Geodesic
Note on Best Approximation of |x|
Canadian mathematical bulletin, Tome 22 (1979) no. 3, pp. 363-366

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In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximation and the extremal transformations U whenever they exist. 
Bennett, Colin; Rudnick, Karl; Vaaler, Jeffrey D. Note on Best Approximation of |x|. Canadian mathematical bulletin, Tome 22 (1979) no. 3, pp. 363-366. doi: 10.4153/CMB-1979-046-x
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[1] 1. Bennett, C., Rudnick, K., and Vaaler, J., Best uniform approximation by linear fractional transformations, Jour, of Approx. Th., to appear. Google Scholar

[2] 2. Bennett, C., Rudnick, K., and Vaaler, J., On a problem of Saff and Varga concerning best rational approximation, Padé and Rational Approximation, (E. B. Saff and R. S. Varga, ed.), Academic Press, 1977. Google Scholar

[3] 3. Ruttan, A., On the cardinality of a set of best complex rational approximations to a real function, Padé and Rational Approximations, (E. B. Saff and R. S. Varga, ed.), Academic Press, 1977. Google Scholar

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