Generalized interpolating refinable function vectors

  • Authors:

  • Bin Han;Soon-Geol Kwon;Xiaosheng Zhuang

  • Affiliations:

  • Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1;Department of Mathematics Education, Sunchon National University, Sunchon 540-742, Republic of Korea;Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

Interpolating scalar refinable functions with compact support are of interest in several applications such as sampling theory, signal processing, computer graphics, and numerical algorithms. In this paper, we shall generalize the notion of interpolating scalar refinable functions to compactly supported interpolating d-refinable function vectors with any multiplicity r and dilation factor d. More precisely, we are interested in a d-refinable function vector @f=[@f"1,...,@f"r]^T such that @f is an rx1 column vector of compactly supported continuous functions with the following interpolation property @f"@?(mr+k)=@d"k@d"@?"-"1"-"m,@?k@?Z,m=0,...,r-1,@?=1,...,r, where @d"0=1 and @d"k=0 for k0. Now for any function f:R@?C, the function f can be interpolated and approximated by f@?=@?@?=1r@?k@?Zf(@?-1r+k)@f"@?(@?-k)=@?k@?Z[f(k),f(1r+k),...,f(r-1r+k)]@f(@?-k). Since @f is interpolating, f@?(k/r)=f(k/r) for all k@?Z, that is, f@? agrees with f on r^-^1Z. Moreover, for r=2 or d2, such interpolating refinable function vectors can have the additional orthogonality property: =r^-^1@d"@?"-"@?"^"'@d"k"-"k"^"' for all k,k^'@?Z and 1=