George Kvernadze - Academia.edu (original) (raw)

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Papers by George Kvernadze

Journal of Approximation Theory - JAT, 1998

Journal of Scientific Computing, 1999

A key step for some methods dealing with the reconstruction of a function with jump discontinuiti... more A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on asymptotic expansion formulae obtained in our earlier work. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function given its truncated Fourier series. We investigate the stability of the method and study its complexity. Finally, we consider several numerical examples in order to emphasize strong and weak points of the algorithm.

Mathematics of Computation, 2003

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locat... more In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients. A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities. In addition, we study the accuracy of the approximations and present some numerical examples.

Mathematical and Computer Modelling, 2006

A simple method is considered for approximating the locations of singularities and the associated... more A simple method is considered for approximating the locations of singularities and the associated jumps of a piecewise constant function. The locations of jump discontinuities of a function are recovered approximately, one by one, by means of ratios of so called higher order Fourier-Jacobi coefficients of the function. It is shown that the location of singularity of a piecewise constant function with one discontinuity is recovered exactly and the locations of singularities of a piecewise constant function with multiple discontinuities are recovered with exponential accuracy. The method is applicable to piecewise smooth functions as well, however the accuracy of the approximation sharply declines. In addition, the stability and complexity of the method is discussed and some numerical examples are presented.

Journal of Approximation Theory, 1998

The well-known identity which determines the jumps of a function of bounded variation by its Four... more The well-known identity which determines the jumps of a function of bounded variation by its Fourier series is extended to larger classes of functions, such as V 8 , 4BV, and V[v], under some conditions on the generalized variations. It is shown as well that the conditions on the generalized variations are definitive in some sense. Based on the above-mentioned results, an identity which determines the jumps of a bounded function by its Fourier series with respect to the system of generalized Jacobi polynomials is obtained for these function classes.

Journal of Approximation Theory, 2002

Necessary and sufficient conditions which imply the uniform convergence of the Fourier-Jacobi ser... more Necessary and sufficient conditions which imply the uniform convergence of the Fourier-Jacobi series of a continuous function are obtained under an assumption that the Fourier-Jacobi series is convergent at the end points of the segment of orthogonality ½À1; 1: The conditions are in terms of the modulus of continuity, Lvariation, and the modulus of variation of a function.

Journal of Approximation Theory, 1996

Necessary and sufficient conditions are obtained for a continuous function guaranteeing the unifo... more Necessary and sufficient conditions are obtained for a continuous function guaranteeing the uniform convergence on the whole interval [ &1, 1] of its Lagrange interpolant based on the Jacobi nodes. The conditions are in terms of 4-variation, 8-variation, the modulus of variation, and the Banach indicatrix of a function.

Computers & Mathematics with Applications, 2000

In the present paper, we pursue the general idea suggested in our previous work. Namely, we utili... more In the present paper, we pursue the general idea suggested in our previous work. Namely, we utilize the truncated Fourier series as a tool for the approximation of the points of discontinuities and the magnitudes of jumps of a 27r-periodic bounded function. Earlier, we used the derivative of the partial sums, while in this work we use integrals. First, we obtain new identities which determine the jumps of a 2n-periodic function of VP, 1 <p < 2, class, with a finite number of discontinuities, by means of the tails of its integrated Fourier series. Next, based on the,se identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of the jump of a 27r-periodic piecewise smooth function with one singularity. By an appropriate linear combination, obtained via integrals of different order, we significantly improve the accuracy of the initial approximations. Then, we apply Richardson's extrapolation methocl to enhance the approximation results. For a function with multiple discontinuities we use simple formulae which "eliminate" all discontinuities of the function but one. Then we treat the function as if it had one singularity. Finally, we give the description of a programmable algorithm for the approximation of the discontinuities, investigate the stability of the method, study its complexity, and present some numerical results.

Applied and Computational Harmonic Analysis, 2001

In our earlier work we developed an algorithm for approximating the locations of discontinuities ... more In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas.

Mathematics of Computation, 2010

In the present paper, we generalize the method suggested in an earlier paper by the author and ov... more In the present paper, we generalize the method suggested in an earlier paper by the author and overcome its main deficiency. First, we modify the well-known Prony method, which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its Fourier coefficients with respect to any system of the classical orthogonal polynomials. Next, we will show that the method is applicable to a wider class of functions, namely, to the class of piecewise smooth functions—for functions which piecewise belong to C 2 [ − 1 , 1 ] C^2[-1,1] , the locations of discontinuities are approximated to within O ( 1 / n ) O(1/n) by means of their Fourier-Jacobi coefficients. Unlike the previous one, the generalized method is robust, since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial. In addition, the error estimate is unif...