Trigonometric rational wavelet bases

We propose a construction of periodic rational bases of wavelets First we explain why this problem is not trivial Construction of wavelet basis is not possible neither for the case of alge braic polynomials nor for the case of rational algebraic functions Of course algebraic polynomials do not belong to L R Nevertheless they can belong to the closure of L R in topology of the generalized convergence However there is not polynomial bases because any shift of polynomial is a polynomial of the same degree So dimension of linear span of a set of polimomial shifts has a nite dimension As for rational bases the reason of non existance is di erent It is clear that the rational function whose shifts f x n g constitutes a basis of the space V cannot have poles in the real line Let d is a maximal distance from poles of x to the real line The function x generates a basis in V V A maximal distance from poles of x to the real line is equal to d It contradicts to the fact that the function x in some sence can be approximated by linear combinations of f x n g Recall classic construction of periodic polynomial wavelets It is based on periodization of non periodic Multiresolution analysis MRA consisting of entire functions As above MRA Meyer s wavelets or any their mod i cations can be taken These examples of triginometric polynomial MRA allowed to resolve many non trivial problems of Analysis relating to con structing orthogonal polynomial bases with special properties We intend to propose MRA that possesses the following three properties the MRA consists of rational trigonometric functions the uniform limit function of the sequence of the periodic interpolating scaling functions n x is the Shannon scaling function