scholarly journals On Fourier Series of Fuzzy-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Uğur Kadak ◽  
Feyzi Başar
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Uniform Convergence ◽  
Level Sets ◽  
Closed Interval ◽  
Complex Form ◽  
Dirichlet Kernel ◽  
Extension Principle ◽  
Science And Engineering ◽  
Interest In Science

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.

scholarly journals On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Uğur Kadak ◽  
Hakan Efe
Keyword(s):  
Power Series ◽  
Uniform Convergence ◽  
Membership Function ◽  
Level Sets ◽  
Radius Of Convergence ◽  
Extension Principle ◽  
Membership Functions ◽  
Function Series ◽  
Convergence Of Sequences ◽  
The Relationship

The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.

scholarly journals Notes on Fourier analysis (XLVIII): Uniform convergence of Fourier series

1951 ◽  
Vol 3 (3) ◽  
pp. 298-305
Author(s):  
Shin-ichi Izumi ◽  
Gen-ichirô Sunouchi
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Uniform Convergence

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

2016 ◽  
Vol 6 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Lei Zhu ◽  
Wei-Wei Xu ◽  
Xing-Dong Yang
Keyword(s):  
Algebraic System ◽  
Condition Numbers ◽  
Linear Algebraic System ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Interest In Science ◽  
Perturbation Bounds

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

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