Bell-shaped nonstationary refinable ripplets
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Bell-shaped nonstationary refinable ripplets Francesca Pitolli1
Received: 10 January 2015 / Accepted: 23 June 2016 / Published online: 6 July 2016 © Springer Science+Business Media New York 2016
Abstract We study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variationdiminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C 2 nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords Total positivity · Variation-diminishing · Refinable ripplet · Bell-shaped function · Nonstationary prewavelet · Nonstationary biorthogonal basis Mathematics Subject Classification (2010) 41A30 · 42C40 · 65T60
Communicated by: Tomas Sauer Dedicated to Laura Gori on the occasion of her 80th birthday Francesca Pitolli
[email protected] 1
Dip. SBAI, Universit`a di Roma “La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy
1428
F. Pitolli
1 Introduction A ripplet is a function f whose integer translates are totally positive [27], i.e., for any ordered real numbers x1 < · · · < xr , and any ordered integers α1 < · · · < αr , r ≥ 1, it holds det f (xi − α ) 1≤i,≤r ≥ 0. (1.1) Total positivity implies that the integer translates of f are variation diminishing, i.e., for any finite sequence c = {cα } − S cα f (· − α) ≤ S − (c) , (1.2) α
where S − denotes the number of strict sign changes of its argument. The inequality (1.2) in turn implies that the system {f (·−α)} has shape-preserving properties, which are known to play a crucial role in several applications, from approximation of data to CAGD [12, 29]. The concept of a ripplet was first introduced by Goodman and Micchelli in [16], where the authors focused their interest on two-scale refinable ripplets, i.e., ripplets that are solution of a two-scale refinable equation ϕ=
aα ϕ(2 · −α) ,
(1.3)
α
where the scaling mask a = {aα } is a suitable real sequence. Well known examples of refinable ripplets are the cardinal B-splines, i.e., the polynomial B-splines on integer nodes. Starting from the seminal paper of Go
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