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Exponential convergence of Chebyshev expansions

We briefly mention the exponential convergence in the analytic case. To this end we employ Bernstein's regularity ellipse, , with foci and sum of its semi axis = r. Denoting
We have

Proof: The transformation takes from the z-plane into the annulus in the -plane. Hence, admits the power expansion
indeed, setting and recalling , the above expansion clearly describes the real interval [-1,1]
Using the Laurent expansion in (app_cheb.36)
and the result follows along the lines of (err_exp.7)-(err_exp.8).

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998