320 LETTER TO THE EDITOR Proof. It follows from(7)that mo(a)2+m1(a)2+…+|mk(a)2=1 m0(u)mo(+丌)+ml(m1(a+丌)+…+mn(a)mn(a+丌)=0. △2o):=∑f(a+2m24+1+272)(2-1-o+2l+x) th(12), fo L∈Z ∑(qm2+∑∑吃 k=ll∈2 te CC(+2 mk(2-L-1(o+2x2D)p(2--1(a+2x2D)da =(2x)2∑/|△1(o)mk( D do +(2x)2∑/|△2()m(2--1a+x)do 2L +(2)2∑ +丌)da L △2(o)mk(2-1-o+x)△1(a)mk(2--o)do CC(+2+0+2 +(2π) )(2-1-a+2rD)do ∑f,qL+1)2<∞ l∈Z Using lemma 2.2 we obtain of Lemma 23.320 LETTER TO THE EDITOR Proof. It follows from (7) that |m0(ω)| 2 + |m1(ω)| 2 + ··· + |mk(ω)| 2 = 1, m0(ω)m0(ω + π ) + m1(ω)m1(ω + π ) + ··· + mn(ω)mn(ω + π ) = 0. So, introducing the notation +1(ω) := l∈Z f (ˆ ω + 2π2 L+1 l)φ(ˆ 2−L−1ω + 2πl), +2(ω) := l∈Z f (ˆ ω + 2π2 L+1 l + 2π2 L )φ(ˆ 2−L−1ω + 2πl + π ), we have, by analogy with (12), for any L ∈ Z  l∈Z |
f,ϕL,l|2 + n k=1  l∈Z |
f,ψ k L,l |2 = (2π )2
π2 L −π2 L      l∈Z  f (ˆ ω + 2π2 L l)φ(ˆ 2−L(ω + 2π2 Ll))     2 dω + (2π )2n k=1
π2 L −π2 L      l∈Z  f (ˆ ω + 2π2 L l)φˆk (2−L(ω + 2π2 Ll))     2 dω = (2π )2n k=0
π2 L −π2 L      l∈Z  f (ˆ ω + 2π2 L l) × mk(2−L−1 (ω + 2π2 Ll))φ(ˆ 2−L−1 (ω + 2π2 Ll))     2 dω = (2π )2n k=0
π2 L −π2 L    +1(ω)mk(2−L−1ω)    2 dω + (2π )2n k=0
π2 L −π2 L    +2(ω)mk(2−L−1ω + π )    2 dω + (2π )2n k=0
π2 L −π2 L +1(ω)mk(2−L−1ω)+2(ω)mk(2 −L−1ω + π ) dω + (2π )2n k=0
π2 L −π2 L +2(ω)mk(2−L−1ω + π )+1(ω)mk(2 −L−1ω) dω = (2π )2
π2 L −π2 L      l∈Z  f (ˆ ω + 2π2 L+1 l)φ(ˆ 2−L−1ω + 2πl)     2 dω + (2π )2
π2 L −π2 L      l∈Z  f (ˆ ω + 2π2 L+1 l + 2π2 L )φ(ˆ 2−L−1ω + 2πl + π )     2 dω = (2π )2
π2 L+1 −π2 L+1      l∈Z  f (ˆ ω + 2π2 L+1 l)φ(ˆ 2−L−1ω + 2πl)     2 dω =  l∈Z |
f,ϕL+1,l|2 < ∞. Using Lemma 2.2 we obtain of Lemma 2.3.