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Тезисы докладов

Аппроксимация функций и квадратурные формулы

The possibility of (nonorthogonal) multiresolution analysis with arbitrary (for example $B$--spline) scaling function is investigated. Let $varphi$ and $psi$ be some functions such that $$ varphi(x)=sum _{kin Z} a_kvarphi(2x-k), psi(x)=sum _{kin Z} b_kvarphi(2x-k) $$ and let $$ V_0=left{varphi(cdot-k)right}_{kin Z};, V_1=left{varphi(2cdot-k)right}_{kin Z};, W_0=left{psi(cdot-k)right}_{kin Z} ;. $$ Let $a*b$ denote the quasiconvolution of sequences $a$ and $b$: $$ (a*b)_p= sum_{sin Z} (-1)^s a_{p-s} b_s, pin Z. $$ smallskipnoindent {bf THEOREM. } {it The decomposition $V_1=V_0 dot+ W_0$ holds if and only if there exist $nin Z$ and $Ane 0$ such that $$ (a*b)_{2p+1}= Adelta_n^p mbox{for all} p in Z. eqno{(1)} $$ If (1) holds then $$ f(x)=sum {alpha_k(f)varphi(x-k)+beta_k(f)psi(x-k)} $$ for every $f in V_1$ of the form $$ f(x)=sum f_kvarphi(2x-k), $$ where $$ alpha_k(f)=frac 1A(f*b)_{2k+2n+1}, beta_k(f)=-frac 1A(f*a)_{2k+2n+1}, kin Z. eqno{(2)} $$ }smallskip For example if $varphiin W_2^m(R)$ is the $m$--order $B$--spline ($(m+1)$--multiple convolution of characteristic function of $[0,1]$) with supp $varphi=[0,m+1]$ then $a_k=2^{-m}C_{m+1}^k$; therefore (1) and (2) take the form $$ nabla ^{m+1}b_{2p+1}=adelta_n^p, eqno{(1')} $$ $$ alpha_k(f)=frac {2^m}a(b*f)_{2k+2n+1}, beta_k(f)=frac {1}a nabla ^{m+1}f_{2k+2n+1}, eqno{(2')} $$ where $nabla f_p=f_p-f_{p-1}$ is the backward finite difference. If (1') holds then $left{psi(2^scdot-k)right}_{s,kin Z}$ is the basis of $W_2^m( R)$. The simplicity of (2') is the compensation for nonorthogonality of this basis. The family of functions $psi$ with property (1') and support $[0,m]$ of minimal measure can be completely described in the constructive manner.

Примечание. Тезисы докладов публикуются в авторской редакции

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