ACM Communications in Computer Algebra, Vol. 42, No. 3, September 2008 ISSAC 2008 Poster Abstracts De Boor-Fix Dual Functionals for Transformation from Polynomial Basis to Convolution Basis Lim Yohanes Stefanus University of Indonesia, Faculty of Computer Science, Depok 16424, Indonesia yohanes@cs.ui.ac.id Abstract Basis transformation plays a central role in data representations and in many problem solving techniques such as Fourier transform and wavelet transform. A pertinent basis gives pertinent information. Dual functionals are a mathematical tool for basis transformation. This work presents, from the perspective of algebraic computing, a generalization of the de Boor-Fix dual functionals for the Bernstein basis functions to the case of the convolution basis functions. The convolution basis functions can be characterized as polar forms of the corresponding Bernstein basis functions. These new dual functionals can be used for computing algebraically the expansion coefficients of any polynomial over the convolution basis which has applications in computer-aided geometric design and secret information encoding. 1 De Boor-Fix Dual Functionals Let {v 1 ,..., v n } be a basis for a vector space. Then we can construct linear functionals { f 1 ,..., f n } such that f i (v j )= δ ij . These linear functionals, which form a basis for the dual space over the same field, are called the dual functionals of {v 1 ,..., v n }. (See, for example, [1] for more details.) Now, if we have w = ∑ n k=1 a k v k , then a k = f k (w). For the Bernstein basis functions, the de Boor-Fix dual functionals μ n j for j = 0, 1,..., n can be expressed as follows [2]: μ n j (Q(u)) = ∑ n r=0 (-1) n-r [Q(u)] (r) [B n n- j (u)] (n-r) (-1) (n- j) n! ( n n- j ) , (1) where Q(u) is any polynomial of degree ≤ n and B n k (u) is the Bernstein basis function ( n k ) u k (1 - u) n-k . The notation [Q(u)] (r) means taking the r-th derivative of Q with respect to u. The functionals μ n j satisfy μ n j (B n k (u)) = δ jk . Example 1. Let Q(u)= 2 - 9u + 21u 2 , then μ 3 0 (Q(u)) = 2, μ 3 1 (Q(u)) = -1, μ 3 2 (Q(u)) = 3, μ 3 3 (Q(u)) = 14. Therefore, in terms of the cubic Bernstein basis functions, Q(u) can be expressed as Q(u)= 2 B 3 0 (u) - B 3 1 (u)+ 3 B 3 2 (u)+ 14 B 3 3 (u). Notation: For brevity, sometimes the arguments of a function are not written explicitly. For example, Q(u) is written just as Q. 2 Convolution Basis The elementary symmetric functions s j (a 1 ,..., a n ), for j = 0, 1,..., n, are defined by the identity ∏ n i=1 (a i t + 1)= ∑ n j=0 s j (a 1 ,..., a n )t j . More general than the elementary symmetric functions are the composite symmetric functions S j ([a 1 , b 1 ],..., [a n , b n ]), j = 0, 1,..., n, defined by the identity ∏ n i=1 (a i t + b i )= ∑ n j=0 S j ([a 1 , b 1 ],..., [a n , b n ])t j . Each S j ([a 1 , b 1 ],..., [a n , b n ]) is a symmetric function with respect to the pairs [a 1 , b 1 ], ..., [a n , b n ]. The polar form [3] of a degree n polynomial p(t ) is the unique symmetric multiaffine polynomial P [ p](u 1 ,..., u n ) for which P [ p](t ,..., t )= p(t ). The convolution basis functions [5] C n k (t ), k = 0, 1,..., n can be obtained by taking the polar forms of the Bernstein basis functions B n k (t ) respectively and evaluating them at the linear functions X j (t )= a j t + b j , j = 1,..., n where a j and b j are constants and a j = 0, i.e., C n k (t )= P [B n k (x)](X 1 ,..., X n ). (2) 146