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 C2 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.

Bell-shaped nonstationary refinable ripplets / Pitolli, Francesca. - In: ADVANCES IN COMPUTATIONAL MATHEMATICS. - ISSN 1019-7168. - STAMPA. - 42:6(2016), pp. 1427-1451. [10.1007/s10444-016-9469-9]

Bell-shaped nonstationary refinable ripplets

PITOLLI, Francesca
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 C2 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.
2016
Bell-shaped function; Nonstationary biorthogonal basis; Nonstationary prewavelet; Refinable ripplet; Total positivity; Variation-diminishing
01 Pubblicazione su rivista::01a Articolo in rivista
Bell-shaped nonstationary refinable ripplets / Pitolli, Francesca. - In: ADVANCES IN COMPUTATIONAL MATHEMATICS. - ISSN 1019-7168. - STAMPA. - 42:6(2016), pp. 1427-1451. [10.1007/s10444-016-9469-9]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/924240
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