LETTER TO THE EDITOR m(2-(+2xk)2 ∏m(2-"(a+2r(k-2)2 ∑∏Im(2"(a+2xk)2 ∑ ∑∏m(2-"(o+2rk)2 pplying the Plancherel and Parseval formulae, we have ∑ k∈Z (2x)2/∑(f(a+2x2/n)(2-(a+2x2/n)de -2JIneZ (2r‖F‖) (12) where F(o)=nez(f(o 2T2Jn)(2-1(o+22n))). Let us introduce the folle ing sequences of functions <2 l≥2 G(o)=∑((+2x2m3(27(0+2x2m) H1()=∑(,(a+2m2027(0+2x2m It is clear that, on the one hand.G川→(27)-1/2fasj→o. On the other hand, n view of(11)318 LETTER TO THE EDITOR ≤ 1 2π 2 l−1 k=0  l+1 n=1 |m(2 −n (ω + 2πk))| 2 +  l+1 n=1 |m(2 −n (ω + 2π(k − 2 l )))| 2  ≤ 1 2π 2 l−1 k=0  l n=1 |m(2 −n (ω + 2πk))| 2 ≤ 1 2π 2 l−1−1 k=0  l n=1 |m(2 −n (ω + 2πk))| 2 +  l n=1 |m(2 −n (ω + 2π(k + 2 l−1 )))| 2  ≤ 1 2π 2 l−1−1 k=0  l−1 n=1 |m(2 −n (ω + 2πk))| 2 ≤ ··· ≤ 1 2π . Applying the Plancherel and Parseval formulae, we have  k∈Z |
f,"j,k |2 = 2π2 −j  k∈Z    
∞ −∞ f (ˆ ω)"( ˆ 2−jω)e i2 −jωk dω     2 = 2π2 −j  k∈Z    
π2 j −π2 j  n∈Z f (ˆ ω + 2π2 j n)"( ˆ 2−j (ω + 2π2 jn)) × e i2 −jωk dω     2 = (2π )2
π2 j −π2 j      n∈Z  f (ˆ ω + 2π2 j n)"( ˆ 2−j (ω + 2π2 jn))     2 dω = (2πFj ) 2 , (12) where Fj (ω) =  n∈Z (f (ˆ ω + 2π2 jn)"( ˆ 2−j (ω + 2π2 jn))). Let us introduce the follow￾ing sequences of functions gˆj (ω) =    f (ˆ ω), |ω| < 2 jπ; , hj = f − gj , j = 0, 1, 2,..., 0, |ω| ≥ 2 jπ; Gj (ω) =  n∈Z  gˆj (ω + 2π2 j n)"( ˆ 2−j (ω + 2π2 jn)) , Hj (ω) =  n∈Z  hˆ j (ω + 2π2 j n)"( ˆ 2−j (ω + 2π2 jn)) . It is clear that, on the one hand. Gj  → (2π )−1/2f  as j → ∞. On the other hand, in view of (11),