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# 02a - a is at most π 2 Note The cosine as the solution to...

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Modern Analysis 2 Test 2 Answer (1 or 2) and (3 or 4). 1. Decide (with proof) on which intervals [ a, b ] an arbitrary continuous func- tion may be uniformly approximated by functions of the form: (i) c 0 + c 1 t 1 / 3 + · · · + c n t n/ 3 ; (ii) c 0 + c 1 t 4 + · · · + c n t 4 n . 2. Let the positive integer N be fixed. By considering 1 /t N (or otherwise) show that each constant function can be uniformly approximated by polyno- mials of the form a N t N + a ( N + 1 t N +1 on any interval [ a, b ] not containing zero. 3. Let the differentiable function f : ( - a, a ) R satisfy f (0) = 0 and f 0 = 1 + f 2 . By considering the function g := 2 /f 0 (or otherwise) show that
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Unformatted text preview: a is at most π/ 2. Note : The cosine as the solution to a second-order IVP may be assumed, as may its least positive zero. 4. Let f and g satisfy the initial value problems f 00 = f ; f (0) = 1 ,f (0) = 0 g 00 = g ; g (0) = 0 ,g (0) = 1 . Establish each of the following: (a) fg-f g = 0; (b) g = f, f = g ; (c) f 2-g 2 = 1; (d) if h satisﬁes the same IVP as f then h = f ; (e) f is even and g is odd. Note : It may be assumed that f is nowhere zero. 1...
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