and their limits. Math. Comp. 74, 1831–1841 (2005)
Kounchev, O., Render, H.: Cardinal interpolation with polysplines on annuli. J. Approx. Theory 137, 89–107 (2005)
Kounchev, O., Wilson, M.: Application of PDE methods to Visualization of Heart Data. In: Wilson, M.J., Martin, R.R. (eds.) Mathematics of Surfaces. Lecture Notes in Computer Science vol. 2768, pp. 377–391. Springer, Berlin Heidelberg New York (2003)
Liu, Y., Lu, G.: Simultaneous approximations for functions in Sobolev spaces by derivates of polyharmonic cardinal splines. J. Approx. Theory 101, 49–62 (1999)
Madych, W.R., Nelson, S.A.: Polyharmonic cardinal splines. J. Approx. Theory 60, 141–156 (1990)
Madych, W.R., Nelson, S.A.: Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70, 94–114 (1992)
Schaback, R.: Multivariate interpolation and approximation by translates of a basis function. In: Chui, C.K., et al. (eds.) Approximation Theory VIII, vol. 1, pp. 491–514. Approximation and Interpolation. World Scientific, Singapore (1995)
Schoenberg, I.J.: Cardinal Spline Interpolation. SIAM, Philadelphia, PA (1973)
Schoenberg, I.J.: Cardinal interpolation and spline functions. J. Approx. Theory 2, 167–206 (1969)
Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)