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Rational Interpolation: Jacobi's Approach Reminiscence

Alexei Uteshev, Ivan Baravy, Elizaveta Kalinina
2021 Symmetry  
We treat the interpolation problem {f(xj)=yj}j=1N for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like {∑j=1Nxjkyj/W′(xj)}k∈N and {∑j=1Nxjk/(yjW′(xj))}k∈N; here, W(x)=∏j=1N(x−xj). We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of
more » ... nkel polynomials. The problem of evaluation of the resultant of polynomials p(x) and q(x) given a set of values {p(xj)/q(xj)}j=1N is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.
doi:10.3390/sym13081401 fatcat:xj5rm5qlyzc2zdgwu3vzchfkmm

Solution of Interpolation Problems via the Hankel Polynomial Construction [article]

Alexei Yu. Uteshev, Ivan Baravy
2016 arXiv   pre-print
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
more » ... ably 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.
arXiv:1603.08752v1 fatcat:ybntk7oczffcdhdzg4ghgu2jka

Solution of rational interpolation problem via the Hankel polynomial construction

Alexei Yu. Uteshev, St. Petersburg State University, Ivan I. Baravy, St. Petersburg State University
2016 Vestnik of Saint Petersburg University Applied Mathematics Computer Science Control Processes  
Baravy SOLUTION OF THE RATIONAL INTERPOLATION PROBLEM VIA THE HANKEL POLYNOMIAL CONSTRUCTION St. Petersburg State University, 7-9, Universitetskaya nab., St.  ... 
doi:10.21638/11701/spbu10.2016.403 fatcat:x7ghxmy2vrdj7kbrns5rzj63uy

Reviews Of Periodicals

Editors The Journal of Belarusian Studies
1972 Journal of Belarusian studies  
Several critical articles deal with the work of contemporary poets and writers, such as Piatrus Brouka, Mik- sim Tank, Anatol As'trejka, Kastus Kirejenka, Janka Bryl, Ivan Samia- kin, Ivan Mielez, Mikola  ...  One member of this family, Ivan, is better known as the Byelorussian poet Janka Lucyna (1851-1897).  ... 
doi:10.30965/20526512-00204024 fatcat:msfabt2z2zc4vmx6xpyjqhxeaq