h08-3 - [-, + ] . All future references to f ( x ) is this...

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CIS6930/4930 Intro to Computational Neuroscience Fall 2008 Home Work Assignment 3: Due Thursday 10/30/08 before class 1. A complex number z is said to be algebraic if there are integers a 0 , a 1 , ..., a n , not all zero, such that a n z n + a n - 1 z n - 1 + ... + a 1 z + a 0 = 0 Prove that the set of all algebraic complex numbers is countable . 2. Consider a metric space ( M , d ( · , · )) . Prove that every convergent sequence is also a Cauchy sequence. 3. Consider a finite dimensional inner product space over the field of complex numbers with the inner product defined through h c 1 e i , c 2 e j i = c 1 c 2 if i = j, and 0 otherwise c 1 , c 2 are complex numbers and e i , e j are basis vectors. Prove that the induced norm satisfies the triangular inequality. 4. Consider the following function over the range [0 , 1] f ( x ) = - 2 × x if x [0 , 1 3 ] f ( x ) = 1 if x ( 1 3 , 2 3 ) f ( x ) = 0 if x [ 2 3 , 1] First translate and scale uniformly the domain of the function so that it now lies on
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Unformatted text preview: [-, + ] . All future references to f ( x ) is this scaled and translated version. Your goal will be to nd an approxima-tion of this function as a Fourier series, and show the graphs of successive approximations overlayed on the actual function. Consider the Fourier basis e inx for n =-N, . .., + N , and the corresponding sum + N X n =-N c n e inx Calculate the values of c n by numerically approximating the integral Z + - f ( x ) e-inx dx , that is, by dividing the range [-, + ] , into small intervals and approximating the integral as a sum. Show graphs of how well f ( x ) is approximated by overlaying the series over f ( x ) for various values of N (for example, N = 5 , 10 , 20 , 50 ). 1...
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This note was uploaded on 01/15/2012 for the course CIS 4930 taught by Professor Staff during the Spring '08 term at University of Florida.

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