Topological and geometric properties of refinable functions and MRA affine frames
Abbreviated Journal Title
Appl. Comput. Harmon. Anal.
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
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.
Applied and Computational Harmonic Analysis
"Topological and geometric properties of refinable functions and MRA affine frames" (2011). Faculty Bibliography 2010s. 1355.