Gibbs phenomenon and lebesgue constants for the Quasi-Hausdorff means of double series Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/vm40xv08j

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  • The topic of summability methods has been studied by many although the name of G. H. Hardy and his classical work "Divergent Series" is best known. Almost all of the early work was done using single sequences or series. In the past 30 years research has been done extending some of these results to double sequences or double series by Cheng for the circular Riesz means and by Ustina for the Hausdorff means. In this paper we extend some of these results for the Quasi-Hausdorff means. The results and methods of attack closely follow those of Ishiguro and Ramanujan, who worked with Quasi-Hausdorff means for single sequences and single series. The terminology is fairly standard although some new definitions are needed. We shall first develop the Quasi-Hausdorff transformation of double sequences and double series, next find conditions to make it a regular transformation, thirdly apply it to the partial sums of a double Fourier series to check the Gibbs phenomenon, and conclude by investigating the Lebesgue constants of the method. It is noted that the class of weight functions used in the definition of the Quasi- Hausdorff means contains the probability distribution functions of two variables. Therefore the results contained in this research could possibly be used in the area of probability.
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