Topological and geometric properties of refinable functions and MRA affine frames

    Authors

    D. G. Han; Q. Y. Sun;W. S. Tang

    Comments

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    Abbreviated Journal Title

    Appl. Comput. Harmon. Anal.

    Keywords

    Refinable functions; Path-connectivity; Nowhere density; Multiresolution; analysis; Affine frame; SHIFT-INVARIANT SUBSPACES; WAVELET DIMENSION FUNCTION; POLY-SCALE; REFINABILITY; SPACES; CONNECTIVITY; L(2)(R(D)); SETS; Mathematics, Applied; Physics, Mathematical

    Abstract

    We investigate some topological and geometric properties of the set R of all refillable functions in L-2(R-d), and of the set of all MRA affine frames. We prove that R is nowhere dense in L-2(R-d); the unit sphere of R is path-connected in the L-2-norm; and for any M-dimensional hyperplane generated by L-2-functions f(0).....f(M), either almost all the functions in the hyperplane are refinable or almost all the functions in the hyperplane are not refinable. We show that the set of all MRA affine frames is nowhere dense in L-2(R-d). We also obtain a new characterization of the L-2-closure of (R) over bar of R, and extend the above topological and geometric results from R to (R) over bar, and even further to the set of all refillable vectors and its L-2-closure. (C) 2010 Elsevier Inc. All rights reserved.

    Journal Title

    Applied and Computational Harmonic Analysis

    Volume

    30

    Issue/Number

    2

    Publication Date

    1-1-2011

    Document Type

    Article

    Language

    English

    First Page

    151

    Last Page

    174

    WOS Identifier

    WOS:000286705400002

    ISSN

    1063-5203

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