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Homework Assignment #6 1: Although our treatment of Wiener’s homogeneous chaos decomposition makes no use of it, Wiener’s own treatment rested on the connection between the decomposition of L2 (W; R) into spaces of homogenious chaos and Hermite polynomials. In order to understand this connection, x2 x2 recall the Hermite polynomials Hm (x) = (−1)m e 2 ∂xm e− 2 , which form an orthonormal basis in L2 (γ0,1 ; R). Next, given L ≥ 1 and µ ∈ NL , define Hµ : RL −→ R so that Hµ x1 , . . . , xL = L Y Hµ` (x` ). `=1 (i) Let L ≥ 1, and show that kHµ k2L2 (γ0,I ;R) = µ! ≡ L Y µj ! j=1 and that {Hµ : µ ∈ NL } is an orthogonal basis in L2 (γ0,I ; R). (ii) Given ξ ∈ RL , show that 1 2 e(ξ,x)RL − 2 |ξ| = X ξµ Hµ (x), µ! L µ∈N where ξ µ ≡ QL µ` `=1 ξ` . (iii) Suppose that {f1 , . . . , fL } is an orthonormal subset of L2 [0, ∞); RM , and, given µ ∈ PL NL with m = kµk ≡ `=1 µ` , show that (m) I˜ ⊗µ1 f1 ⊗µ ⊗···⊗fL L (∞) = Hµ If1 (∞), . . . , IfL (∞) . (iv) Let {gj : j ≥ 1} be an orthonormal basis in L2 [0, ∞); R , and, for α ∈ A = {α ∈ Nx : kαk < ∞}, set S(α) = {j : αj ≥ 1} and Y 1 . Hα,j ≡ √ Hα` Ig(1) j α! j∈S(α) show that, for each m ≥ 1, {Hα : kαk = m} is an orthonormal basis for Z (m) .