Modulus Of Smoothness Research Articles

Equivalence between K-functionals and modulus of smoothness on the quaternion algebra

Equivalence between K-functionals and modulus of smoothness on the quaternion algebra

On the equivalence of K-functionals and modulus of smoothness constructed by the generalized Fourier–Bessel transform

On the equivalence of K-functionals and modulus of smoothness constructed by the generalized Fourier–Bessel transform

The Approximation of Weighted Hölder Functions by Fourier-Jacobi Polynomials to the Singular Sturm-Liouville Operator

In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.

On Shape Parameter α‐Based Approximation Properties and q‐Statistical Convergence of Baskakov‐Gamma Operators

We construct a novel family of summation‐integral‐type hybrid operators in terms of shape parameter α ∈ [0,1] in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre’s K‐functional, Lipschitz class, and second‐order modulus of smoothness. The approximation results are then obtained in weighted space. Finally, for these operators q‐statistical convergence is also investigated.

A strong converse inequality for the iterated Boolean sums of the Bernstein operator

"We establish a two-term strong converse estimate of the rate of approximation by the iterated Boolean sums of the Bernstein operator. The characterization is stated in terms of appropriate moduli of smoothness or K-functionals."

A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

<p style='text-indent:20px;'>In this paper, we introduce a bi-variate case of a new kind of <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [<xref ref-type="bibr" rid="b31">31</xref>]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.</p>

On Stancu operators depending on a non-negative integer

In this paper, we deal with Stancu operators which depend on a non-negative integer parameter. Firstly, we define Kantorovich extension of the operators. For functions belonging to the space Lp [0, 1] , 1 ? p < ?, we obtain convergence in the norm of Lp by the sequence of Stancu-Kantorovich operators, and we give an estimate for the rate of the convergence via first order averaged modulus of smoothness. Moreover, for the Stancu operators; we search variation detracting property and convergence in the space of functions of bounded variation in the variation seminorm.

Approximation properties by some modified Szasz-Mirakjan-Kantorovich operators

The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by Mishra and Yadav (Tbilisi Mathematical Journal. 11(3), (2018), 175-91). Further, we determine the rate of convergence in the term of the with derivative of bounded variation and for the quantitative means of the defined operators, we establish the quantitative Voronovskaya type and Gr$\ddot{\text{u}}$ss type theorems. Moreover, the examples are given with graphical representation to support the main results.

Kantorovich variant of Stancu operators

Stancu type operators play a crucial role in convergence estimates. The present article concerns the convergence estimates for certain Stancu type Kantorovich operators. We first establish some direct formulas giving the local approximation theorem, Voronovskaja type asymptotic formula, bound for the second central moment with some curtailment, and the global approximation theorem by means of modulus of continuity and the Ditzian-Totik Modulus of smoothness. We also study the difference estimates between Stancu-Bernstein operators and its Kantorovich variant. Further, we show the convergence of these operators by graphics to certain functions.

Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

<p style='text-indent:20px;'>The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B<inline-formula><tex-math id="M1">\begin{document}$ \ddot{o} $\end{document}</tex-math></inline-formula>gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.</p>

Approximation by α-Bernstein-Schurer operator

In this paper, we introduce a new family of generalized Bernstein-Schurer operators and investigate some approximation properties of these operators. We obtain a uniform approximation result using the well-known Korovkin theorem and give the degree of approximation via second modulus of smoothness. Also, we present Voronovskaya and Grüss-Voronovskaya type results for these operators.

Approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces

In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(\cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${\mathcal{\widetilde{M}}}_{p(\cdot),\lambda(\cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.

Best neural approximation for 2-quasi inner product spaces

Many researchers studied two-normed spaces for objects of function approximation. Also, neural networks are great instruments for function approximation from function spaces, especially, Lebesque spaces (Lp ). In this paper, the aim is to approximate Lebesque integrable functions via two-quasi inner product that is defined here in terms of two-norm. Moreover, there is a definition of a special type of neural network with special weights to estimate the approximation error, which is equivalent to the modulus of smoothness of functions of study. Estimates of infimum and supremum bounds so that the modulus of smoothness is involved in each bound. It is interesting to investigate more work related to what we presented in this study, not only for those fields of neural network function approximation, but also to approximate functions by various types of operators.

The order of comonotone approximation of differentiable periodic functions

Let $\Dely$ be a set of all $2\pi$-periodic functions $f$ that are continuous on the real axis $R$\ and\ change their monotonicity at various fixed points $y_{i}\in\lbrack-\pi,\pi),\ i=1,...,2s,\ s\in N$ (i.e., there is a set $Y:=\{y_{i}\}_{i\in\mathbb{Z}}$ of points $y_{i}=y_{i+2s}+2\pi$ on $R$ such that $f$ are nondecreasing on $[y_{i},y_{i-1}]$ if $i$ is even, and nonincreasing if $i$ is odd). In the article, a function $f_{Y}=f\in C^{(1)}\cap\Dely$ has been constructed such that \[ \lim_{n\rightarrow\infty}\sup\frac{n\,E_{n}^{(1)}(f)}{\omega_{4}(f^{\prime },\pi/n)}=\infty, \] where $E_{n}^{(1)}(f)$ is the error of the best uniform approximation of the function $f\in\Dely$ by trigonometric polynomials of order $n\in N$, which also belong to the set $\Dely$, and $\omega_{4}(f^{\prime},\cdot)$ is the $4$-th modulus of smoothness of the function $f^{\prime}.$ So, for a certain constant $c$, the inequality $E_{n}^{(1)}(f)\leq\frac{c}{n}\omega _{3}(f^{\prime},\pi/n)$ is the best with respect to the order of the modulus of smoothness.

Optimal stochastic Bernstein polynomials in Ditzian–Totik type modulus of smoothness

We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian–Totik type modulus of smoothness on the interval [0,1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust.

On a New Construction of Generalized q-Bernstein Polynomials Based on Shape Parameter λ

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter λ on the symmetric interval [−1,1]. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables.

Equivalence of K-functionals and modulus of smoothness constructed by first Hankel–Clifford transform

Equivalence of K-functionals and modulus of smoothness constructed by first Hankel–Clifford transform

Rectifiable Curves in Proximally Smooth Sets

In this article we study some geometric properties of proximally smooth sets. First, we introduce a modification of the metric projection and prove its existence. Then we provide an algorithm for constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space, with the moduli of smoothness and convexity of power type. Our algorithm returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses our modification of the metric projection. We estimate the length of the constructed curve and its deviation from the segment with the same endpoints. These estimates coincide up to a constant factor with those for the geodesics in a proximally smooth set in a Hilbert space.

Approximation and localized polynomial frame on conic domains

Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone.The main results provide construction of semi-discrete localized tight frame in weighted L2 norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a K-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the K-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain.

Why moduli of p–smoothness?

For the increasing importance of discovering new types of moduli of smoothness, more suitable measurements are provided by the moduli of p–smoothness. Measuring fractional smoothness of functions by p–variation is used for many purposes in approximation theory. In this paper, we express ωk -1/p (f; d) in terms of ωk -1/q (f; d) for any 1 < p, q < ∞ to get fractional modulus of smoothness of functions with bounded kth p-varation. Also, embedding of the space is proved with necessity and sufficient conditions.