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In this paper, we propose an integrability condition under which we prove C 1 linearization for planar contractions in the sense of modulus of continuity and we establish the regularity of corresponding diffeomorphisms. Besides, we explain that our integrability condition is indeed a sharp condition for C 1 linearization by constructing a counterexample, i.e., we construct certain planar contraction F which fails to satisfy the sharp intergrability condition, so that it can not be C 1 linearized. [ABSTRACT FROM AUTHOR]

This article explores a Chlodowsky-type extension of the Kantorovich variant of Szász operators based on the associated λ-polynomials. The convergence properties of these operators are established by employing the universal Korovkin-type property, and the order of approximation is determined using the classical modulus of continuity and the Ditzian-Totik modulus of continuity. Additionally, the αβ-statistical convergence properties of the operators are derived. Additional approximation outcomes are also formulated within the context of the polynomial weighted space. Finally, the theoretical results are supported by numerical and graphical examples for different parameters. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the rate of the approximation by the matrix transform of partial sums of double Vilenkin-Fourier series in Lp( G m 2 ) space (1 ≤ p < ∞) and in C( G m 2 ). We give an application for Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of blending-type Bernstein–Kantorovich operators. These operators depend on three parameters: α , γ , and s. We establish results on the uniform convergence and rate of convergence of these operators in terms of the first and second order modulus of continuity. We also investigate the shape-preserving properties of the operators, such as monotonicity and convexity, for each choice of α , γ , and s. Finally, we provide graphical and numerical results to illustrate the accuracy of the operators and to demonstrate how they approach certain functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the nonlinear Meyer-König and Zeller operators based on q-integers. Firstly, we describe the q-Meyer-König and Zeller operators of max-product type. Then, we give an error estimation for the q-Meyer-König and Zeller operators of max-product kind by using a modulus of continuity. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the approximation properties of the biharmonic Poisson integral in the upper half-plane. The problem of approximating functions by biharmonic Poisson operators in the upper half-plane in the metric space Lp (−∞, +∞) is considered. The main result of the paper is based on the representation of the integral kernel of the biharmonic Poisson integral obtained by applying the parameterization approach. It was found that the considered integral kernel belongs to the class of delta-shaped kernels. The upper bound is obtained for the approximation of functions by biharmonic Poisson operators in terms of the first-order modulus of continuity. [ABSTRACT FROM AUTHOR]

In this article, a novel Schurer form of λ -Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ -Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin's type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre's K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new sequence of Riemann-Liouville type fractional Bernstein-Kantorovich operators $ K_{n}^{\alpha}(f;x) $ depending on a parameter $ \alpha $. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first and second order modulus of continuity of these operators. Moreover, we introduce a new operator that preserves affine functions from Riemann-Liouville type fractional Bernstein-Kantorovich operators. Further, we define the bivariate case of Riemann-Liouville type fractional Bernstein-Kantorovich operators and investigate the order of convergence. Some numerical results are given to illustrate the convergence of these operators and its comparison with the classical case of these operators. [ABSTRACT FROM AUTHOR]

In this article, the (p, q)–Stancu–Schurer–Bleimann–Butzer–Hahn ((p, q)‐SSBBH) operators are introduced. The Korovkin‐type theorem is obtained to show the approximation properties of these operators. Then, the rate of convergence of these operators with the help of the modulus of continuity and Lipschitz‐type maximal functions is calculated, respectively. Finally, for the asymptotic behavior of these operators, the Voronovskaja‐type theorem is given. Furthermore, the convergence of these operators to the considered function f by plotting the graphs is demonstrated. And, this convergence is compared with the convergence of the (p, q)–Bleimann–Butzer–Hahn ((p, q)‐BBH) operators to the same function. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions ν s , q (z) = (z − 1 2 [ s ] q ) ϱ on [ 0 , ∞) , we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators' uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre's K-functional. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a generalization of the Szász-Baskakov operators with the help of A(2) class Appell polynomials. For every compact subset of [0, 8), the uniform convergence of these operators is provided. We also mention the convergence rate of our new operators and then we find some approximation results. The rate of convergence is obtained with the help of the Steklov function. [ABSTRACT FROM AUTHOR]

We introduce blended variant of Kantorovich-Stancu type Lupaş operators and study convergence and q-statistical convergence properties using Korovkin theorem. We investigate rate of convergence interms of modulus of continuity, Peetre's K-functional and Lipschitz functions. Some Direct results and Voronovskaja type theorem are established. Moreover, graphical analysis and error estimations are presented using MATLAB. [ABSTRACT FROM AUTHOR]

In this study, a new kind of modified λ -Bernstein-Stancu operators is constructed. Compared with the original λ -Bézier basis function, the newly operator basis function is more concise in form and has certain symmetry beauty. The moments and central moments are computed. A Korovkin-type approximation theorem is presented, and the degree of convergence is estimated with respect to the modulus of continuity, Peetre's K-functional, and functions of the Lipschitz-type class. Moreover, the Voronovskaja type approximation theorem is examined. Finally, some numerical examples and graphics to show convergence are presented. [ABSTRACT FROM AUTHOR]

Local continuity is established for locally bounded, weak solutions to a doubly non-linear parabolic equation that models the temperature of a material undergoing a multi-phase transition. The enthalpy, as a maximal monotone graph of the temperature, is allowed to possess several jumps and/or infinite derivatives at the transition temperatures. The effect of the p-Laplacian-type diffusion is also considered. As an application, we demonstrate a continuity result for the saturation in the flow of two immiscible fluids through a porous medium, when irreducible saturation is present. [ABSTRACT FROM AUTHOR]

In this paper, we present a Kantorovich-type Szász–Mirakjan operators. Initially, we establish the recurrence relationship for the moments of these operators and provide the central moments up to the fourth degree. Subsequently, we analyze the local approximation properties of these operators using Peetre’s K-function. We investigate the rate of convergence, by utilizing the ordinary modulus of continuity and Lipschitz-type maximal functions. Additionally, we prove weighted approximation theorems and Voronoskaja-type theorems specific to these new operators. Following this, we introduce bivariate extension of these operators and investigate some approximation properties. Lastly, we include several numerical illustrative examples. [ABSTRACT FROM AUTHOR]

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in L p (G 2) space ( 1 ≤ p < ∞ ) and in C (G 2) . It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods. [ABSTRACT FROM AUTHOR]

In this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function χ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function σ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, L p -approximation is also established. At the end of the paper, examples of activation functions are discussed. [ABSTRACT FROM AUTHOR]

In this article, we study the approximation properties of Durrmeyer-type sampling operators. We consider the composition of generalized sampling operators and Durrmeyer sampling operators. For the new composition operators, we provide the pointwise and uniform convergence, as well as the quantitative estimates in terms of the first-order modulus of continuity and K-functional. Moreover, we investigate the rate of convergence using weighted modulus of continuity. Also, difference estimates are provided for the operators. Additionally, we provide approximation results for the linear combinations of the composition operators. Finally, we discuss the rate of convergence for the operators via graphical example. [ABSTRACT FROM AUTHOR]

This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre- K functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables. [ABSTRACT FROM AUTHOR]

Abstract This article explores a Chlodowsky-type extension of the Kantorovich variant of Szász operators based on the associated λ-polynomials. The convergence properties of these operators are established by employing the universal Korovkin-type property, and the order of approximation is determined using the classical modulus of continuity and the Ditzian-Totik modulus of continuity. Additionally, the αβ-statistical convergence properties of the operators are derived. Additional approximation outcomes are also formulated within the context of the polynomial weighted space. Finally, the theoretical results are supported by numerical and graphical examples for different parameters.

Abstract In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions ν s , q ( z ) = ( z − 1 2 [ s ] q ) ϱ $\nu _{s,q}(z)=\Bigl(z- \frac{1}{2[s]_{q}}\Bigr)_{\varrho}$ on [ 0 , ∞ ) $[0, \infty )$ , we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function ψα$$ {\psi}_{\alpha } $$ and investigate their approximation properties. By choosing an appropriate function ψα$$ {\psi}_{\alpha } $$, the order of approximation of our operators to a function f$$ f $$ is at least as good as the classical Bernstein–Kantorovich operators on the interval [0,1]$$ \left[0,1\right] $$. We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided. [ABSTRACT FROM AUTHOR]

In the present article, we introduce a Kantorovich variant of the neural network interpolation operators activated by smooth ramp functions proposed by Qian and Yu (2022). We discuss the convergence of these operators in the spaces, C([c,d])$$ C\left(\left[c,d\right]\right) $$ and Lp([c,d]),1≤p<∞$$ {\mathtt{L}}&amp;#x0005E;{\mathtt{p}}\left(\left[c,d\right]\right),1\le \mathtt{p}&lt;\infty $$, and establish some direct approximation theorems. Further, we derive the converse results by means of Berens–Lorentz lemma and Peetre's K‐functional. We present a multivariate version of the aforementioned Kantorovich neural network interpolation operators and investigate the direct and converse results in the continuous and Lp,1≤p<∞$$ {\mathtt{L}}&amp;#x0005E;{\mathtt{p}},1\le \mathtt{p}&lt;\infty $$, spaces. [ABSTRACT FROM AUTHOR]

This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $ L^p $ initial vorticity, provided that $ p\geq 4 $. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $ |\log\nu|^{-\alpha/2} $ with $ \alpha > 0 $.

Abstract Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w $L^{p,w}$ , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w $f\in L^{p,w}$ . We use the modulus of continuity of the functions to measure the rate of convergence.

We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\hat{f}(n)$, $n\in\mathbb{N}$, of functions $f\in L^p(G)$ for some $1

The known logarithmic-type modulus of continuity is improved for weak solutions to the two-phase Stefan problem in low space dimensions. In the two dimensional problem, the modulus becomes \boldsymbol \omega (r)\approx \operatorname {exp}\{-c|\operatorname {ln} r|^{\frac 12}\}, for some c\in (0,1). In the one dimensional problem, the modulus becomes Hölder-type. The main merit lies in a delicate, potential theoretical estimate that sharpens the expansion of positivity for non-negative super-solutions to parabolic equations. Our argument offers a unified approach to the state-of-the-art moduli in all dimensions. [ABSTRACT FROM AUTHOR]

The primary objective of this research article is to introduce and study an approximation operator involving the Tricomi function by using Korovkin’s theorem and a conventional method based on the modulus of continuity. In Lipschitz-type spaces, we demonstrate the rate of convergence, and we are also able to determine the convergence properties of our operators. In addition, we illustrate the convergence of our proposed operators using various graphs and error-estimating tables for numerical instances. [ABSTRACT FROM AUTHOR]

The almost-sure sample path behavior of the operator fractional Brownian motion with exponent D, including multivariate fractional Brownian motion, is investigated. In particular, the global and the local moduli of continuity of the sample paths are established. These results show that the global and the local moduli of continuity of the sample paths are completely determined by the real parts of the eigenvalues of the exponent D, as well as the covariance matrix at some unit vector. These results are applicable to multivariate fractional Brownian motion. [ABSTRACT FROM AUTHOR]

Beyond Hölder’s type, this paper mainly concerns the persistence and the remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allowing the non-integrable part to be critically finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only the modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps the prescribed universal Diophantine frequency unchanged. Further, the non-Hölder regularity for the invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of Durrmeyer variant of generalized Ismail operators based on Sheffer polynomials. These type of polynomials are a generalization of Appell polynomials and a special case of Boas-Buck-type polynomials. We study the convergence of these operators with the help of universal Korovkin type theorem, modulus of continuity, Peetre's $ K $-functional and the class of Lipschitz type functions. Furthermore, We extend this study to include Voronovskaja-type asymptotic results, quantitative-Voronovskaja and Gr$ \ddot{u} $ss-Voronovskaja type theorems and the convergence rate of these operators for functions in a polynomial weighted space by means of the weighted modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators. [ABSTRACT FROM AUTHOR]

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail [ABSTRACT FROM AUTHOR]

In the present article, we introduce a Durrmeyer variant of certain approximation operators. We estimate the moment-generating function and moments of these operators employing the Lambert W function and establish some direct results. We further provide a composition of these operators with Szász–Mirakjan operators and estimate direct results for the composition operator. Additionally, we provide a graphical comparison of the approximation properties of the operators. [ABSTRACT FROM AUTHOR]

A method is given for finding the exact value of the argument of the modulus of continuity in estimates of the rate of convergence of Fourier series. [ABSTRACT FROM AUTHOR]

In this research paper, we construct a new sequence of Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem. [ABSTRACT FROM AUTHOR]

Singular integral operators play an important role in approximation theory and harmonic analysis. In this paper, we consider a weighted Lebesgue space L p , w , define a modified Gauss–Weierstrass singular integral on it, and study direct and inverse approximation properties of the operator followed by a Korovkin-type approximation theorem for a function f ∈ L p , w . We use the modulus of continuity of the functions to measure the rate of convergence. [ABSTRACT FROM AUTHOR]

This paper introduces a new idea to obtain a better order of approximation for the Baskakov operator. We conclude two new operators from orders one and two of the Baskakov type. Also, we prove some directed results concerning the rate of convergence of these operators. We apply the Maple software to demonstrate how the operators converge to a specific function. [ABSTRACT FROM AUTHOR]

It is usually not possible to solve partial differential equations, especially the delay type, with analytical methods. Therefore, in this article, we present an efficient method for solving differential equations of the difference delayed reaction-diffusion type, which can be generalized to other delayed partial differential equations. In the proposed approach, we first convert the delayed equation into an equivalent non-delayed equation by inserting the corresponding delay function with an effective technique. Then, using a pseudo-spectral method, we discretize the obtained equation in the Legendre-Gauss-Lobatto collocation points and present an algebraic system with an equal number of equations and unknowns which can be solved by quasi-Newton methods such as Levenderg-Marquardt algorithm. The approximate solutions can be obtained with exponential accuracy. The convergence analysis of the method is fully discussed and four examples are presented to evaluate the results and compare with one of the conventional methods used to solve partial differential equations, that is, the compact finite difference method. [ABSTRACT FROM AUTHOR]

In this study, a different generalization of q-Bernstein operators with the parameter λ ∈ [ − 1 , 1 ] is created. The moments and central moments of these operators are calculated, a statistical approximation result for this new type of (λ , q) -Bernstein operators is obtained, and the convergence properties are analyzed using the Peetre K-functional and the modulus of continuity for this new operator. Finally, a numerical example is given to illustrate the convergence behavior of the newly defined operators. [ABSTRACT FROM AUTHOR]

This paper deals with the approximations of the functions by generalized Durrmeyer operators of Szász–Mirakjan, which are linear positive operators. Several approximation results are presented well, and we estimate the approximation properties along with the order of approximation and the convergence theorem of the proposed operators. For an explicit explanation of the operators, we determine the properties using the weight function. A quantitative approach is discussed for the operators; quantitative Voronovskaya type and Grüss type theorems are established, showing the operators' more efficient work. We investigate the A$$ A $$‐statistical convergence properties for the said operators, including the rate of approximation in a statistical sense. An important property for the rate of convergence of the operators is obtained in terms of the function with a derivative of the bounded variation. At last, the graphical representations and numerical analysis are discussed and shown to support our theoretical findings. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators. [ABSTRACT FROM AUTHOR]

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with L p initial vorticity, provided that p ≥ 4. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order | log ⁡ ν | − α / 2 with α > 0. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the rate of the approximation by the de la Vallée Poussin type Marcinkiewicz matrix transform of Walsh-Fourier series in Lp(G²) spaces (1 ≤ p < ∞) and in C(G²). Namely, we prove ... in some special cases. Moreover, we give an application for functions in Lipschitz classes Lip(α,p,G²) (α > 0, 1 ≤ p < ∞) and Lip(α, C(G²)) (α > 0). [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]