Riesz multiwavelet bases generated by vector refinement equation |
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Authors: | Song Li ZhiSong Liu |
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Institution: | (1) Department of Mathematics, Zhejiang University, Hangzhou, 310027, China;(2) School of Mathematics, Physics & Information Science, Zhejiang Ocean University, Zhoushan, 316004, China |
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Abstract: | In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L
2(ℝ
s
). Suppose ψ = (ψ1,..., ψ
r
)
T
and are two compactly supported vectors of functions in the Sobolev space (H
μ(ℝ
s
))
r
for some μ > 0. We provide a characterization for the sequences {ψ
jk
l
: l = 1,...,r, j ε ℤ, k ε ℤ
s
} and to form two Riesz sequences for L
2(ℝ
s
), where ψ
jk
l
= m
j/2ψ
l
(M
j
·−k) and , M is an s × s integer matrix such that lim
n→∞
M
−n
= 0 and m = |detM|. Furthermore, let ϕ = (ϕ1,...,ϕ
r
)
T
and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr
)
T
and , ν = 1,..., m − 1 such that two sequences {ψ
jk
νl
: ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ
s
} and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ
s
} form two Riesz multiwavelet bases for L
2(ℝ
s
). The bracket product f, g] of two vectors of functions f, g in (L
2(ℝ
s
))
r
is an indispensable tool for our characterization.
This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123) |
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Keywords: | vector refinement equations Riesz multiwavelet base biorthogonal wavelets |
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