We obtain the rate of approximation of the bivariate operators by using the complete and partial moduli of continuity and the degree of approximation with the aid of a Lipschitz-type space and the Peetre K-functional.
The concern of this paper is to obtain Voronoskaja-type asymptotic result by calculating an estimate of fourth order central moment for these operators and discuss the rate of convergence for the bivariate case by using the complete and partial moduli of continuity and the degree of approximation by means of a Lipschitz-type function and the Peetre K-functional.
Taşdelen et al. [6] introduced a Kantorovich variant of the Szász operators based on Brenke type polynomials and discussed the order of convergence with the aid of the moduli of continuity and Peetre's K-functional.
This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process.
Also, we estimate the approximation order in terms of Peetre's K-functional and partial moduli of continuity.
Also, they estimated the approximation order in terms of Peetre's K-functional and partial moduli of continuity.
We have also calculated the rate of convergence of operators by means of Peetre's K-functional and partial moduli of continuity.
In order to use a classical result of Vitanza (see, Theorem 2.1 in [13]) it is necessary to make an appropriate change of variables which allows to transform the operator L ˜ into a differential operator L ˜ ∗ whose lower order coefficients, in particular, belonging to Lebesgue spaces and their moduli of continuity can be estimated by moduli of continuity of the corresponding coefficients of L ˜.
For (fin C(I_{1}times I_{2})) and (delta>0), the partial moduli of continuity with respect to x and y are given by bar{omega}_{1}(f delta)=sup bigl{ biglvert f(x_{1},y -f(x_{2},y -fgrvert :yin I_{2} mbox_{2}d } vert x_{1}-x_{2}vert leqdelta bigr} and bar{omega}_{2}(f;delta)=sup bigl{ biglvert f(x,y bigrvert,yin2})bI_{2ert :xin I_{1} mbox{ and } vert y_{1}-y_{2}vert leqdelta bigr}.
Then the degree of approximation of a signal (function) f, using Hausdorff means of its Fourier series, is given by E_{n}(f)= inf_{n} Vert H_{n}-f Vert _{p}^{(v)} = O biggl( (n+1)^{-1} int_{(n+1)^{-1}}^{pi} frac{t^{-2} omega(t)}{ v(t)},dt biggr), (3.1) where ω and v denote the Zygmund moduli of continuity such that (omega(t)/v(t)) is positive and non-decreasing.
