Ukrainian Mathematical Journal, Vol. 59, No. 12, 2007 ESTIMATES FOR WAVELET COEFFICIENTS ON SOME CLASSES OF FUNCTIONS V. F. Babenko 1, 2 and S. A. Spector 2 UDC 517.5 Let ψ m D be orthogonal Daubechies wavelets that have m zero moments and let W p k 2, = f L i f k p ∈ ≤ { } 2 1 ( ): ( ) ˆ ( ) R ωω , k ∈ N . We prove that lim sup ( , ) ( ) : , m m D m D q p k f f W →∞ ′ ∈ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ψ ψ Ÿ 2 = ( ) ( ) / / / / / 2 1 2 1 2 1 12 1 1 1 12 π π π p k pk p q pk - - - - - ⎛ ⎝ ⎞ ⎠ . Let L p = L p ( R ) , 1 ≤ p ≤ ∞, be the space of measurable functions f : R → C with finite norm f p , where f p = f L p ( ) R = fx dx p p () / R ∫ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 if p <∞ and f ∞ = f L ∞ ( ) R = vrai sup () x fx ∈R . For f L p ∈ ( ) R and g L q ∈ ( ) R , where pq , [; ] ∈∞ 1 , 1 1 p q + = 1, we set ( f, g ) = fxgxdx () () R ∫ . We consider the following classes of functions f L ∈ 2 ( ) R : For k ∈ N and p ∈ ( 1, ∞ ), we set W p k 2, = f L i f k p ∈ ≤ { } 2 1 ( ): ( ) ˆ ( ) R ωω , where 1 Dnepropetrovsk National University, Dnepropetrovsk. 2 Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1594–1600, December, 2007. Original article submitted June 19, 2006. 0041–5995/07/5912–1791 © 2007 Springer Science+Business Media, Inc. 1791