Algorithmic formulation of wavelet analysis

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The contents of researches published during two years from 2000 till 2001 can be classified into two parts, which are stated as follows :(1). The solution to an open problem related to Hilbert's 13th problem. The 13th problem formulated by Hilbert, which asks if any continuous functions of several variables can be represented as superposition representation constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after fifty years. Actually, the analytic function theoretic problem asking if any analytic functions of several variables can be represented as superposition representation constructed from several analytic functions of fewer variables has remained to be solved. The representative of this research project has succeeded in giving the solution to this problem which is based on entropy theoretic methods.(2). Application of the result (1) to the theory of data compression. It is known that the 13th problem is closely related to the problem of nomographs, which belongs to the theory of data compression. Exactly speaking, the set of all analytic functions of several variables, which can be represented as superpositions constructed from several analytic functions of fewer variables, is a subset of the first category. This solution implies that few analytic functions can be represented as superpositions constructed from several functions of fewer variables. Moreover, both this result and the wavelet analysis are also applied to the solution to the problem of generalized nomographs.