Periodic Wavelets

We give de nitions of Multiresolution analysis and wavelet decomposition for a wide range of quasi Banach spaces of periodic distributions Elementary properties of such MRA are investigated Economical algorithms of wavelet decomposition and reconstruction are represented x Introduction We consider a general approach to de nition of Multiresolution analysis MRA fV g j of a broad range of quasi Banach spaces of periodic distributions generalized functions In spite of the fact that most of the known MRA of L satisfy our de nition it does not coincide with known ones see for example even for the space L Our constraints for MRA are more severe The result of these constraints is possibility to recover MRA by any of the spaces V j Periodizations of practically all classical MRA of the space L see possess such property We know only two examples of MRA introduced by Y Meyer p and C K Chui and H N Mhaskar which do not possess this property and consequently are not MRA from our point of view In x we introduce a special class H of quasi Banach spaces For these spaces we de ne the notion of MRA We note that most of classical spaces of distributions such as Lebesgue and Hardy spaces the Besov Lizorkin Triebel spaces the Orlicz spaces the space of Borel measures and many others belongs to H Because each of the spaces fV g are invariant with respect to the translate by j then it is clear that bases of eigen functions of an operator of the translate are of fundamental importance in the study of these spaces Using a representation in these bases we obtain a description of properties of the spaces V j In particular we nd a full description of all possible MRA of X H In x we de ne wavelet spaces for xed MRA ofX H Wavelet decomposition of the space X is generated by dual MRA of the spaces of bounded convolution operators acting from the space X to L Two numerical algorithms of wavelet decomposition and reconstruction are considered in x The rst of them is based on the transfer to the frequency domain For a function f V j this algorithm is of complexity O j j The second one is applicable only for a case analogous to the case of compactly supported generally speaking nonorthogonal wavelets for MRA of function spaces on a real line The complexity of such algorithm is equal to O j where is a size of the support In x we show that in every MRA there exist many bases of translates with nite masks of re nement equations For such bases we construct algorithms of wavelet expansion and recovery those are realized as convolutions with nite windows of a size at most a size of the mask x Multiresolution analysis of spaces of periodic functions and distributions We denote by D the space of periodic distributions We recall some principal properties of distributions Full constructing a theory of periodic distributions can be found for example in Distributions can be considered either as bounded linear functionals on the space C of in nitely di erentiable functions or as formal trigonometric series f x X