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m425b-sm1soln-s08

m425b-sm1soln-s08 - MATH 425b SAMPLE MIDTERM 1 SOLUTIONS...

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MATH 425b SAMPLE MIDTERM 1 SOLUTIONS SPRING 2008 Prof. Alexander (1)(a) The problem does not actually say if we are dealing with real or complex-valued functions (though C [0 , 1] implicitly means real-valued), so I will give solutions for both cases. For the real case we can do the following. Let E = { all even polynomials } . Then P 1 ( x ) 1 is in E , so E vanishes nowhere. E contains 1-1 functions, such as P 2 ( x ) = x 2 on [0 , 1], so E separates points. By Stone-Weierstrass, E is dense in C [0 , 1]. For the complex case we cannot use Stone-Weierstrass, so we need something a little more complicated (which actually covers the real-valued case as well.) Given f C [0 , 1] let ˜ f ( x ) = f ( x ). Let > 0. By Theorem 7.26 there exists a polynomial ˜ P with sup x | ˜ P ( x ) - ˜ f ( x ) | < . Then also sup x | ˜ P ( x 2 ) - f ( x ) | = sup x | ˜ P ( x 2 ) - ˜ f ( x 2 ) | < . But P ( x ) = ˜ P ( x 2 ) is an even polynomial, and || P - f || < , which shows the even polynomials are dense. (b) We show that a typical odd function in C [ - 1 , 1] cannot be approximated too closely by an even polynomial. Take for example f ( x ) =
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