By Edgar Collins Bain, Harold W Paxton

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Other variations of the PG algorithm have been introduced either to speed up convergence or to improve convergence in the presence of noise or aliasing. For example, Chen [70] introduced a modiﬁcation of the basic algorithm fi = Pgi , gi+1 = QT f + QT fi , g0 = QT f , where QT = I − QT , by replacing fi , expressed in terms of its sample series, by fiChen where fiChen = ∑|k|≤T f (k) sinc(t − k) + ∑|k|>T fi (k) sinc(t − k). In other words, the terms for |k| ≤ T of the sampling series for fi are replaced by those of the sampling series of the band-limited function Chen .

34) Tk = =k 2 − k cos 2π W , ⎪ ⎩1 = k+1 2 (k + 1)(N − 1 − k), and vanishing at all other integer pairs. 33) except for the eigenvalue μ in the sn (k)-term. Consequently, it is possible to have a solution satisfying sn (k) = 0 for all k < 0 and k ≥ N provided that, for some μ ∈ R, N−1 ∑ Tk sn ( ) = μ sn (k), =0 (k = 0, . . , N − 1) . Let TN denote the N × N block of T corresponding to 0 ≤ k, < N. One ﬁnds that TN has N real eigenvalues which, of course, are also eigenvalues of M . In fact, these eigenvalues correspond to the N largest eigenvalues of M , as will be shown now.

To see this, note that x2 F2π e− 2 +2xt−t 2 (ξ ) = e− ξ2 2 2 −2iξ t+t . Equating powers of t in the generating function relationship x2 e− 2 +2xt−t = 2 ∞ x2 ∑ e− 2 n=0 Hn (x) tn n! then shows that F2π (hn ) = (−i)n hn . PSWFs and Hermite Functions √ The asymptotic resemblance between the c-scaled operator Ps and the operator H suggests that φn (t; c) ≈ D√c hn (t) as c → ∞ as the following lemma, proved in [231, Theorem 9, p. 243] (cf. 2]), indicates. 9. If |t| < (log c)1/3 then √ D√c φn (t; c)(t) = hn (t) + o(1) and √ d √ d D c φn (t; c) = hn (t) + o(1) .

### Alloying elements in steel by Edgar Collins Bain, Harold W Paxton

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