LETTER TO THE EDITOR m0(+r)m1(a+丌) ind mo(o), mI(o) are essentially bounded functions mo(-o)=mo(o); i.e., Fourier series of these functions have real coefficients. It is known(see [4]that for any scalin function (r)and the associated wavelet y(x), generating an orthogonal wavelet basi the corresponding symbols mo(o), mI(o) satisfy(2). Any refinable function whose symbol mo is solution to(2), generates a tight frame(see [5] for the case when mo is polynomial, the general case was proved in [2]) We cannot independently look for the functions mo and mI. In fact, usually we find solution of the equation m(o)2+|mo(a+m)2=1 and then all possible functions mI can be represented in the form mI(o=a(@mo(o+T) where a(o)is an arbitrary T-periodic function, satisfying la(o)l=l,a(-o)=a(o) Now suppose we have an arbitrary refinable function (o) with the symbol mo which does not satisfy (3). Then the set lo(r-n)Inez does not constitute an orthonormal basis If this set forms a Riesz basis, then we can use orthogonalization, proposed in [6] However, in this case, when the function o has a compact support, this property fails for the orthogonalized basis. This argues for construction other systems keeping compactness of support. It will be shown in Section 4 that tight frame of wavelets leads to one of the possible compactly supported systems. We note that sometimes the orthogonalization can be conducted even if our set Is not a Riesz basis. The simplest example gives a refinable function 1/2,|x|≤ 0 x|>1 with the symbol mo(o)=cos 20. In this case the MRa coincides with the Haar MRA Thus. the function 1,0≤x≤1; g(x)= x>lor x <0: is the natural orthogonalization Nevertheless, it is easy to design a refinable function such that its mRa does not allow orthogonalization. Indeed, let us introduce a refinable function (x)=sinax/tx where< a< 1. It generates the space Vo which consists of functions of L2(R)with Fourier transform supported on [-a, a]. Thus, for any function f E v the function chez If(o 2k)l vanishes on the set [-, T]a, ar]. Hence, its integer translates do not form an orthonormal bases(see [4D). In this case the traditional procedure of constructing an orthonormal wavelet basis cannot be applied. We note that by the same reason even a biorthogonal pair with this MRa cannot be constructed In the case when the symbol mo of a refinable function p does not satisfy (3)we cannot onstruct an orthonormal bases of v of the form (x-k), y(x-k)). However, we carLETTER TO THE EDITOR 315 where M(ω) = m0(ω) m1(ω) m0(ω + π ) m1(ω + π )  , and m0(ω), m1(ω) are essentially bounded functions m0(−ω) = m0(ω); i.e., Fourier series of these functions have real coefficients. It is known (see [4]) that for any scaling function ϕ(x) and the associated wavelet ψ(x), generating an orthogonal wavelet basis, the corresponding symbols m0(ω), m1(ω) satisfy (2). Any refinable function ϕ, whose symbol m0 is solution to (2), generates a tight frame (see [5] for the case when m0 is polynomial, the general case was proved in [2]). We cannot independently look for the functions m0 and m1. In fact, usually we find a solution of the equation |m0(ω)| 2 + |m0(ω + π )| 2 = 1, (3) and then all possible functions m1 can be represented in the form m1(ω) = α(ω)e iωm0(ω + π ), (4) where α(ω) is an arbitrary π-periodic function, satisfying |α(ω)| = 1, α(−ω) = α(ω). Now suppose we have an arbitrary refinable function ϕ(ω) with the symbol m0 which does not satisfy (3). Then the set {ϕ(x − n)}n∈Z does not constitute an orthonormal basis of V 0 . If this set forms a Riesz basis, then we can use orthogonalization, proposed in [6]. However, in this case, when the function ϕ has a compact support, this property fails for the orthogonalized basis. This argues for construction other systems keeping compactness of support. It will be shown in Section 4 that tight frame of wavelets leads to one of the possible compactly supported systems. We note that sometimes the orthogonalization can be conducted even if our set is not a Riesz basis. The simplest example gives a refinable function ϕ(x) =  1/2, |x| ≤ 1; 0, |x| > 1; with the symbol m0(ω) = cos 2ω. In this case the MRA coincides with the Haar MRA. Thus, the function ϕ(x) =  1, 0 ≤ x ≤ 1; 0, x > 1 or x < 0; is the natural orthogonalization. Nevertheless, it is easy to design a refinable function such that its MRA does not allow orthogonalization. Indeed, let us introduce a refinable function ϕ(x) = sinπax/πx, where 0 < a < 1. It generates the space V 0 which consists of functions of L 2 (R) with Fourier transform supported on [−aπ, aπ]. Thus, for any function f ∈ V 0 the function  k∈Z |f (ˆ ω + 2kπ)| 2 vanishes on the set [−π,π]\[aπ, aπ]. Hence, its integer translates do not form an orthonormal bases (see [4]). In this case the traditional procedure of constructing an orthonormal wavelet basis cannot be applied. We note that by the same reason even a biorthogonal pair with this MRA cannot be constructed. In the case when the symbol m0 of a refinable function ϕ does not satisfy (3) we cannot construct an orthonormal bases of V 1 of the form {ϕ(x − k),ψ(x − k)}. However, we can