68.4. THE SKOROKHOD INTEGRAL 2343

where G = G(W (k1) , · · · ,W (kp)) and for simplicity, ∥h∥H = 1.Consider the vectors

{h,h1, · · · ,hn,k1, · · · ,kp

}. Starting with the left and moving to-

ward the right, delete vectors which are dependent on the preceding vectors, obtaining alinearly independent set of vectors which includes h. Then let

{h,e1, · · · ,eq

}be an or-

thonormal basis having the same span as the original vectors{

h,h1, · · · ,hn,k1, · · · ,kp}

.Then from the fact that W is linear, there are smooth functions having polynomial growthĜ, F̂ such that

G(W (k1) , · · · ,W (kp)) = Ĝ(W (h) ,W (e1) , · · · ,W (eq))

F (W (h1) , · · · ,W (hn)) = F̂ (W (h) ,W (e1) , · · · ,W (eq))

Note that hi = ∑qj=1 (hi,e j)e j +(hi,h)h. Thus

F (W (h1) , · · · ,W (hn)) =

F

(W

(q

∑j=1

(h1,e j)e j +(h1,h)h

), · · · ,W

(q

∑j=1

(hn,e j)e j +(hn,h)h

))

= F

(q

∑j=1

(h1,e j)W (e j)+(h1,h)W (h) , · · · ,q

∑j=1

(hn,e j)W (e j)+(hn,h)W (h)

)and so, D1F̂ is given by

D1F̂ =n

∑i=1

Di (F (W (h1) , · · · ,W (hn)))(hi,h)

Then by Lemma 68.4.2

E ⟨DG,F (W (h1) , · · · ,W (hn))h⟩= E⟨DĜ, F̂h

⟩= E

⟨D1(Ĝ)

h+q

∑k=1

Dk(Ĝ)

ek, F̂h

⟩= E

(D1(Ĝ)

F̂)

=1(√

2π)q+1

∫Rq

∫R

D1Ĝ(x) F̂ (x)e−12 |x|

2dx1dx̂1

=−1(√2π)q+1

∫Rq

∫R

Ĝ(x)D1

(F̂ (x)e−

12 |x|

2)

dx1dx̂1

=−1(√2π)q+1

∫Rq

∫R

Ĝ(x)((

D1F̂)(x)e−

12 |x|

2− F̂ (x)x1e−

12 |x|

2)

dx1dx̂1

= E((

F̂W (h)−D1F̂)

Ĝ)= E

((FW (h)−

n

∑i=1

Di (F)(hi,h)

)G

)Thus Fh ∈ D(δ ) and

δ (Fh) = FW (h)−n

∑i=1

Di (F)(hi,h)