The present paper is the detailed exposition of the results announced in [1]. Introduction. In 1985, Meyer constructed [2] an infinitely smooth function ~b(t), t 9 N 1, with compact spectrum, such that the system 2J/2~b(2Jt -k), j,k 9 Z, forms an orthonormalized basis in .L2(]R~).Following Oskolkov, we call such functions wavelets. At present there are known wavelets decreasing exponentially at infinity [3,4] and compactly supported wavelets [5]. However, these functions have finite smoothness. From the papers [6] and [7, p. 93] it follows that there do not exist infinitely smooth compactly supported wavelets.In this paper we consider the system kl/ :: {qP0k,~)jk, j = 0,1 .... ; k 9 Z} of infinitely smooth functions that have the following properties:(1) @ is an orthonormalized basis in L2(N1); (2) ~0k(t) = ~00(~ -k), Cjk(t) = Cj0(~ -k2-J);Unlike wavelets, the system @ is not generated by contractions and translations of one function. However, We denote by F and F -1 direct and inverse Fourier transforms, respectively.Voronezh Civil Engineering Institute.

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