Differential Equations and Mathematical Physics

Optimization of the error in exponential-trigonometric interpolation formula

Kh. M. Shadimetov^ab, A. K. Boltaev<sup>bc

a</sup> Tashkent State Transport University, Tashkent, 100167, Uzbekistan
^b V.I. Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, 100174, Uzbekistan
^c International Nordic University, Tashkent, 100043, Uzbekistan

Abstract: In engineering geodesy, point clouds obtained through area measurement methods, such as terrestrial laser scanning or photogrammetry, need to be approximated by a curve or surface that can be described by using a continuous mathematical function, often employing splines and optimal interpolation formulas.
This work is devoted to the construction of an optimal interpolation formula that is exact for exponential-trigonometric functions in a Hilbert space. The optimal interpolation formula is obtained by minimizing the norm of the error functional with respect to the coefficients. The article proves the existence and uniqueness of the optimal interpolation formula and provides explicit analytical expressions for the optimal coefficients of the interpolation formula. Using the constructed optimal interpolation formula, specific functions were interpolated, and a comparison was made with known results from other authors.

Keywords: interpolation formula, error of the formula, optimal coefficients, Sobolev method

UDC: 519.652

MSC: 65D05, 65D07, 65D15

Received: May 15, 2024
Revised: November 16, 2024
Accepted: November 19, 2024
First online: December 26, 2024

DOI: 10.14498/vsgtu2094

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