scholarly journals Degrees of the approximation of integrable functions by some special matrix means of fourier series

2019 ◽  
Vol 105 (119) ◽  
pp. 39-47
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka ◽  
Ewa Sylwestrzak-Maślanka
Keyword(s):  
Fourier Series ◽  
Integrable Functions ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix ◽  
Approximation Of Function

We consider the pointwise and normwise approximation of function by some special matrix means of its Fourier series. The results corresponding to the theorem of ?enski and Szal in [5] and the results of Saini and Singh in [6] are shown. Some special cases as corollaries are also formulated.

scholarly journals On the approximation of integrable functions by some special matrix means of Fourier series

2017 ◽  
Vol 50 (1) ◽  
pp. 351-359
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Summability Methods ◽  
Integrable Functions ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix

Abstract The results concerninig pointwise approximation and product of summability methods corresponding to the theorems of Xh. Z. Krasniqi [Poincare J. Anal. Appl., 2014, 1, 1-8] and W. Łenski and B. Szal [Math. Slovaca, 2016, 66(6), 1-12] are generalized. Some special cases are also formulated as corollaries.

scholarly journals On the Approximation of Functions From Lp by Some Special Matrix Means of Fourier Series

2015 ◽  
Vol 55 (1) ◽  
pp. 81-90
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Special Matrix ◽  
Norm Approximation ◽  
Appl Math

Abstract The results corresponding to some theorems of S. Lal [Appl. Math. and Comput. 209 (2009), 346-350] and the results of W. Łenski and B. Szal [Banach Center Publ., 95, (2011), 339-351] are shown. The better degrees of pointwise approximation than these in mentioned papers by another assumptions on summability methods for considered functions are obtained. From presented pointwise results the estimation on norm approximation are derived. Some special cases as corollaries are also formulated.

scholarly journals On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
W. Łenski ◽  
B. Szal
Keyword(s):  
Fourier Series ◽  
Conjugate Functions ◽  
Pointwise Approximation ◽  
Conjugate Fourier Series ◽  
Summability Methods ◽  
Matrix Means ◽  
Special Cases ◽  
Norm Approximation

The results generalizing some theorems onN, pnE, γsummability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.

scholarly journals On Pointwise Approximation of Functions by Some Matrix Means of Conjugate Fourier Series

2015 ◽  
Vol 55 (1) ◽  
pp. 91-108
Author(s):  
W. Lenski ◽  
B. Szal
Keyword(s):  
Fourier Series ◽  
Pointwise Approximation ◽  
Approximation Of Functions ◽  
Conjugate Fourier Series ◽  
Riesz Method ◽  
Matrix Means ◽  
Special Cases ◽  
Norm Approximation

Abstract The results corresponding to some theorems of S. Lal [Tamkang J. Math., 31(4)(2000), 279-288] and the results of the authors [Banach Center Publ. 92(2011), 237-247] are shown. The same degrees of pointwise approximation as in mentioned papers by significantly weaker assumptions on considered functions are obtained. From presented pointwise results the estimation on norm approximation with essentialy better degrees are derived. Some special cases as corollaries for iteration of the Nörlund or the Riesz method with the Euler one are also formulated.

scholarly journals Degrees of the approximations by some special matrix means of conjugate Fourier series

2019 ◽  
Vol 52 (1) ◽  
pp. 116-129
Author(s):  
Radosława Kranz ◽  
Aleksandra Rzepka ◽  
Ewa Sylwestrzak-Maślanka
Keyword(s):  
Fourier Series ◽  
Conjugate Fourier Series ◽  
Matrix Means ◽  
Special Matrix

Abstract In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

scholarly journals Definition of Set of diagnostic Parameters of System based on the Functional Spaces Theory

2019 ◽  
Vol 18 (4) ◽  
pp. 949-975 ◽  
Author(s):  
Valentin Senchenkov ◽  
Damir Absalyamov ◽  
Dmitriy Avsyukevich
Keyword(s):  
Fourier Series ◽  
Experimental Studies ◽  
Numerical Procedure ◽  
Physical Nature ◽  
Convergent Series ◽  
Technical Condition ◽  
Partial Sums ◽  
Diagnostic Tools ◽  
Diagnostic Parameters ◽  
Integrable Functions

The development of methodical and mathematical apparatus for formation of a set of diagnostic parameters of complex technical systems, the content of which consists of processing the trajectories of the output processes of the system using the theory of functional spaces, is  considered in this paper. The trajectories of the output variables are considered as Lebesgue measurable functions. It ensures a unified approach to obtaining diagnostic parameters regardless  a physical nature of these variables and a set of their jump-like changes (finite discontinuities of trajectories). It adequately takes into account a complexity of the construction, a variety of physical principles and algorithms of systems operation. A structure of factor-spaces of measurable square Lebesgue integrable functions, ( spaces) is defined on sets of trajectories. The properties of these spaces allow to decompose the trajectories by the countable set of mutually orthogonal directions and represent them in the form of a convergent series. The choice of a set of diagnostic parameters as an ordered sequence of coefficients of decomposition of trajectories into partial sums of Fourier series is substantiated. The procedure of formation of a set of diagnostic parameters of the system, improved in comparison with the initial variants, when the trajectory is decomposed into a partial sum of Fourier series by an orthonormal Legendre basis, is presented. A method for the numerical determination of the power of such a set is proposed. New aspects of obtaining diagnostic information from the vibration processes of the system are revealed. A structure of spaces of continuous square Riemann integrable functions ( spaces) is defined on the sets of vibrotrajectories. Since they are subspaces in the afore mentioned factor-spaces, the general methodological bases for the transformation of vibrotrajectories remain unchanged. However, the algorithmic component of the choice of diagnostic parameters becomes more specific and observable. It is demonstrated by implementing a numerical procedure for decomposing vibrotrajectories by an orthogonal trigonometric basis, which is contained in spaces. The processing of the results of experimental studies of the vibration process and the setting on this basis of a subset of diagnostic parameters in one of the control points of the system is provided. The materials of the article are a contribution to the theory of obtaining information about the technical condition of complex systems. The applied value of the proposed development is a possibility of their use for the synthesis of algorithmic support of automated diagnostic tools.

scholarly journals Conformable Derivatives in Laplace Equation and Fractional Fourier Series Solution

2019 ◽  
Vol 9 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Ronak Pashaei ◽  
Mohammad Sadegh Asgari ◽  
Amir Pishkoo
Keyword(s):  
Fourier Series ◽  
Laplace Equation ◽  
Series Solution ◽  
Fractional Integral ◽  
Fourier Coefficients ◽  
Separation Of Variables ◽  
Special Cases ◽  
Separation Of Variables Method ◽  
Conformable Derivatives ◽  
Conformable Fractional Integral

In this paper the solution of conformable Laplace equation, \frac{\partial^{\alpha}u(x,y)}{\partial x^{\alpha}}+ \frac{\partial^{\alpha}u(x,y)}{\partial y^{\alpha}}=0, where 1 < α ≤ 2 has been deduced by using fractional fourier series and separation of variables method. For special cases α =2 (Laplace's equation), α=1.9, and α=1.8 conformable fractional fourier coefficients have been calculated. To calculate coefficients, integrals are of type "conformable fractional integral".

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