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   -> Volume 6, Issue 4


Preprint: Three papers on M-band wavelets.
 
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sqiyu@email.ims.cuhk.hk (Visitor Sun Qiyu)
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PostPosted: Sat Apr 05, 1997 6:37 am    
Subject: Preprint: Three papers on M-band wavelets.
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#8 Preprint: Three papers on M-band wavelets.

Enclosed please find three abstract on M-band wavelets.
The address to contact with is: Qiyu Sun, Center for Mathematical Sciences,
Zhejiang University, Hangzhou, Zhejiang 310027, P.R.China.

Title:M-band scaling function with filter having vanishing moments two and
minimal length

Author: Qiyu Sun and Zeyin Zhang

Abstract: Let $Mge 2$ be a fixed integer. A compactly supported and
square integrable function $phi$ is called a scaling function if it
satisfies $int phi(x)dx=1$, $int_R phi(x)phi(x-k)dx=delta_k $
and a refinement equation $phi(x)=sum_{kin Z} c_k
phi(Mx-k)$. Denote its symbol of the refinement equation by
$H(z)={1over M} sum_{kin Z} c_k z^k$. In this paper, we consider
some properties of the scaling function with its symbol
$H(z)=({1-z^Mover M(1-z)})^2 ({1+ hetaover 2}+{1- hetaover 2}z),
where $ heta=sqrt{2M^2+1over 3}$. It is to easy that that scaling
function is minimal supported scaling function which reproducing
polynomial with degree at most one. We consider the Holder
continuiuty, local linearity, linear independence and interpolation
problem.Especially we find that the filter is rational coefficient
when $M=11$, the corresponding scaling function is locally linear
function when $Mge 3$ which leads to exact Holder index of the
scaling function, and is locally linearly dependent. Differentiablity
at adjoint $M$-adic points is slightly different when $M=3$. Also the
interpolation is slightly different when $M=11$.

Title: Asymptotic Behaviour of M-band scaling functions of Daubechies type

Author: Ning Bi and Qiyu Sun

Abstract: Let $Mge 2$ be a fixed integer. A compactly supported and
square integrable function $phi$ is called a scaling function if it
satisfies $int phi(x)dx=1$, $int_R phi(x)phi(x-k)dx=delta_k $
and a refinement equation $phi(x)=sum_{kin Z} c_k
phi(Mx-k)$. Denote its symbol of the refinement equation by
$H(z)={1over M} sum_{kin Z} c_k z^k$. Let
{}^M_Na(s)=sum_{s_1+s_2+cdots+s_{M-1}=sprod^{M-1}_{j=1} inom
{N-1+s_j}{s_j} (2sin {jpiover m})^{-2s_j}, 0le sle N-1$, and
$P(t)=sum_{s=0}^{N-1} {}^M_N a(s) t^s$.Set $|{}_NH(xi)|^2=({sin
Mxi/2over sin xi/2})^{2N} P(2-22cosxi)$. We consider the
asymptotic behaviour of the scaling function with its symbol ${}_NH$
and the asymptotic behaviour of ${}_NH$ as $M$ tend to infinity. It is
proved that its limit function is linear combination of a B-spline and
its deritives.

Title: Integral Representation of M-band filters of Daubechies type

Author: Daren Huang, Qiyu Sun and Zeyin Zhang

Remark: this paper wqill appear in Ke Xue Tong Bao 1997.(Chinese Bulletin
of Sciences)

Abstract: Let ${}^M_Na(s)=sum_{s_1+s_2+cdots+s_{M-1}=sprod^{M-1}_{j=1}
inom {N-1+s_j}{s_j} (2sin {jpiover m})^{-2s_j}, 0le sle N-1$,
and $P(t)=sum_{s=0}^{N-1} {}^M_N a(s) t^s$.Set $|{}_NH(xi)|^2=({sin
Mxi/2over sin xi/2})^{2N} P(2-22cosxi)$. Meyer gives a formula
$$|{}_NH(xi)|^2=1-C_Nint^xi_0sin^{2N-1}omegadomega$$ when
$M=2$. We give similar result for $|{}_NH(xi)|^2$ when $Mge 2$. Also
similar formula is proved for interpolation filter.

Qiyu Sun
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