{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

m425b-sm1soln-s08

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

This preview shows pages 1–2. Sign up to view the full content.

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 ) =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern