By Arieh Iserles

The 5th quantity of Acta Numerica offers "state of the paintings" research and strategies in numerical arithmetic and medical computing. This assortment encompasses numerous vital elements of numerical research, together with eigenvalue optimization; conception, algorithms and alertness of point set tools for propagating interfaces; hierarchical bases and the finite aspect process. will probably be a precious source for researchers during this very important box.

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Satisﬁes n r = CR r for any r ∈ [1, ∞], where C = φ r . Also, by the chain rule ∇ψ (58) 1 = R φ 1. We know that f = ψ ∗ f . In the case of ﬁrst derivatives, 1. therefore follows from (57) and (24). The general case of 1. then follows by induction. , let r satisfy 1q = 1p − r1 . Apply Young’s inequality obtaining f ψ∗f q ψ r f p = ≤ q n Rr f = R p n( p1 − 1q ) f p. We now extend the Lp → Lq bound to ellipsoids instead of balls, using change of variables. An ellipsoid in Rn is a set of the form (59) E = {x ∈ Rn : j |(x − a) · ej |2 ≤ 1} rj2 for some a ∈ Rn (called the center of E), some choice of orthonormal basis {ej } (the axes) and some choice of positive numbers rj (the axis lengths).

J 1Actually the order is exactly 2j but we have no need to know that. 6. THE STATIONARY PHASE METHOD 41 (ii)Assume that ∇φ(p) = 0, and Hφ (p) is invertible. Then, for a supported in a small neighborhood of p, n dk πiλφ(p) (e I(λ)) ≤ Ck λ−( 2 +k) . k dλ Proof. 1 after diﬀerentiating under the integral sign as in the proof below. For (ii) we need the following. Claim. Let {φi }M i=1 be real valued smooth functions and assume that φi (p) = 0, ∇φi (p) = 0. Let Φ = ΠM i=1 φi . Then all partial derivatives of Φ of order less than 2M also vanish at p.

All the basic formulas for the L1 Fourier transform extend to the L1 +L2 Fourier transform by approximation arguments. This was done above in the case of the duality relation. Let us note in particular that the transformation formulas in Chapter 1 extend to L1 + L2 . For example, in the case of (5), one has (36) f ◦ T = |det (T )|−1 fˆ ◦ T −t if f ∈ L1 + L2 . Since we already know this when f ∈ L1 , it suﬃces to prove it when f ∈ L2 . Choose {fk } ⊂ L1 ∩ L2 , fk → f in L2 . Composition with T is continuous on L2 , as is Fourier transform, so we can write down (36) for the {fk } and pass to the limit.