Modulus Of Smoothness Research Articles

On Banaś modulus of smoothness and Gao Pythagorean constant of L(μ)

On Banaś modulus of smoothness and Gao Pythagorean constant of L(μ)

Max-Product Sampling Kantorovich Operators: Quantitative Estimates in Functional Spaces

In this paper, we study the order of approximation for max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. We establish a quantitative estimate for the considered family of sampling-type operators using the Orlicz-type modulus of smoothness, which involves the modular functional of the space. From this result, it is possible to obtain the qualitative order of convergence when functions belonging to suitable Lipschitz classes are considered. On the other hand, in the compact case, we exploit a suitable definition of K-functional in Orlicz spaces in order to provide an upper bound for the approximation error of the involved operators. The treatment in the general framework of Orlicz spaces allows one to obtain a unifying theory on the rate of convergence, as the proved results can be deduced for a wide range of functional spaces, such as L p -spaces, interpolation spaces and exponential spaces.

Best one sided approximation of unbounded functions trigonometrical polynomials for one dimensional weighted-space

The aim of studying this research is to find the best one sided trigonometrical approximation of unbounded function of one variable in weighted space. Moreover we estimated the degree of the best one-sided trigonometrical approximation by averaged modulus of smoothness in same space.

Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

UDC 517.5 We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E s ( f ) and the moduli of smoothness generated by the spherical mean operator defined on the L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.

Convergence and high order of approximation by Steklov sampling operators

In this paper we introduce a new class of sampling-type operators, named Steklov sampling operators. The idea is to consider a sampling series based on a kernel function that is a discrete approximate identity, and which constitutes a reconstruction process of a given signal f, based on a family of sample values which are Steklov integrals of order r evaluated at the nodes k/w, k∈Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\in {\\mathbb {Z}}$$\\end{document}, w>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$w>0$$\\end{document}. The convergence properties of the introduced sampling operators in continuous functions spaces and in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-setting have been studied. Moreover, the main properties of the Steklov-type functions have been exploited in order to establish results concerning the high order of approximation. Such results have been obtained in a quantitative version thanks to the use of the well-known modulus of smoothness of the approximated functions, and assuming suitable Strang-Fix type conditions, which are very typical assumptions in applications involving Fourier and Harmonic analysis. Concerning the quantitative estimates, we proposed two different approaches; the first one holds in the case of Steklov sampling operators defined with kernels with compact support, its proof is substantially based on the application of the generalized Minkowski inequality, and it is valid with respect to the p-norm, with 1≤p≤+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le p \\le +\\infty $$\\end{document}. In the second case, the restriction on the support of the kernel is removed and the corresponding estimates are valid only for 1<p≤+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < p\\le +\\infty $$\\end{document}. Here, the key point of the proof is the application of the well-known Hardy–Littlewood maximal inequality. Finally, a deep comparison between the proposed Steklov sampling series and the already existing sampling-type operators has been given, in order to show the effectiveness of the proposed constructive method of approximation. Examples of kernel functions satisfying the required assumptions have been provided.

The Degree of Best Downsampled Convolutional Neural Network Approximation in Terms of Orlicz Modulus of Smoothness

The Degree of Best Downsampled Convolutional Neural Network Approximation in Terms of Orlicz Modulus of Smoothness

On Kantorovich variant of Brass-Stancu operators

Abstract In this study, we deal with Kantorovich-type generalization of the Brass-Stancu operators. For the sequence of these operators, we study L p {L}^{p} -convergence and give some upper estimates for the L p {L}^{p} -norm of the approximation error via first-order averaged modulus of smoothness and the first-order K K -functional. Moreover, we show that the Kantorovich generalization of each Brass-Stancu operator satisfies variation detracting property and is bounded with respect to the norm of the space of functions of bounded variation on [ 0 , 1 ] \left[0,1] . Finally, we present graphical and numerical examples to compare the convergence of these operators to given functions under different parameters.

Degree of Lp Approximation Using Activated Singular Integrals

In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.

Ulyanov inequalities for the mixed moduli of smoothness in mixed metrics

Abstract In this paper, mixed moduli of smoothness of functions of two variables are studied. We prove Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in different metrics. Estimates for the mixed moduli of smoothness of the derivative of a function are also obtained in terms of the mixed moduli of smoothness of the function itself.

The second order modulus revisited: remarks, applications, problems

Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.

Convergence of \(\lambda\)-Bernstein - Kantorovich operators in the \(L_p\)- norm

We show the convergence of \(\lambda\)-Bernstein - Kantorovich operators defined by Acu et al. [J. Ineq. Appl. 2018], for functions in \(L_p[0,1],\, p\geq 1\). We also determine the convergence rate via integral modulus of smoothness.

Quantitative Uniform Approximation by Activated Singular Operators

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

Approximation by Stancu variant of [formula omitted]-Bernstein shifted knots operators associated by Bézier basis function

The current paper presents the λ-Bernstein operators through the use of newly developed variant of Stancu-type shifted knots polynomials associated by Bézier basis functions. Initially, we design the proposed Stancu generated λ-Bernstein operators by means of Bézier basis functions then investigate the local and global approximation results by using the Ditzian–Totik uniform modulus of smoothness of step weight function. Finally we establish convergence theorem for Lipschitz generated maximal continuous functions and obtain some direct theorems of Peetre’s K-functional. In addition, we establish a quantitative Voronovskaja-type approximation theorem.

Generalized Modulus of Smoothness and K-functional.

Generalized Modulus of Smoothness and K-functional.

A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

Weighted integrability of the Cherednik-Opdam transform in terms of the moduli of smoothness

Weighted integrability of the Cherednik-Opdam transform in terms of the moduli of smoothness

Developing poplar wood into a green structure-decoration integrated material for prefabricated wooden building application

Fast growing poplar wood poses significant challenges including low mechanical strength, poor dimensional stability and weak texture, limiting its direct application in the field of building structures and decoration. We treated poplar wood using compression, digital printing and waterproofing techniques to develop a green structure-decoration integrated material. The compression treatment significantly improved the wood density, fiber orientation and surface smoothness. At a compression ratio of 60 %, poplar veneer demonstrates remarkable improvements in modulus of elasticity (MOE), modulus of rupture (MOR) and surface smoothness, with increases of 222.53 %, 233.53 % and 93.20 %, respectively, compared to uncompressed wood. Notably, this compression treatment reduces the surface roughness of poplar to 0.75 μm, meeting the flatness requirements for digital printing. Consequently, clear textures and vibrant colors are directly printed onto the flat surface of compression-densified poplar wood, eliminating the need for traditional surface sanding pre-treatment process, and accordingly improving the efficiency of the material preparation. Furthermore, a final waterproofing emulsion spraying treatment creates a dense micro/nanoscale hierarchical structure on the surface, enabling the compression-densified poplar wood samples to achieve MOE and MOR of 14.15 GPa and 123.00 MPa respectively, even after water absorption and re-dried processes. These values consistently exceed the standard value of MOE (12.50 GPa) and MOR (35.00 MPa) for the highest grade (TCT40) specified in GB50005-2017, indicating significantly improved structural stability. The structure-decoration integrated materials developed in this study offer high strength and dimensional stability, making them promising candidates for high value utilization in prefabricated wooden buildings.

On the Mellin-Gauss-Weierstrass operators in the Mellin-Lebesgue spaces

In this paper, we present the modulus of smoothness of a function $f∈X_{c}^{p}$, which the Mellin-Lebesgue space, and later we state some properties of it. In this way, the rate of convergence is gained. Moreover, we elucidate some pointwise convergence results for the Mellin-Gauss-Weierstrass operators. Especially, we acquire the pointwise convergence of them at any Lebesgue point of a function $f$.

Convolutional Neural Networks Approximation in Quasi-Orlicz Spaces on Sphers

It is necessary to study the theoretical bases of an approximation deep convolutional neural networks, because of its interesting developments in vital domains. The approximation abilities of deep-convolution neural networks produced by downsampling operators in quasi- Orlicz spaces have been studied, since this space is wider and more important than other spaces. In this paper, we define quasi-Orlicz norm on spherical spaces. In addition, modulus of smoothness is also studied in terms of quasi-Orlicz norm. Finally, Function approximation theorems are studied by using convolution neural networks with

Approximation by modified (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-gamma-type operators

The main object of this paper is to construct a new class of modified (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven.