In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithm to reconstruct a signal x from its noisy measurement F (x) +...

We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally Lipschitzian operator in Rn is based on Rademacher’s theorem which does not hold in function spaces. We introduce a concept of slant differentiability and us...

In this paper we report on a method for regularizing a nonlinear Hammerstein type operator equation in Hilbert scales. The proposed method is a combination of Lavrentieve regularization method and a Modified Newton’s method in Hilbert scales . Under the assumptions that the operator F is continuously differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulf...

We discuss a number of novel steplength selection schemes for proximal-based convex optimization algorithms. In particular, we consider the problem where the Lipschitz constant of the gradient of the smooth part of the objective function is unknown. We generalize two optimization algorithms of Khobotov type and prove convergence. We also take into account possible inaccurate computation of the ...

LetM be a compact spin manifold with a smooth action of the ntorus. Connes and Landi constructed θ-deformations Mθ of M , parameterized by n×n real skew-symmetric matrices θ. TheMθ’s together with the canonical Dirac operator (D,H) on M are an isospectral deformation of M . The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show th...

A holomorphic self-map φ of the unit disk is constructed such that the composition operator induced by φ is bounded on the Hardy-Sobolev space H1 2 of order 2 as well as on the ordinary holomorphic Lipschitz space Lip1 but unbounded on the Zygmund class Λ1. Among these three function spaces we have embedding relations H1 2 ⊂ Lip1 ⊂ Λ1. So, the main points here are that our construction provides...

Let U be an open subset of \(\mathbb {C}\) with boundary point \(x_0\) and let \(A_{\alpha }(U)\) the space functions analytic on that belong to lip\(\alpha (U)\), “little Lipschitz class”. We consider 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, \(0<\lambda <1\), \(M_*^{1+\alpha }\) lower \(1+\alpha \) di...

Let Ω⊂ Rd be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space H1(Ω) up to functions in the Sobolev space H1(Rd) via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains Ω and Rd \Ω . Another estimate of...