Cube formulas on spaces of continuously differentiable functions

Siberian Mathematical Journal -     

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.     

Multiplicative functionals on algebras of differentiable functions

Research supported in part by D.G.I.C.Y.T Project PB-87-1031, Spain.     

Variational functionals defined on spaces ofBV functions and their dependence on boundary data

Summary We extend in a suitable way a class of functionals defined on W ^1,1 (Ω) to the space BV_b(Ω) ⊗ (C(∂Ω))*. Optimization theorems for the extended functional are given. As an application we obtain a known result about minimal surface problems. Entrata in Redazione il 18 luglio 1978. This work was supported in part by Laboratorio per la Matematica Applicata del C.N.R., Italy.     

Local mappings on spaces of differentiable functions

We study conditions under which a functionalF(u, B) defined for everyu∈C ^k(Ω;R ^m ) and every Borel subsetB of Ω admits the integral representation

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.     

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.     

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

We characterize finite codimensional linear isometries on two spaces, C ⁽ⁿ⁾[0; 1] and Lip [0; 1], where C ⁽ⁿ⁾[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C ⁽ⁿ⁾[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.     

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.     

Interpolation of weighted spaces of differentiable functions onR d

Summary Let W _p ^m (w) be the Sobolev space of functions f such thatD fL _p(w) for ||m.     

Constructing regularization methods in the spaces of differentiable functions

The method of finding approximations to a solution and its derivatives is constructed for a certain class of integral Volterra equations of the first kind. Matching conditions for the regularization parameter and the error in the initial data are presented. Sharp (in terms of order) error estimates are obtained for approximate solutions on certain compact classes.