Abstractthat fix the function $e^{-2x} $
e
−
2
x
for $x\geq 0 $
x
≥
0
. Then, we provide the approximation properties of these newly defined operators for different types of function spaces. In addition, we focus on the rate of convergence utilizing appropriate moduli of continuity. Then, we provide the Voronovskaya-type theorem for these new operators. Finally, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATLAB complier.
In this paper, we continue the study of approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{\Omega}_{p,\theta}$ in the space $L_{q},$ $1\leq p<q<\infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.
We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.
In this paper we find sharp inequalities of Jackson-Stechkin type between the best approximations of periodic differentiable functions by trigonometric polynomials and generalized moduli of continuity of m-th order in the space L_2. The exact values of various n-widths of classes of functions from L_2 defined by the generalized modus of continuity of the $r$-th derivative of the function f are calculated.
The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.
AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.
On approximation of Bernstein–Chlodowsky–Gadjiev type operators that fix $e^{-2x}$
