Lipschitz Class Research Articles

Averaging principle for stochastic differential equations with monotone condition

In this paper, we consider the averaging principle for a class of stochastic differential equations (SDEs) with the nonlinear terms only satisfying the local Lipschitz and monotone conditions. Compared with the classical Lipschitz and linear growth conditions, conditions given in this paper are relatively weak, so the conclusions obtained in this paper are suitable for a wider class of SDEs.

Approximation by modified q-Gamma type operators via A-statistical convergence and power series method

In this article, we introduce a q-analogue of the operators defined by Usta and Betus (Linear Multilinear Algebra, DOI: 10.1080/03081087.2020.1791033 (2020)) and find a general expression of moments for the q- operators. Next, we study the approximation properties of these operators in a weighted space via A-statistical method and the power series method. Later, we define a k-th order generalization of the q- operators by using the approach of Kirov and Popova (Math. Balk. 7, 149–162 (1993)) and estimate the error in the approximation for the functions in a Lipschitz class. We modify these operators so that they reproduce the constant as well as a monomial and provide better approximation. Finally, we introduce some graphical results to validate our theoretical claims.

On length measures of planar closed curves and the comparison of convex shapes

In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle \(\mathbb {S}^1\) that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski–Fenchel–Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski–Fenchel–Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach.

Free interpolation for the Zygmund class

Abstract We introduce interpolation sets for the Zygmund class 𝒵 {\mathcal{Z}} in the unit disc of the complex plane. This space lies between the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the class of order α = 1 {\alpha=1} , whose interpolation sets are given in a different way. We prove that the interpolation sets for 𝒵 {\mathcal{Z}} are interpolation sets for the Lipschitz classes of order α, 0 < α < 1 {0<\alpha<1} , and the latter are interpolation sets for a space slightly larger than 𝒵 {\mathcal{Z}} .

Hilbert and Poincaré-Bertrand Formulas in Polyanalytic Function Theory Involving Higher Order Lipschitz Classes

Hilbert and Poincaré-Bertrand Formulas in Polyanalytic Function Theory Involving Higher Order Lipschitz Classes

Approximation Theorem for New Modification ofq-Bernstein Operators on (0,1)

In this work, we extend the works of F. Usta and construct new modifiedq-Bernstein operators using the second central moment of theq-Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’sK-functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces

In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.

Modified ρ-Bernstein Operators for Functions of Two Variables

In this manuscript, we consider a bivariate extension of modified ρ-Bernstein operators and obtain Voronovskaya type and Grüss Voronovskaya type theorems for these operators. Further, we determine the rate of convergence of these operators in terms of the complete and partial moduli of continuity and compute an estimate of the error in terms of the Peetre’s K-functional. Also, we define the associated Generalized Boolean Sum (GBS) operators and study the rate of convergence of these operators with the aid of the mixed modulus of smoothness for the Bögel continuous and Bögel differentiable functions and the degree of approximation for the Lipschitz class of Bögel continuous functions.

Approximation of functions of Lipschitz class and solution of Fokker-Planck equation by two-dimensional Legendre wavelet operational matrix

In this paper, Lipschitz class of two-variables is considered. This is the genralization of well-known Lipschitz class of functions. A new estimator of functions belonging to generalized Lipschitz class has been obtained. Also, the solutions for the Fokker-Planck equations have been obtained for two different cases by two-dimensional Legendre wavelet operational matrix method. The approximated solutions of the time-and space-Fokker Planck equation have been compared with the exact solutions and the solutions obtained by homotopy perturbation method. The proposed scheme is simple, effective and suitable for the solution of Fokker-Planck equation.

Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Quantitative Estimates for Nonlinear Sampling Kantorovich Operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in L^{p}-spaces, 1le p<infty , and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming f in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the L^p-case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.

Summation-Integral Type Operators Based on Lupas-Jain Functions

We introduce a genuine summation-integral type operators based on Lupaş-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and ????-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators.

Approximation by Phillips operators via q-Dunkl generalization based on a new parameter

In the present article we study the approximation properties of Phillips operators by q-Dunkl generalization. We construct the operators in a new q-Dunkl form and obtain the approximation properties in weighted function space. We give the rate of convergence in terms of Lipschitz class by initiate the modulus of continuity and finally, we present some direct theorems in Peetre’s K -functional.

Hölder conditions and $$\tau $$-spikes for analytic Lipschitz functions

Let U be an open subset of \(\mathbb {C}\) with boundary point \(x_0\) and let \(A_{\alpha }(U)\) be the space of functions analytic on U that belong to lip\(\alpha (U)\), the “little Lipschitz class”. We consider the condition \(S= \sum _{n=1}^{\infty }2^{(t+\lambda +1)n}M_*^{1+\alpha }(A_n \setminus U)< \infty ,\) where t is a non-negative integer, \(0<\lambda <1\), \(M_*^{1+\alpha }\) is the lower \(1+\alpha \) dimensional Hausdorff content, and \(A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}\). This is similar to a necessary and sufficient condition for bounded point derivations on \(A_{\alpha }(U)\) at \(x_0\). We show that \(S= \infty \) implies that \(x_0\) is a \((t+\lambda )\)-spike for \(A_{\alpha }(U)\) and that if \(S<\infty \) and U satisfies a cone condition, then the t-th derivatives of functions in \(A_{\alpha }(U)\) satisfy a Hölder condition at \(x_0\) for a non-tangential approach.

Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials

In the present article, we establish a link between the theory of positive linear operators and the orthogonal polynomials by defining Bernstein-Chlodowsky-Kantorovich operators based on Gould-Hopper polynomials (orthogonal polynomials) and investigate the degree of convergence of these operators for unbounded continuous functions having a polynomial growth. In this connection, the moments of the operators are derived first, and then the approximation degree of the considered operators is established by means of the complete and the partial moduli of continuity. Next, we focus on the rate of convergence of these operators for functions in a weighted space. The associated Generalized Boolean Sum (GBS) operator of the operators under study is defined, and the degree of approximation is studied with the aid of the mixed modulus of smoothness and the Lipschitz class of Bögel continuous functions.

Approximation by a new generalization of Szász-Mirakjan operators via (p,q )-calculus

In this work, we obtain the approximation properties of a new generalization of Szász-Mirakjan operators based on post-quantum calculus. Firstly, for these operators, a recurrence formulation for the moments is obtained, and up to the fourth degree, the central moments are examined. Then, a local approximation result is attained. Furthermore, the degree of approximation in respect of the modulus of continuity on a finite closed set and the class of Lipschitz are computed. Next, the weighted uniform approximation on an unbounded interval is showed, and by the modulus of continuity, the order of convergence is estimated. Lastly, we proved the Voronovskaya type theorem and gave some illustrations to compare the related operators' convergence to a certain function.

Observers Design for a Class of Nonlinear Stochastic Discrete-Time Systems

This paper mainly studied the observer design of Lipschitz stochastic discrete system. For the first time, generalized Lipschitz conditions are introduced into the observer design of a class of nonlinear stochastic discrete systems. From this paper, it can be seen that the generalized Lipschitz condition can better reflect the structural information of the nonlinear part, and thus has more advantages than the classical Lipschitz condition in the observer design. The stability criterion and observer design method of nonlinear stochastic discrete systems were proposed. Numerical examples illustrated the advantages and feasibility of the theoretical results obtained.

Absolute convergence of Fourier integrals and Lipschitz classes

The necessary and sufficient conditions, in terms of Fourier transforms $\hat{f}$ of functions $f \in L^1(\mathbb{R})$, are obtained for $f$ to belong to the Lipschitz classes $H^{\omega}(\mathbb{R})$ and $h^{\omega}(\mathbb{R})$.

Dunkl-type generalization of the second kind beta operators via (p,q)-calculus

The main purpose of this research article is to construct a Dunkl extension of (p,q)-variant of Szász–Beta operators of the second kind by applying a new parameter. We obtain Korovkin-type approximation theorems, local approximations, and weighted approximations. Further, we study the rate of convergence by using the modulus of continuity, Lipschitz class and Peetre’s K-functionals.

НЕРАВЕНСТВА, УТОЧНЯЮЩИЕ СВЯЗИ МЕЖДУ СМЕШАННЫМИ МОДУЛЯМИ ГЛАДКОСТИ В МЕТРИКАХ $L_p$ и $L_{\infty}$

The problem of estimating the moduli of smoothness of functions from Lq in terms of their moduli of smoothness from Lp is well known. The first stage in the estimation of moduli of smoothness was the study of the properties of functions from the Lipschitz classes and obtaining the corresponding embeddings in the works of Titchmarsh, Hardy, Littlewood, Nikol’skii. The classical Hardy-Littlewood embedding for Lipschitz spaces can be obtained as a consequence of the Ulyanov’s inequality for the moduli of continuity of a function of one variable. In the works of Ulyanov, the modulus of smoothness of natural order was considered. The introduction of fractional moduli of smoothness made it possible in the works of Potapov, Simonov, Tikhonov to strengthen the Ulyanov’s inequality. Later, the same authors were able to generalize Ulyanov’s inequality to functions of two variables, obtaining estimates for mixed moduli of smoothness. The sharpness of these inequalities was proved in the case when 1 &lt; 𝑝 &lt; 𝑞 &lt; ∞ or 1 = 𝑝 &lt; 𝑞 = ∞. In this article, we study mixed moduli of smoothness of fractional orders of a function of two variables. Inequalities are obtained that refine the previously known estimates of the Ulyanov type inequalities between mixed moduli of smoothness in the metrics Lp and Lq for values 1 &lt; 𝑝 &lt; 𝑞 = ∞. The accuracy of the obtained estimates is investigated. The relationship between these and previously known estimates has been studied.