Better approximation by a Durrmeyer variant of $ \alpha- $Baskakov operators

<p style='text-indent:20px;'>The aim of this paper is to study some approximation properties of the Durrmeyer variant of <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Baskakov operators <inline-formula><tex-math id="M3">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> proposed by Aral and Erbay [<xref ref-type="bibr" rid="b3">3</xref>]. We study the error in the approximation by these operators in terms of the Lipschitz type maximal function and the order of approximation for these operators by means of the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja and Gr<inline-formula><tex-math id="M4">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type theorems are also established. Next, we modify these operators in order to preserve the test functions <inline-formula><tex-math id="M5">\begin{document}$ e_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ e_2 $\end{document}</tex-math></inline-formula> and show that the modified operators give a better rate of convergence. Finally, we present some graphs to illustrate the convergence behaviour of the operators <inline-formula><tex-math id="M7">\begin{document}$ M_{n,\alpha} $\end{document}</tex-math></inline-formula> and show the comparison of its rate of approximation vis-a-vis the modified operators.</p>

Splines of the Fourth Order Approximation and the Volterra Integral Equations

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

Image Compression and Enlargement Algorithms

In some cases, there are problems associated with the compression and enlargement of images. The use of splines is quite effective in some cases. In this paper, a new image compression algorithm is presented. The features of increasing the size of an image when using local polynomial or non-polynomial splines are considered. The main method for enlarging an image is based on the use of splines of the second and third order of approximation. Polynomial and trigonometric splines are considered. To speed up the process of enlarging the image, we used the parallelization techniques

Sharing Asymmetric Tail Risk: Smoothing, Asset Prices and Terms of Trade

Crises and tail events have asymmetric effects across borders, raising the value of arrangements improving insurance of macroeconomic risk. Using a two-country DSGE model, we provide an analytical and quantitative analysis of the channels through which countries gain from sharing (tail) risk. Riskier countries gain in smoother consumption but lose in relative wealth and average consumption. Safer countries benefit from higher wealth and better average terms of trade. Calibrated using the empirical distribution of moments of GDP-growth across countries, the model suggests non-negligible quantitative effects. We offer an algorithm for the correct solution of the equilibrium using DSGE models under complete markets, at higher order of approximation.

Blending type Approximations by Kantorovich variant of α-Schurer operators

In the present manuscript, we present a new sequence of operators, i:e:, -Bernstein-Schurer-Kantorovich operators depending on two parameters 2 [0; 1] and > 0 foe one and two variables to approximate measurable functions on [0:1+q]; q > 0. Next, we give basic results and discuss the rapidity of convergence and order of approximation for univariate and bivariate of these sequences in their respective sections . Further, Graphical and numerical analysis are presented. Moreover, local and global approximation properties are discussed in terms of rst and second order modulus of smoothness, Peetre’s K-functional and weight functions for these sequences in dierent spaces of functions.

Adapted Exponential Type Estimator in the Presence of Non-response

In this article, we propose an estimator using the exponential function for the population mean in the case of non-response on both the study and the auxiliary variables. The equations for the Bias and Mean Square Error (MSE) are derived to the first order of approximation and theoretical comparisons are made with existing estimators in literature. Besides, we examine the efficiency of the proposed estimator according to the classical ratio and regression estimator, Hansen-Hurwitz unbiased estimator, and the estimator of Singh et al. (2009). Following theoretical comparisons, we infer that the proposed estimator is more efficient than compared estimators under the obtained conditions in theory. Moreover, these theoretical results are supported numerically by providing an empirical study on five different data sets.

Simultaneous Approximation by a New Sequence of Integral Type Based on Two Parameters

This paper introduces a generalization sequence of positive and linear operators of integral type based on two parameters to improve the order of approximation. First, the simultaneous approximation is studied and a Voronovskaja-type asymptotic formula is introduced. Next, an error of the estimation in the simultaneous approximation is found. Finally, a numerical example to approximate a test function and its first derivative of this function is given for some values of the parameters.

Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $$L^{p}$$ L p -spaces, $$1\le p<\infty $$ 1 ≤ p < ∞ , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $$L^p$$ L p -case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

ON THE DEVELOPMENT OF RATIO TYPE ESTIMATORS USING AUXILIARY INFORMATION

The paper produces some new modified forms of the ratio estimators using the auxiliary information. The large sample properties, that is, the bias and mean squared error up to the first order of approximation are determined. The comparison is made with other existing estimators by using an applied data. It has been observed that the proposed estimators have a fewer mean squared error and leads to the efficient results as compared to the classical ratio estimator, Sisodia and Dwivedi, Singh and Kakran, Upadhyaya and Singh estimators.