Abstract
The close relationship between Chebyshev, Fourier and Laurent series is defined. Approximants to real Chebyshev series are defined as real parts of Pade approximants. Using the doubly-complex ‘JI-numbers’, these definitions are extended to complex Chebyshev and complex Fourier series. It is shown that these approximants are all also defined by the generating function method. For m ≥ n, our Chebyshev series approximants are equal to the Clenshaw-Lord approximants, and our Fourier and Laurent series approximants are equal to the related approximants of Gragg and Johnson; for m < n, our approximants differ from the other approximants. For m < n, the properties of the two types of approximant are compared.
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