By F. Borceux, et al.,

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On a matrix algebra related to the discrete Hartley transform. SIAM J. Matrix Anal. Appl. 14(2), 500–507 (1993) 6. : Multigrid methods. volume 294 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, (1993) 7. : Convergence estimates for product iterative methods with applications to domain decomposition. Math. Comput. 57(195), 1–21 (1991) 8. : Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57(195) 23–45 (1991) 9.

43) Each term of the last sum is nonnegative around 0, and at least two are infinite whenever two or more components of s are equal to 1. ,d r=r c 2q(d−1) 1 − cos(xr ) 1 − cos(xr ) q ψi(r) (x) (r) ψi (0) . c 2q(d−1) The proof is trivial if s = 0 since p(x + π s) converges to 2qd > 0 if x goes to 0, while f (x) vanishes. 546 A. Aricò, M. Donatelli Proof of Lemma 4 The proof follows from Fubini’s theorem for rectangles, with the advantage that φq[r] can be factorized thanks to (38). ,d s=r d = 22q(d−1) π [1 + cos(xs )]2q e−iks xs dxs −π q[r] s bks s=1 which completes the proof.

T. t. xi = i N (−1)i i N (−1)i i N (−1)i i N (−1)i 8 8 7 7 4 4 4 4 31 31 31 31 7 7 7 7 9 8 7 7 7 7 6 6 82 92 96 99 13 9 6 4 the proposed AMG in the circulant case. Furthermore, also in the Toeplitz case, using the technique described in the previous subsection, our AMG seems to maintain about a constant number of iterations increasing the size of the problem. This is a good news especially for the second test function r, since it has a zero of order 4 and, in order to preserve the Toeplitz structure at each level, in the restriction process we have to remove 2n2 n3 components.