# Decomposition of Riesz operators by Smyth M. R., West T. T.

By Smyth M. R., West T. T.

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Example text

Greenberg with the limit taken over n-partitions t = tm > tm−1 > · · · > t1 > t0 = s. Using the Lie product formula and the uniform boundedness of each of the exponentials, one can represent Tf (t, s) as the double limit Tf (t, s) = s. lim lim tm − tm−1 n UA m→∞ n→∞ × · · · × UA t1 − t0 n exp − exp t1 t0 − tm tm−1 L(f (s))ds L(f (s))ds n n n n . (26) But L is diagonal and positive on T+ , and therefore so are the exponentials in (26). Thus Tf (t, s)T+ ⊂ T+ , completing the proof of (a). To prove (b), one employs the important identity N3 (Am )ij = 0, ∀m ∈ Z+ (27) i=1 and the expansion n−1 n es(−I+E) = 1 es(wi −1) w−αi Eα .

Almost nothing is known in this venue. How do the discrete models compare? What are the eﬀects of the square well potential? Does a deep well lead to clustering? What are the possibilities of non-periodic boundary conditions? What would a fully discrete Vlasov–Enskog system look like? The author encourages further speculation and research. 32 W. Greenberg References 1. R. Gatignol, Theorie cinetique de gaz a repartition discrete de vitesses, Lecture Notes in Physics 36, Springer-Verlag, New York, 1975.

We discretize the domain Ω. Set ωi,j = {(x, y); x ∈]xi , xi+1 [; y ∈]yj , yj+1 [}. Over the cell ωi,j , we have f (x, y) = f (xi , yj ) + (x − xi ) ∂f ∂f (xi , yj ) + (y − yj ) (xi , yj ). ∂x ∂y (56) We assume that for x ∈]xi , xi+1 [ and y ∈]yj , yj+1 [, u0 is of the form u0 (x, y) = w0 (x, y) + (x − xi )w1 (y) + (y − yj )w2 (x). (57) From equality (57) we draw ∆u0 = ∆w0 + (y − yj ) d2 w2 d2 w1 (x) + (x − x ) (y). i dx2 dy 2 (58) Put together this equality above and system (48), and identify alike terms in the factors (x − xi ) and (y − yj ) to obtain the following equations depending on the coeﬃcient functions w0 , w1 , and w3 , to be determined, ∆w0 (x, y) = f (xi , yj ) d2 w1 ∂f (xi , yj ) (y) = dy 2 ∂x d2 w2 ∂f (xi , yj ) (x) = dx2 ∂y x ∈]xi , xi+1 [; y ∈]yj , yj+1 [.