Hermite-Like Interpolating Refinable Function Vector and Its Application in Signal Recovering

作者：Li; Youfa

来源：Journal of Fourier Analysis and Applications, 2012, 18(3): 517-547.

DOI：10.1007/s00041-011-9208-z

• Hermite-like interpolating refinable function vectors
• Symmetry
• Interpolating theorem
• Approximation orders
• Truncated error estimate
• Computing derivatives
• Recovering signals

Interpolating refinable function vectors with compact support are of interest in applications such as sampling theory, numerical algorithm, and signal processing. Han et al. (J. Comput. Appl. Math. 227: 254-270, 2009), constructed a class of compactly supported refinable function vectors with (d, r)-interpolating property. A continuous d-refinable function vector phi = (phi(1),..., phi(r))(T) is (d, r)-interpolating if
phi(l)(m/r + k) = delta(k)delta(l-1-m), for all k is an element of Z, m = 0,1,..., r-1, l = 1,..., r.
In this paper, based on the (d, r)-interpolating refinable function vector phi is an element of (C-1(R))(r), we shall construct r functions phi(r+ 1),..., phi(2r) such that the new d-refinable function vector phi(t) = (phi(T), phi(r+1),..., phi(2r))(T) belongs to (C-1(R))(2r) and has the Hermite-like interpolating property:
phi l(m/r + k) = 0, phi(l)'(m/r + k) = delta(k)delta(l-r-1-m),
for all k is an element of Z, m = 0,1,..., r - 1, l = r + 1,..., 2r.
Then any function f is an element of C-1(R) can be interpolated and approximated by
(f) over tilde = Sigma(r)(l=1)Sigma(k is an element of Z)f(k + l-1/r)phi(l)(.-k) + Sigma(j is an element of Z)Sigma(2r)(l'=r+1)[f'(j + l'-r-1/r) - Sigma(r)(l=1)Sigma(k is an element of Z)f(k + l-1/r)phi(l)' (j + l'-r-1/r - k)]phi(l')(.-j).
That is, (f) over tilde ((kappa))( k + m/r) = f((kappa))(k + m/r), for all kappa is an element of {0, 1}, for all k is an element of Z, and m = 0, 1,..., r - 1. When phi has symmetry, it is proved that so does phi(t) by appropriately selecting some parameters. Moreover, we address the approximation order of f phi(t). A class of Hermite-like interpolating refinable function vectors with symmetry are constructed from phi such that they have higher approximation order than it. Several examples of Hermite-like interpolating refinable function vectors are given to illustrate our results. The truncated error estimate of the interpolating series above is given in Sect. 3. A numerical example of recovering signal is given in Sect. 5 to check the efficiency of the interpolating formula above.

• 出版日期2012-6
• 单位广西大学