Solution of Interpolation Problems via the Hankel Polynomial Construction
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We treat the interpolation problem {f(x_j)=y_j}_j=1^N for polynomial and rational functions. Developing the approach by C.Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences {∑_j=1^N x_j^ky_j/W^'(x_j) }_k∈ N and {∑_j=1^N x_j^k/(y_jW^'(x_j)) }_k∈ N ; here W(x)=∏_j=1^N(x-x_j) . The obtained results are applied for the error correction problem, i.e. the problem of reconstructing the polynomial from a redundant set of its values some of which are probably erroneous. The problem of evaluation of the resultant of polynomials p(x) and q(x) from the set of values {p(x_j)/q(x_j) }_j=1^N is also tackled within the framework of this approach.
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