By Arieh Iserles
Acta Numerica is an annual book containing invited survey papers by way of best researchers in numerical arithmetic and clinical computing. The papers current overviews of modern advancements of their zone and supply 'state of the paintings' innovations and research.
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Extra resources for Acta Numerica 2011: Volume 20
D} be the Jacobian matrix of the map (t1 , . . , td ) → (z1 , . . , zd ). 18. The Fourier transform of the Jacobian determinant, J := F(det(J)(t)), is the alternating function J = cD ∈ E∧W , where D is the Weyl denominator and the constant c = 4πi d |W | . Proof. Let sα be any reﬂection in W . Since z(t) = z(sα t) we have J(t) = J(sα t)sα . Hence det(J(t)) = − det(J(sα t)) and we conclude that J ∈ E∧W . 16, D divides J. We need to conﬁrm that J/D is constant. We compute 4πi α , wλk ewλk . Jk, = |W | + w∈W Thus, Jk, is supported on W + λk and J is supported on the set w j λj .
28) (−1)|w| f (wt). 22). Note that these are not all distinct functions. Since they possess symmetries cλ (t) = cwλ (t), sλ (t) = (−1)|w| swλ (t), for every w ∈ W , we need only one λ from each orbit of W . The weights in the Weyl chamber, Λ+ = C+ ∩ Λ, form a natural index set of orbit representatives, and we ﬁnd L2 -orthogonal bases by taking the corresponding cλ (t) and sλ (t). 9 also holds in the following more general case. 14. Let φ be a rank d root system with weights lattice Λ. Let W be the Weyl group and let denote the fundamental domain of the aﬃne ˜ = W Weyl group W L∨ .
Z. Munthe-Kaas and B. Owren The skew subspace E∧W does not form an algebra, but this can be corrected by dividing out the Weyl denominator. Deﬁne the Weyl vector ρ ∈ Λ as d λj = ρ= j=1 1 2 α. α∈Φ+ We deﬁne the Weyl denominator D ∈ E∧W as (−1)|w| ewρ . 16. , there exists a unique b ∈ E∨W such that a = bD. Proof. 2). Any a ∈ E∨W can be written as a polynomial in z1 , . . , zd , and hence the following polynomials are well-deﬁned. 17. For λ ∈ Λ we deﬁne multivariate Chebyshev polynomials of the ﬁrst and second kind, Tλ and Uλ , as the unique polynomials that satisfy Tλ (z1 , .
Acta Numerica 2011: Volume 20 by Arieh Iserles