Series that can be differentiated term - wise m times if the function is m - smooth
نویسنده
چکیده
Abstract Let f ∈ C(−π, π), where m > 0 is an integer. An algorithm is proposed for representing f as a convergent series which admits m times term-wise differentiation. This algorithm is illustrated by numerical examples. It can be used, for example, for acceleration of convergence of Fourier series. The algorithm is generalized to the case when f is piecewise-C(−π, π) 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 discontinuity point s is a point at which at least one of the quantities dj := f (s− 0)− f (s+ 0) 6= 0, where 0 ≤ j ≤ m.
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