By expanding any polynomial Pn e Pn in the monomial basis, n Pn(x) = ctx",...
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By expanding any polynomial Pn e Pn in the monomial basis, n Pn(x) = ctx", , = k=0 the problem of interpolation through distinct absciss is equivalent to solving the linear system Ac = f defined by: [1 2 n 0 CO 1 . Xox . X1 x X2 X3 X2 x xn x2 1 C1 fo fi f2 2 n 1 C2 = : : 1 2 n Xnxh n 2-1 det(A) = II II(X; x;) 2 X i=1 j=0 Using the monomial basis and the determinantal result above: = 1 1. prove that there exists a unique polynomial that satisfies Pn+1 (xi) = f' for i= 0,..., n, and Pn+1(0) = fn+1, with distinct absciss; find conditions on the absciss such that there exists a unique even polynomial of degree 2n that interpolate n + 1 points: P2n(-xi) = P2n (xi) = fi for i = 0,...,n. 3. find conditions on the absciss such that there exists a unique odd polynomial of degree 2n + 1 that interpolate n +1 points: -P2n+1(-x;) = P2n+1(x;) = f; for i = 0, ..., n. - = = 2
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