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The Approximation Problem and specifically, "direct" rational Chebyshev approximation is discussed. A brief summary is made of "direct" Chebyshev approximation. The remainder of the thesis is devoted to various aspects of a "Remes-type" Algorithm for rational Chebyshev approximation, as proposed by Fraser and Hart. It is finally concluded that the inherent difficulties of the method would generally outweigh the advantages of the rational approximation which it obtains.doi:10.14288/1.0080568 fatcat:zrnnbid4vzf6dgaqtkgf6xgynq