Algorithm for Decomposition of Integers and Smooth Approximation of Functions

The problem of expansion in powers is generalized into decomposition positive integers the sequence degrees different orders, con-ditions are determined, and algorithm for decomposi-tion constructed. based on two procedures: 1) achieve-ment a minimum residual at each step; 2) speeding decom-position through expanding local base by reducing in-dex, which ensures finiteness algorithm. has such effi-ciency factors as high rate decomposition, ease implementation, availability options numbers ex-tended, narrowed, sparse bases, protects encoded information from external influences. can be used to encode large amounts digital under basic systems small dimensions. Decomposition opti-mal correct. Optimality follows condition that step value disjunction space mixed parameters x∈N,y∈Ris achieved. Correctness algo-rithm due fact when reduced, expands basis degree indicators one. By switching discrete model continuous replac-ing with power functions, we obtain smooth approximation ill-conditioned function neighborhood decomposition. construction posinomial polynomials polynomi-als one promising directions integration non-differentiable functions replacement variables catas-trophe theory.Posinomials (functions variable exponent) predict splitting interval parts, since they determine loga-rithmic change an arbitrary monotonic function. method provides optimal sum powers, therefore transition replacing well allows achieve accura-cy approximation.