Twice continuously differentiable semilocal smoothing spline

Twice continuously differentiable periodic local and semilocal smoothing splines, or S-splines from the class C ² are considered. These splines consist of polynomials of 5th degree, first three coefficients of each polynomial are determined by values of the previous polynomial and two its derivatives at the point of splice, coefficients at higher terms of the polynomial are determined by the least squares method. These conditions are supplemented by the periodicity condition for the spline function on the whole segment of definition or by initial conditions. Uniqueness and existence theorems are proved. Stability and convergence conditions for these splines are established.     

Finely continuously differentiable functions

In Lávi?ka [A remark on fine differentiability, Adv. Appl. Clifford Algebras 17 (2007) 549–554], it is observed that finely continuously differentiable functions on finely open subsets of the plane are just functions which are finely locally extendable to usual continuously differentiable functions on the whole plane. In this note, it is proved that, under a mild additional assumption, this result remains true even in higher dimensions. Here the word “fine” refers to the fine topology of classical potential theory.     

Finely continuously differentiable functions

This paper establishes an explicit characterization of those real-valued functions on a finely open set in Euclidean space which are continuously differentiable with respect to the fine topology of classical potential theory. It improves and extends previous work of several authors. Differentiability of all orders is considered, and some consequences of the characterization are deduced.     

Cube formulas on spaces of continuously differentiable functions

Siberian Mathematical Journal -     

A continuously differentiable filled function method for global optimization

In this paper, a new filled function method for finding a global minimizer of global optimization is proposed. The proposed filled function is continuously differentiable and only contains one parameter. It has no parameter sensitive terms. As a result, a general classical local optimization method can be used to find a better minimizer of the proposed filled function with easy parameter adjustment. Numerical experiments show that the proposed filled function method is effective.     

Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation φ(f(x))=g(φ(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise defining. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one.     

Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions

The author examines matrices and bordered matrices having exactly one or at most one negative eigenvalue. These results are then used to state matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions. For quadratic functions, a new criterion for quasiconvexity is established, and its equivalence to the already known criterion [11,23] is also shown.     

Approximation by cubic splines in the classes of continuously differentiable functions

The problem of approximating continuously differentiable periodic functionsf(x) by cubic interpolation splines s_n(f; x) with equidistant nodes is considered. Asymptotically exact estimates for f(x)-s_n(f; x)_C are obtained in the classes of functions W¹H.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 215–226, February, 1972.In conclusion, I am deeply grateful to N. P. Korneichuk for a number of valuable remarks and conjectures utilized while working on this paper.     

Korovkin type approximation theorems on the space of continuously differentiable fnctions

It is studied Korovkin type approximation theorems on C⁽¹⁾ ([0, 1]) the space of continuously differentiable functions on the unit interval. It is proved that test functions for which Korovkin type approximation theorems hold depending on norms of C⁽¹⁾ ([0, 1]).     

On the functional form of convex underestimators for twice continuously differentiable functions

The optimal functional form of convex underestimators for general twice continuously differentiable functions is of major importance in deterministic global optimization. In this paper, we provide new theoretical results that address the classes of optimal functional forms for the convex underestimators. These are derived based on the properties of shift-invariance and sign- invariance.     

Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions

The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on Cⁿ[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cⁿ[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.     

On weak convergence of functionals of stochastic processes with continuously differentiable paths

We obtain a criterion for weak convergence of a sequence of stochastic processes _n(t), t [0, 1],n N, _n(t) R ^m in the spaceC _m ^k [0, 1] of continuously differentiable functions. We consider several examples of weakly convergent sequences of stochastic processes inC _m ^k [0, 1] and several integer functionals defined on these random variables.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 85–90, 1987.     

On the role of continuously differentiable exact penalty functions in constrained global optimization

The aim of this paper is to show that the new continuously differentiable exact penalty functions recently proposed in literature can play an important role in the field of constrained global optimization. In fact they allow us to transfer ideas and results proposed in unconstrained global optimization to the constrained case.First, by drawing our inspiration from the unconstrained case and by using the strong exactness properties of a particular continuously differentiable penalty function, we propose a sufficient condition for a local constrained minimum point to be global.Then we show that every constrained local minimum point satisfying the second order sufficient conditions is an attraction point for a particular implementable minimization algorithm based on the considered penalty function. This result can be used to define new classes of global algorithms for the solution of general constrained global minimization problems. As an example, in this paper we describe a simulated annealing algorithm which produces a sequence of points converging in probability to a global minimum of the original constrained problem.     

On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

In this article we study the validity of the Whitney \(C^1\) extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the \(C^1\) extension property. We conclude by showing that the \(C^1\) extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.     

Approximation of smooth functions by S-splines

The paper contains a method for recovering a function from a table of its values. The method ensures smoothing of measured data, can be easily transformed into an algorithm and used in automatic information processing. The reconstruction method described in the paper is related to the approximation of smooth functions by means of some special splines, called S-splines, introduced by one of the authors. Conditions for stability and convergence of S-splines are obtained.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 197–206, 1984.The authors would like to thank Prof. V. M. Tikhomirov for advice and help with this work; thanks are also due to V. A. Gruzdev and E. V. Logacheva who provided their programs and computations.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

A minimization method for the sum of a convex function and a continuously differentiable function

This paper presents a method for finding the minimum for a class of nonconvex and nondifferentiable functions consisting of the sum of a convex function and a continuously differentiable function. The algorithm is a descent method which generates successive search directions by solving successive convex subproblems. The algorithm is shown to converge to a critical point.The authors wish to express their appreciation to the referees for their careful review and helpful comments.