TY - JOUR

T1 - Reconstruction of piecewise smooth multivariate functions from fourier data

AU - Levin, David

N1 - Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2020/9

Y1 - 2020/9

N2 - In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Pade-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case.

AB - In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Pade-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case.

KW - Fourier data

KW - Multivariate approximation

KW - Piecewise smooth

KW - Reconstruction

UR - http://www.scopus.com/inward/record.url?scp=85089520296&partnerID=8YFLogxK

U2 - 10.3390/AXIOMS9030088

DO - 10.3390/AXIOMS9030088

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AN - SCOPUS:85089520296

SN - 2075-1680

VL - 9

JO - Axioms

JF - Axioms

IS - 3

M1 - 88

ER -