Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Author: Knut S. Eckhoff
Journal: Math. Comp. 64 (1995), 671-690
MSC: Primary 65T20; Secondary 65D15
MathSciNet review: 1265014
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Knowledge of a truncated Fourier series expansion for a $ 2\pi $-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the $ 2\pi $-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiable up to the specified order.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65T20, 65D15

Retrieve articles in all journals with MSC: 65T20, 65D15

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society