316 LETTER TO THE EDITOR hope that there exists a collection of several framelets y, v2,.,w"E V, satisfying the following conditions (4(=2m每 (2)for any fEL(R), algorithms of decomposition and reconstruction the recurrent formulac (k,f=c=∑c+1.-2 k∈Z 1 (5) (q3k,f)= Cj+. kgk c+1=∑ckh-k+∑∑k8,- k∈z where hk, gi are coefficients of the expansions mo(o)=2-Ekez hke-iko and k∈z8ke~k The goal of Section 2 is to show that this problem can be solved with at most two framelets and to present explicit formulae for symbols of the framelets. In Sections 3 and 4 we prove that in the case when mo(o) is either a rational function or a polynomial we can choose mI(o), m2(o)as rational functions or polynomials respectively 2. GENERAL FRAMELETS Let be a refinable function with a symbol mo, v(o)=mk(o /2)(o/2)E V,where each symbol mk is a 2T-periodic and essentially bounded function for k= l, 2,..., n. It is well known that for constructing practically important tight frames the matrix m0(u) mn(o) m0(a+丌)m1(a+丌) plays an important role It is easy to see that the equality M(o)M(o)=I (7) is equivalent to(5)and(6) It turns out that(7)also implies the tightness of the corresponding frame THEOREM 2.1. If(7) holds, then the functions y el generate a tight frame Remark. For n= l this theorem was proved in [5] for polynomial symbols and in [2] for the general case. For an arbitrary n it was proved in [8] under some additional decay assumption for and in [3] for an arbitrary polynomial symbol. In[8] Theorem 2. 1 wa alled the unitary extension principle316 LETTER TO THE EDITOR hope that there exists a collection of several framelets ψ 1 , ψ2 ,...,ψ n ∈ V 1 , satisfying the following conditions: (1) functions {{ψ l j,k }j,k∈Z} n l=1 , where ψ l j,k (x) = 2 j/2ψl(2 j x − k), form a tight frame of the space L 2 (R); (2) for any f ∈ L 2 (R), algorithms of decomposition and reconstruction the recurrent formulae
ϕj,k , f = cj,l =  k∈Z cj+1,kh¯ k−2l, 1 ≤ q ≤ n, (5)
ϕ g j,k , f = d q j,l =  k∈Z cj+1,kg¯ q k−2l , and cj+1,l =  k∈Z cj,khl−k + n q=1  k∈Z d q j,k g q l−k , (6) where hk , g q k are coefficients of the expansions m0(ω) = 2 −1/2 k∈Z hke −ikω and mq (ω) = 2 −1/2  k∈Z g q k e −ikω, q = 1,..., n take place. The goal of Section 2 is to show that this problem can be solved with at most two framelets and to present explicit formulae for symbols of the framelets. In Sections 3 and 4 we prove that in the case when m0(ω) is either a rational function or a polynomial we can choose m1(ω), m2(ω) as rational functions or polynomials respectively. 2. GENERAL FRAMELETS Let ϕ be a refinable function with a symbol m0, ψˆ k (ω) = mk(ω/2)φ(ˆ ω/2) ∈ V 1 , where each symbol mk is a 2π-periodic and essentially bounded function for k = 1, 2,..., n. It is well known that for constructing practically important tight frames the matrix M(ω) = m0(ω) m1(ω) ... mn(ω) m0(ω + π ) m1(ω + π) ... mn(ω + π )  , plays an important role. It is easy to see that the equality M(ω)M∗ (ω) = I (7) is equivalent to (5) and (6). It turns out that (7) also implies the tightness of the corresponding frame. THEOREM 2.1. If (7) holds, then the functions {ψ k } n k=1 generate a tight frame of L 2 (R). Remark. For n = 1 this theorem was proved in [5] for polynomial symbols and in [2] for the general case. For an arbitrary n it was proved in [8] under some additional decay assumption for φˆ and in [3] for an arbitrary polynomial symbol. In [8] Theorem 2.1 was called the unitary extension principle.