Abstract
Formulas are given for the fast expansion of a smooth function on a segment, which allows one to increase the rate of convergence of the Fourier series unlimitedly depending on the order of expansion. The theorems on the convergence of a series in fast expansion, the possibility of its multiple termwise differentiation, the uniqueness of fast expansion, are proved, and the error in the residual series in fast expansion is estimated. Algorithms for applying fast expansion are given, the concept of the fast expansion operator, which is necessary for solving nonlinear integro-differential problems, is introduced.
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