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Asymptotic Expansions of Integrals by Norman Bleistein

25 February 2017 adminCalculus

By Norman Bleistein

Excellent introductory textual content, written by way of specialists, offers a coherent and systematic view of ideas and techniques. issues include integration by way of components, Watson's lemma, LaPlace's process, desk bound part, and steepest descents. extra topics include the Mellin remodel process and no more trouble-free facets of the tactic of steepest descents. 1975 edition.

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THE RECONSTRUCTION THEOREM 21 induced transformation. 3, S = T (T )−1 = [U, V ] = U1 V1 , for some U, V, U1 , V1 ∈ [E], U1 , V1 involutions. Also supp(T ) = A and µ(A) ≤ 12 µ(supp(T )) = 12 , as A ∩ T (A) = ∅, so δu (T , 1) ≤ 12 . ✷ Fix now T ∈ [E]. All transformations below are in [E]. 4, T = S0 T1 , where S0 is a commutator and δu (T1 , 1) = µ(supp(T1 )) ≤ 12 . We concentrate on T1 . Split X = X1 ∪ X2 ∪ . . , with µ(Xi ) = 2−i , supp(T1 ) ⊆ X1 . 4 again, T1 = S1 T2 , where S1 is a commutator of elements with support contained in X1 , T2 has support also contained in X1 , and δu (T2 , 1) ≤ 14 .

Fix a free Borel action (n, x) → n · x of Z on (X, µ) which generates E. To show that F is smooth, it is enough to show that if (Y, ν) is one of 4. THE RECONSTRUCTION THEOREM 29 the ergodic components of the action of Γ, with ν non-atomic, then F |Y is smooth. Consider the cocycle α from Γ × Y into Z : α(γ, x) = (the unique n such that γ · x = n · x). Then there is Borel f : Y → Z such that α(γ, x) = f (γ · x) − f (x). Fix a set A of positive ν-measure, thus meeting every F -class in Y , on which f is constant.

The following simple result was originally proved for residually finite groups. Ben Miller then noticed that the argument really shows the following stronger fact. 13. Let E be an ergodic, hyperfinite equivalence relation. Given a countable group Γ, if for every γ ∈ Γ \ {1} there is a homomorphism π : Γ → [E] such that γ ∈ ker(π), then Γ embeds into [E]. In particular every residually amenable Γ embeds into [E] and for every countable group Γ there is unique normal subgroup N such that Γ/N embeds into [E] and every homomorphism from N into [E] is trivial.

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