Differentiable Lipschitz Research Articles

A q‐Erkuş–Srivastava polynomials operator

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a ‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the ‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted ‐statistical convergence and the power series method. We also define a th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of ‐Erkuş–Srivastava multivariable polynomials operator and study its ‐statistical convergence by using four‐dimensional matrix transformation.

The Cost of Nonconvexity in Deterministic Nonsmooth Optimization

We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using [Formula: see text] calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves [Formula: see text] for any difference-of-convex objective and [Formula: see text] for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense. Funding: This work was supported by the National Science Foundation [Grant DMS-2006990].

On Vector-Valued Banach Function Algebras

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Silov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.

Bivariate positive linear operators constructed by means of q-Lagrange polynomials

In this paper we propose an extension of the linear positive operators constructed by means of the q-Lagrange polynomials and investigate the order of convergence by means of the complete and partial moduli of continuity and the Peetre's K-functional. We also define the associated Generalized Boolean Sum (GBS) operators and investigate the rate of convergence of these operators with the aid of mixed modulus of smoothness for Bögel continuous and Bögel differentiable functions. Finally, we introduce an sth (s being a non-negative integer) order generalization of the above operators and study the degree of approximation of these operators for sth order continuously differentiable Lipschitz class functions.

φ-contractibility of some classes of Banach function algebras

In this paper, we study ?-contractibility of natural Banach function algebras on a compact Hausdorff space. As a consequence, we characterize ?-contractibility of the Lipschitz algebra Lip(X,d?), for a compact metric space (X,d). We also characterize ?-contractibility of certain subalgebras of Lipschitz functions including rational Lipschitz algebras, analytic Lipschitz algebras and differentiable Lipschitz algebras.

Approximation of o-minimal maps satisfying a Lipschitz condition

Consider an o-minimal expansion of the real field. We show that definable Lipschitz continuous maps can be definably fine approximated by definable continuously differentiable Lipschitz maps whose Lipschitz constant is close to that of the original map.

Radon–Nikodym Property and Area Formula for Banach Homogeneous Group Targets

We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable Lipschitz mappings defined on a Carnot group.

Normed algebras of infinitely differentiable Lipschitz functions

Let X be a perfect compact plane set, ω ={w n } be a weight sequence of positive numbers, and 0 < α ≤ 1. Then Lip(X, α, ω) denotes the algebra of infinitely differentiable functions f on X such that f (n), for n = 0, 1, …, satisfies the Lipschitz condition of order α, and such that ,where ||·||α is the Lipschitz norm on X. Using some formulae from combinatorial analysis, we show that, under certain conditions on ω, if f ∈ Lip(X, α, ω) and f (z)≠0 for every z ∈ X, then 1/f ∈ Lip(X, α, ω). We then conclude that if, moreover, the maximal ideal space of (X, α, ω), the uniform closure of Lip(X, α, ω), equals X then every non-zero continuous complex homomorphism on Lip(X, α, ω) is an evaluation character at some point of X.

Compact and quasicompact homomorphisms between differentiable Lipschitz algebras

In this note we consider homomorphisms between differentiable Lipschitz algebras $Lip^n(X,\alpha)$ ($0<\alpha \leq 1$) and $lip^n(X,\alpha)$ ($0<\alpha <1$), where $X$ is a perfect compact plane set. We give sufficient conditions implying the compactness and power compactness of these homomorphisms. Moreover, we investigate under what conditions a quasicompact homomorphism between these algebras is power compact. We also give a necessary condition for a homomorphism between these algebras to be quasicompact and in certain cases to be power compact. Finally, using these results, by giving an example we show that there exists a quasicompact homomorphism between these algebras which is not power compact.

Compact Endomorphisms of Infinitely Differentiable Lipschitz Algebras

Compact Endomorphisms of Infinitely Differentiable Lipschitz Algebras

Porosity and differentiability in smooth Banach spaces

We improve a result of Preiss, Phelps and Namioka, showing that every submonotone mapping in a Gateaux smooth Banach space is single-valued on the complement of a $\sigma$-cone porous subset. If a Banach space $E$ has a uniformly $\beta$-differentiable Lipschitz bump function (with respect to some bornology $\beta$), then we show with a much simpler argument (localization of $\delta$-minimum of a perturbed function) that every continuous convex function on $E$ is $\beta$-differentiable on the complement of a $\sigma$-uniformly porous set.

Graphs of Gâteaux Derivatives are &lt;em&gt;w&lt;/em&gt;∗-Connected

We show that if $(X, \|. \|)$ is a separable Banach space, $\Omega \subset X$ is open, connected and $f: \Omega \to \mathbb{R}$ is an everywhere Gâteaux differentiable Lipschitz continuous function, then the graph of the derivative of $f$ is connected in $(\Omega,\| . \|)\times(X^\star , w^\star)$.

Not too well differentiable Lipschitz isomorphisms

Several examples of Lipschitz isomorphisms are given with non-surjective derivatives. In particular there is a Lipschitz isomorphism on l2 that maps a cube affinely into a hyperplane.

Inverse functions of pseudo regular mappings and regularity conditions

We show that, even for monotone directionally differentiable Lipschitz functionals on Hilbert spaces, basic concepts of generalized derivatives identify only particular pseudo regular (or metrically regular) situations. Thus, pseudo regularity of (multi-) functions will be investigated by other means, namely in terms of the possible inverse functions. In this way, we show how pseudo regularity for the intersection of multifunctions can be directly characterized and estimated under general settings and how contingent and coderivatives may be modified to obtain sharper regularity conditions. Consequences for a concept of stationary points as limits of Ekeland points in nonsmooth optimization will be studied, too.