Estimates of Functions with Transformed Double Fourier Series

2021 ◽  
Vol 58 (1) ◽  
pp. 32-83
Author(s):  
Boris V. Simonov
Keyword(s):  
Fourier Series ◽  
Double Fourier Series ◽  
Moduli Of Smoothness ◽  
Best Approximations

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

Steady-state modelling method for matrix-reactance frequency converter with boost topology

2011 ◽  
Vol 60 (2) ◽  
pp. 137-148
Author(s):  
Igor Korotyeyev ◽  
Beata Zięba
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Differential Equations ◽  
Numerical Method ◽  
Frequency Converter ◽  
Double Fourier Series ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Modelling Method ◽  
Three Phase

Steady-state modelling method for matrix-reactance frequency converter with boost topologyThis paper presents a method intended for calculation of steady-state processes in AC/AC three-phase converters that are described by nonstationary periodical differential equations. The method is based on the extension of nonstationary differential equations and the use of Galerkin's method. The results of calculations are presented in the form of a double Fourier series. As an example, a three-phase matrix-reactance frequency converter (MRFC) with boost topology is considered and the results of computation are compared with a numerical method.

Double Fourier Series

2018 ◽  
pp. 1
Author(s):  
Amira Ali Een Faied
Keyword(s):  
Fourier Series ◽  
Double Fourier Series

scholarly journals Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M

2019 ◽  
Vol 69 (6) ◽  
pp. 1367-1380 ◽  
Author(s):  
Stanislav Chaichenko ◽  
Andrii Shidlich ◽  
Fahreddin Abdullayev
Keyword(s):  
Fractional Order ◽  
Moduli Of Smoothness ◽  
Best Approximations ◽  
Approximation Theorems ◽  
Inverse Approximation ◽  
Type Spaces

Abstract In the Orlicz type spaces 𝓢M, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre K-functionals in the spaces 𝓢M.

scholarly journals The double Fourier series of a discontinuous function

1924 ◽  
Vol 106 (737) ◽  
pp. 299-314 ◽  
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Double Fourier Series ◽  
General Term ◽  
Discontinuous Function ◽  
Lebesgue Integral ◽  
Function Of Two Variables

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function f ( x, y ) possesses a double Lebesgue integral over the square (– π < π ; – π < y < π ). Then the general term of the double Fourier series of this function is given by cos = є mn { a mn cos mx cos ny + b mn sin mx sin ny + c mn cos mx sin ny + d mn sin mx cos ny } There є 00 = ¼, є m0 = ½ ( m > 0), є 0n = ½ ( n > 0), є ms = 1 ( m > 0, n >0). the coefficients are given by the formulæ a mn = 1/ π 2 ∫ π -π ∫ π -π f ( x, y ) cos mx cos ny dx dy , obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫ ms = Σ m μ = 0 Σ n v = 0 A μ v = 1/π 2 ∫ π -π ∫ π -π f (s, t) sin( m +½) ( s - x ) sin ( n + ½) ( t - y )/2 sin ½ ( s - x ) 2 sin ½ ( t - y ) if f ( s , t ) is defined outside the original square by double periodicity, we have sub S ms = 1/π 2 ∫ π 0 ∫ π 0 f ( x + s , y + t ) + f ( x + s , y - t ) + f ( x - s , y + t ) + f ( x - s , y - t ) sin ( m + ½) s / 2 sin ½ s sin ( n + ½) t / 2 sin ½ t ds dt .

scholarly journals Threading Dislocations Piercing the Free Surface of an Anisotropic Hexagonal Crystal: Review of Theoretical Approaches

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Salem Neily ◽  
Sami Dhouibi ◽  
Roland Bonnet
Keyword(s):  
Free Surface ◽  
Fourier Series ◽  
Dislocation Core ◽  
Surface Region ◽  
Double Fourier Series ◽  
Threading Dislocations ◽  
Core Size ◽  
Hexagonal Crystal ◽  
Integral Formalism ◽  
Theoretical Approaches

Inclined threading dislocations (TDs) piercing the oriented free surface of a crystal are currently observed after growth of oriented thin films on substrates. Up to date the unique way to treat their anisotropic elastic properties nearby the free surface region is to use the integral formalism, which assumes no dislocation core size and needs numerical double integrations. In a first stage of the work, a new and alternative approach to the integral formalism is developed using double Fourier series and the concept of a finite core size, which is often observed in high-resolution transmission electron microscopy. In a second stage, the integral formalism and the Fourier series approaches are applied to the important case of a TD piercing the basal free surface of a hexagonal crystal. For this particular geometry, easy-to-use expressions are derived and compared to a third approach previously known for a plate-like crystal. Finally, the numerical interest and the convergence of these approaches are tested using the basal free surface of the GaN compound, in particular for TDs with Burgers vectors c and (a + c).

Damping of floor vibrations by constrained viscoelastic layers

1977 ◽  
Vol 4 (4) ◽  
pp. 405-411
Author(s):  
A. Farah ◽  
I. M. Ibrahim ◽  
R. Green
Keyword(s):  
Fourier Series ◽  
Loss Modulus ◽  
Optimization Technique ◽  
Double Fourier Series ◽  
Viscoelastic Layer ◽  
Sandwich Beams ◽  
Algorithm Optimization ◽  
Floor Systems ◽  
Floor Vibrations ◽  
Viscoelastic Layers

Formulations are presented for enhancing the serviceability of one-way floor systems subjected to dynamic loading through the use of constrained viscoelastic layers. The constrained layers are combined with the floors and the resulting systems are analyzed as sandwich structures using a double Fourier series approach. Results indicate that the damping of the resulting sandwich beams is governed by factors related to the elastic and geometric properties of the constrained layer and the constraining system (i.e. cover plates and beams) and the loss modulus of the viscoelastic material, and is highly influenced by the location of the viscoelastic layer in the sandwich beams. Optimum designs of the sandwich beams are obtained using the box algorithm optimization technique.

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