Let R a, b Atilde c, d be a rectangle in R2 A function f is said to have separated variables if for some scalars ck and functions fk C a, b , gk C c, d Prove that given f C(R) there is a sequence of functions with separated variables, Pn, such that Pn f uniformly on R as n CA ft(x)gx(y) P(x, y) C fi C)g(y) 3D k 1

Question: Let R = [a, b] Ã [c, d] be a rectangle in R2. A function f is said to have separated variables if for some

Let R = [a, b] Ã— [c, d] be a rectangle in R2. A function f is said to have separated variables if

for some scalars ck and functions fk ˆˆ C[a, b], gk ˆˆ C[c, d]. Prove that given f ˆˆ C(R) there is a sequence of functions with separated variables, Pn, such that Pn †’ f uniformly on R as n †’ ˆž.

CA ft(x)gx(y) P(x, y) = C fi C)g(y) %3D k=1

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