Vol. 18 No. 10 (2026): Asymptotic Behavior of Eckhoff's Method for Convergence Acceleration of Dirac Eigenfunction Expansions

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  Asymptotic Behavior of Eckhoff's Method for Convergence Acceleration of Dirac Eigenfunction Expansions

  Rafayel Barkhudaryan, Gegham Gevorkyan, Lusine Poghosyan

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  Abstract

  The current paper considers the problem of recovering a vector-function on $[-1,1]$ from a limited number of coefficients of its expansion into a series of eigenfunctions of a one-dimensional Dirac system. The Krylov-Lanczos-Eckhoff-Gottlieb acceleration method is examined in the situation when the boundary values it requires have to be computed from the generalized Fourier coefficients themselves. This leads to a $2q\times 2q$ linear system whose matrix is a block Vandermonde matrix; its determinant and inverse are computed explicitly, and the asymptotic $L_2$-error constant of the method is found, paralleling the classical trigonometric case.

  References

  A. Barkhudaryan, R. Barkhudaryan and A. Poghosyan, Asymptotic behavior of Eckhoff's method for Fourier series convergence acceleration. Anal. Theory Appl., 23 (2007), no. 3, pp. 228-242.

  R. Barkhudaryan, Acceleration of convergence of expansions in eigenfunctions of the Dirac system (in Russian). Mathematics in Higher School, 2 (2006), no. 3, pp. 54-62.

  R. Barkhudaryan, Convergence acceleration of eigenfunction expansions of the one-dimensional Dirac system. In: Numerical Analysis and Applied Mathematics ({ICNAAM} 2007), volume 936 of AIP Conference Proceedings, pp. 86-89, 2007.

  R.H. Barkhudaryan, On the Gibbs phenomenon for expansions by eigenfunctions of the boundary problem for Dirac system. J. Contemp. Math. Anal., 48 (2013), no. 4, pp. 139-144.

  Å. Björck and V. Pereyra, Solution of Vandermonde systems of equations. Math. Comput., 24 (1970), no. 112, pp. 893-903.

  K.S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Math. Comput., 61 (1993), no. 204, pp. 745-763.

  K.S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions. Math. Comput., 64 (1995), no. 210, pp. 671-690.

  K.S. Eckhoff, On a high order numerical method for functions with singularities. Math. Comput., 67 (1998), no. 223, pp. 1063-1087.

  K.S. Eckhoff and C.E. Wasberg, On the numerical approximation of derivatives by a modified Fourier collocation method. Technical Report 99, Department of Mathematics, University of Bergen, 1995.

  F.R. Gantmacher, The Theory of Matrices, volume 1, Chelsea Publishing Company, New York, 1959.

  A. Gelb and D. Gottlieb, The resolution of the Gibbs phenomenon for ``spliced'' functions in one and two dimensions. Comput. Math. Appl., 33 (1997), no. 11, pp. 35-58.

  D. Gottlieb, C.-W. Shu, A. Solomonoff and H. Vandeven, On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function. J. Comput. Appl. Math., 43 (1992), no. 1-2, pp. 81-98.

  A.N. Krylov, On Approximate Calculations. Tipolitography of Birkenfeld, St. Petersburg, 1907. Lectures delivered in 1906 (in Russian).

  C. Lanczos, Evaluation of noisy data. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal., 1 (1964), pp. 76-85.

  C. Lanczos, Discourse on Fourier Series. Oliver and Boyd, Edinburgh, 1966.

  B.M. Levitan and I.S. Sargsyan, Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators, volume 39 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1975.

  I.G. Macdonald, textit{Symmetric Functions and Hall Polynomials}. Oxford Mathematical Monographs. Oxford University Press, New York, 2nd edition, 1995.

  A. Nersessian and A. Poghosyan, Bernoulli method in multidimensional case (in Russian). Technical Report 20 Ar-00, ArmNIINTI, 2000.

  A. Nersessian and A. Poghosyan, Accelerating the convergence of trigonometric series. Cent. Eur. J. Math., 4 (2006), no. 3, pp. 435-448.

  A. Nersessian and A. Poghosyan, On a rational linear approximation of Fourier series for smooth functions. J. Sci. Comput., 26 (2006), no. 1, pp. 111-125.

  A. Poghosyan, L. Poghosyan and R. Barkhudaryan, On an exact convergence of quasi-periodic interpolations for the polyharmonic-neumann eigenfunctions. Algorithms, 17 (2024), article 497.

  A.V. Poghosyan, L.D. Poghosyan and R.H. Barkhudaryan, On some quasi-periodic approximations. Armen. J. Math., 12 (2020), no. 10, pp. 1-27.

  A.V. Poghosyan, L.D. Poghosyan and R.H. Barkhudaryan, On the convergence of the quasi-periodic approximations on a finite interval. Armen. J. Math., 13 (2021), no. 10, pp. 1-44.

  J.K. Shaw, L.W. Johnson and R.D. Riess, Accelerating convergence of eigenfunction expansions. Math. Comput., 30 (1976), no. 135, pp. 469-477.