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We may conclude that it is always computationally advantageous to use the center series instead of the corresponding DP Fourier series. In the case of the slower-converging DP Fourier series the improvement in the speed of convergence can be very large indeed, to the point where it may represent a qualitative difference in our ability to deal with the function in a practical manner. With the use of a sufficiently high-order center series, essentially any real function that gives origin to a set of finite DP Fourier coefficients, and that results in an inner analytic function that has a finite number of sufficiently soft singularities on the unit circle, can be very well represented numerically by that center series.
The one limitation to the use of the center series is the need to know, in order to derive the form of the coefficients of the series, the positions of all the dominant singularities of the corresponding inner analytic function on the unit circle. However, the construction process is safe, in the sense that in the worst case scenario all that happens is that one fails to obtain a better convergence speed. Moreover, this fact can be verified analytically during the construction of the center series, before any computer time is actually spent. In effect, the fact of the failure to obtain improvement may itself serve as a guide to search for the correct positions of the singularities, by what are essentially algebraic means, as explained in [2].
The position of the singularities can be induced by the qualitative analytical properties of the function to be represented, such as that it is discontinuous or non-differentiable at certain points. In physics applications these characteristics are bound to be reflected in the structure of the problem being dealt with, so that a physical analysis may suffice to determine the singularities. In any case, whenever it is possible to use them, the center series constitute a significant improvement in our ability to represent real functions numerically in practical applications.