scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

Functions of Minimal Norm with the Given Set of Fourier Coefficients

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 651
Author(s):  
Pyotr Ivanshin
Keyword(s):  
Fourier Series ◽  
Existence And Uniqueness ◽  
Fourier Coefficients ◽  
Nonnegative Function ◽  
Trigonometric Fourier Series ◽  
Minimum Norm ◽  
Minimal Norm ◽  
Uniqueness Of The Solution ◽  
The Given

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

scholarly journals Reconstruction of Piecewise Smooth Multivariate Functions from Fourier Data

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 88
Author(s):  
David Levin
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Order Approximation ◽  
High Accuracy ◽  
Rectangular Domain ◽  
High Order ◽  
Piecewise Smooth ◽  
Multidimensional Case ◽  
Order Spline ◽  
The Given

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 Padé-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.

scholarly journals Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

2017 ◽  
Vol 4 (1) ◽  
pp. 119-133 ◽  
Author(s):  
V.V. Zozulya
Keyword(s):  
Fourier Series ◽  
Couple Stress ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Couple Stress Theory ◽  
Order Theory ◽  
Theory Of Elasticity ◽  
High Order ◽  
Stress Theory ◽  
Curved Rods

AbstractNew models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

scholarly journals Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 276 ◽  
Author(s):  
Taekyun Kim ◽  
Dae Kim ◽  
Lee-Chae Jang ◽  
Gwan-Woo Jang
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Bernoulli Polynomials ◽  
Series Expansions ◽  
Linear Combinations ◽  
Lucas Polynomials ◽  
Finite Products ◽  
Derive Fourier Series

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

scholarly journals Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

2017 ◽  
Vol 4 (1) ◽  
pp. 104-118 ◽  
Author(s):  
V.V. Zozulya
Keyword(s):  
Fourier Series ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Order Theory ◽  
Theory Of Elasticity ◽  
Micropolar Elasticity ◽  
Elasticity System ◽  
Thin Walled Structures ◽  
Curved Rods ◽  
Linear Micropolar Elasticity

AbstractNew models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin.High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors,vectors of displacements and rotation and body force shave been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking in to account micropolar couple stress and rotation effects.

EXPLICIT FORMULAS FOR THE FOURIER COEFFICIENTS OF A CLASS OF ETA QUOTIENTS

2012 ◽  
Vol 09 (02) ◽  
pp. 487-503 ◽  
Author(s):  
OLIVIA X. M. YAO ◽  
ERNEST X. W. XIA ◽  
J. JIN
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Fourier Coefficients ◽  
Series Expansions ◽  
Explicit Formulas

Recently, Williams discovered explicit formulas of the coefficients c(n) in the Fourier series expansions of a class of eta quotients. Motivated by the results obtained by Williams, we find that the coefficients c(2n) in the Fourier series expansions of another class of eta quotients can be represented as a linear combination of σ3(n), σ3(n/2), σ3(n/3) and σ3(n/6). One example is [Formula: see text] where [Formula: see text]

scholarly journals Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

2017 ◽  
Vol 4 (1) ◽  
pp. 221-236 ◽  
Author(s):  
V.V. Zozulya
Keyword(s):  
Fourier Series ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Constitutive Relations ◽  
Nonlocal Theory ◽  
Order Theory ◽  
Theory Of Elasticity ◽  
High Order ◽  
Middle Line ◽  
Curved Rods

Abstract New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.

scholarly journals On the Use of Piecewise Standard Polynomials in the Numerical Solutions of Fourth Order Boundary Value Problems

2014 ◽  
Vol 33 ◽  
pp. 53-64 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Md. Bellal Hossain
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Linear Combinations ◽  
Piecewise Continuous ◽  
The Given

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

scholarly journals A simple Spectral Observer

2018 ◽  
Author(s):  
Lizeth Torres ◽  
Javier Jiménez- Cabas ◽  
José Francisco Gómez-Aguilar ◽  
Pablo Pérez-Alcazar
Keyword(s):  
Fourier Series ◽  
Edge Detection ◽  
State Estimation ◽  
Linear Combination ◽  
Fourier Coefficients ◽  
Design Procedure ◽  
Recursive Identification ◽  
Market Data

The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. This paper proposes a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients and (4) the fitting of the historical Bitcoin market data from 2014-12-01 to 2018-01-08.

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