Note di Matematica 27, n. 1, 2007, 131–137. Construction of a function using its values along C 1 curves Oltin Dogaru Department of Mathematics I, University Politehnica of Bucharest, Spaiul Independentei 313, 060042 Bucharest, Romania udriste@mathem.pub.ro oltin.horia@yahoo.com Received: 4/3/2005; accepted: 28/7/2006. Abstract. Let G : D ⊂ R n → R be a function. Any parametrized curve α in D determines the composition gα = G ◦ α. If α belongs to a family of curves, the family {gα} satisfies some conditions. Our goal is to find the conditions in which the families {α}, {gα} determine the function G. Section 1 emphasizes the origin of the problem. Section 2 defines and studies the notion of the Γ-function. Section 3 presents the construction of a function using a Γ-function. Keywords: extension of maps, function spaces, Γ-function, generated functions MSC 2000 classification: primary 54C20, secondary 54C35 1 The origin of the problem In the theory of nonholonomic optimization [6] it appears the following types of problems. Let D be an open set of R n and ω = ∑ n i=1 ω i dx i be a C 0 Pfaff form on D. For every parametrized C 1 curve α : I → D, we consider g α : I → R, g α (t)=  t t 0 〈ω(α(u)),α ′ (u)〉du + c α (a primitive of ω along α). In this way we obtain a family of functions {g α }, called system of ω-primitives which depends on the family of constants {c α }. Question: is it possible to choose the family {c α } such that g α◦ϕ = g α ◦ ϕ for any α and for any diffeomorphism ϕ?. If ω = dG, with G : D → R a C 1 function, the answer is positive, because we can consider g α = G ◦ α. In this way, it appears a more general problem. Let us suppose that for any parametrized curve α : I → D, a function g α : I → R is given. What conditions we must impose to the family {g α } in order to exist a unique function G : D → R, having certain properties (like continuity, with partial derivatives, class C 1 ) and such that G ◦ α = g α ? Recall that two C k parametrized curves α : I → D and β : J → D are said to be equivalent, if there exists a C k diffeomorphism ϕ : J → I such that β = α ◦ ϕ. We say that ϕ is a change of parameter on α. An equivalence class ˜ α of a given C k parametrized curve α is called curve. Then α is called a representative of ˜ α. Let I =[a, b] be a closed interval in R. A continuous mapping α : I → D is brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by ESE - Salento University Publishing