In this work, we construct Stancu type modification of the generalization of Meyer-König and Zeller operators (MKZ) defined in [12]. We show that the Lipschitz constant of a Lipschitz continuous function and the properties of the function of modulus of continuity can be retained by these operators.

We propose a general scheme of reduction of a Lipschitz programming problem to a problem of minimizing increasing convex-along-rays function. It is based on the positively homogeneous extension of degree p of the objective function and projective transformation of IR n + onto the unit simplex. The application of cutting angle method to Lipschitz programming is considered.

We show that rough isometries between metric spacesX,Y can be lifted to the spaces of real valued 1-Lipschitz functions over X and Y with supremum metric and apply this to their scaling limits. For the inverse, we show how rough isometries between X and Y can be reconstructed from structurally enriched rough isometries between their Lipschitz function spaces.

We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.

Consider a bounded open set U ⊂ Rn and a Lipschitz function g : ∂U → Rm. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variant...

New concepts related to approximating a Lipschitz function between Banach spaces by a ne functions are introduced Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X Y for which any Lipschitz function from X to Y can be so approx imated is obtained This is applied to the study of Lipschitz and uniform quotient mapp...

We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α (0 < α < 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of mod...

The goal of this article is to develop a framework for large margin classification in metric spaces. We want to find a generalization of linear decision functions for metric spaces and define a corresponding notion of margin such that the decision function separates the training points with a large margin. It will turn out that using Lipschitz functions as decision functions, the inverse of the...