Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002
Applying an Euler type transformation for accelerating convergence of trigonometric series
Faton M. Berisha and Muharrem Q. Berisha
Faculty of Mathematical and Natural Sciences, University of Prishtina, Prishtina,Kosova
For a sequence of real or complex numbers and a given sequence the linear operator is defined by
Theorem 1. Let be a convergent real or complex trigonometric series and a sequence of real or complex numbers such that . Then the following equation holds true
where are .
Corollary 1. If there exist the finite limits , then series on the left-hand side of (1) converges faster then the series
on the right-hand side.
The acceleration of convergence is increased with the value of p.
Example 1. Let . Then Obviously, for every integer p the sequence satisfies the conditions of Corollary 1. In particular, in order to calculate the approximate sum of the series with an error not greater than we must compute the sum of first 9 terms. Applying the transformation (1), the same accuracy is obtained by computing the sum of 5 terms for , 3 terms for and 1 term for .