2011/11/21 by Yilmaz Simsek
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebra over a field #Approximation Theory and Sequence Spaces #Basis (linear algebra) #Basis function #Bernstein polynomial #Differential equation #Laplace transform #Mathematical proof #Monomial basis #math.CA #semigroups and automata theory
paper · pdf · doi:10.1186/1687-1812-2013-80
published as Fixed Point Theory and Applications 2013 · 12
arxiv created 2011/11/21 · openalex publication_date 2013/04/02 · openalex created_date 2017/03/16 · arxiv updated 2018/11/19 · openalex updated_date 2026/08/05
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions. MSC: 14F10, 12D10, 26C05, 26C10, 30B40, 30C15, 44A10.