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Since we assumed f (x) to be bounded in D2 , we take | f (x)| ≤ B( f ) in D2 . 22) where L(∂ D2 ) ≤ 4 + 2π r is the length of ∂ D2 . ρ −1 Since ρ (x) = 1+x 1−x , we have x = ρ +1 ; this transformation maps ρ ∈ (0, ∞) to x ∈ (−1, 1). We estimate that the maximum value of g takes place approximately at ρ = exp( h2 ) (see [16]). Under these assumptions, we have g ρ −1 ρ +1 = N ∏ j=−N = N ∏ j=−N = ρ − 1 e jh − 1 − ρ + 1 e jh + 1 2ρ − 2e jh (ρ + 1)(1 + e jh) 22N ρ N+1 (1 + ρ )2N+1 N ∏ j=1 1 − ρ e− jh 1 + e− jh 2 |1 − e−(N+1)h|.

In Sect. 5 we present the transformation PE−M , initially for approximation f and its derivative on [−1, 1], and we then extend these results to approximation over a more general arc Γ , via use of the methods of Sect. 4. In Sect. 6 we present several examples of applications of our results. 2 Sinc Notation and Interpolation Formulas In this section we recall some Sinc notation, and we also discuss polynomial interpolation. Our polynomial interpolation presentation differs somewhat from that of [16], where the interval [0, 1] was the starting point of derivation.

Only about 10 % improvement was possible, mainly because this transformation is unable to remove the singularity. 2. 32). Here, too, only about 10 % improvement was possible. 3. 38). Fantastic improvement was possible, with proper choice of α and β , as is illustrated in Figs. 12 below. 387. Evidently, this transformation has possibilities. We were not able to determine a priori the parameters α and β . 4. , whenever we tried it, best approximation always occurred with q very close to 1 . , we first fixed q and we then determined α and β .

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