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Series that can be differentiated term-wise m times if the function is m-smooth

A. G. Ramm

Let f ∈ Cm(−, ), where m > 0 is an integer. An algorithm is pro- posed for representing f as a convergent series which admits m times term-wise differentiation. This algorithm is illustrated by numerical ex- amples. It can be used, for example, for acceleration of convergence of Fourier series. The algorithm is generalized to the case when f is piecewise-Cm(−, ) function with known positions of finitely many jump discontinuities and the sizes of the jumps and to the case when these positions and the sizes of the jumps are unknown. A jump dis- continuity point s is a point at which at least one of the quantities dj := f(j)(s − 0) − f(j)(s + 0) 6= 0, where 0 ≤ j ≤ m.