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# dft - DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1 2...

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DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1. Representations of polynomials 1 2. Comparison of the representations 2 3. Roots of unity 2 4. Discrete Fourier Transform 2 5. Fast Fourier Transform 2 1. Representations of polynomials A polynomial f ( x ) of degree less than n can be represented in coefficient form as f ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n - 1 x n - 1 , or, more succinctly, as the sequence of n numbers a = ( a 0 , . . . , a n - 1 ). This is not the only representation. If we choose n distinct points x = x 0 , . . . , x n , we can calculate the values y 0 = f ( x 0 ) , . . . , y n - 1 = f ( x n - 1 ) . The sequence of n numbers y = ( y 0 , . . . , y n - 1 ) is called the point-value form of f with respect to the points x . The coefficient form and the point-value form are equivalent. To recover a from y , we find the the unique polynomial of degree less than n that interpolates the points ( x 0 , y 0 ) , . . . , ( x n - 1 , y n - 1 ). Note that a and y are related by x 0 0 x 1 0 . . . x n - 1 0 . . . . . . x 0 n - 1 x 1 n - 1 . . . x n - 1 n - 1 a 0 .

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