Myron S. Henry

On extremal sets and strong unicity constants for certain C∞ functions

Abstract For each f continuous on the interval I , let B n ( f ) denote the best uniform polynomial approximation of degree less than or equal to n . Let M n ( f ) denote the corresponding strong unicity constant. For a certain class of nonrational functions F , it is shown that there exist positive constants α and β and a natural number N such that αn ⩽ M n ( f ) ⩽ βn for n ⩾ N . The results of the present paper also provide concise estimates to the location of the extreme points of f − B n ( f ). The set F includes the functions f α ( x ) = e αx , α ≠ 0.

Uniform strong unicity for rational approximation

Abstract Let R n m denote the class of rational functions defined on a closed interval I with numerators in the class of polynomials of degree at most n and positive valued denominators in the class of polynomials of degree at most m . If f ϵ C ( I ) is normal, the well-known strong unicity theorem asserts that there is a smallest positive constant γ n , m ( f ) such that ∥ f − R ∥ ⩾ ∥ f − R f ∥ + γ n , m ( f )∥ R − R f ∥ for all R ϵ R n m , where R f is the best uniform approximation to f from R n m . In this paper, the dependence of γ n , m ( f ) on f is investigated. Sufficient conditions are given to insure that inf fϵΓ γ n , m ( f ) > 0, where Γ is a subset of C( I ). Necessity of these conditions is investigated and examples are given to show that known results for R n 0 do not directly extend to R n m for m > 0.

Local and global Lipschitz constants

Abstract Let X be a closed subset of I = [−1, 1], and let B n ( f ) be the best uniform approximation to f ϵ C [ X ] from the set of polynomials of degree at most n . An extended global Lipschitz constant is defined for f, and it is shown that this constant is asymptotically equivalent to the strong unicity constant. Estimates of the size of the local Lipschitz constant for f are given when the cardinality of the set of extremal points of f − B n ( f )is n + 2. Examples which illustrate that the local and extended global Lipschitz constants may have very different asymptotic behavior are constructed.

Precise orders of strong unicity constants for a class of rational functions

Abstract Let R ⊆ C[∞-1, 1] denote a certain class of rational functions. For each f ϵ R, consider the polynomial of degree at most n that best approximates f in the uniform norm. The corresponding strong unicity constant is denoted by M n ( f ). Then there exist positive constants α and β, not depending on n , such that an ⩽ M n ( f )⩽ βn , n = 1,2,….

Orders of strong unicity constants

Abstract Given f ϵ C ( I ), the growth of the strong unicity constant M n ( f ) for changing dimension is considered. Under appropriate hypotheses it is shown that 2 n + 1 ⩽ M n ( f ) ⩽ βn 2 . Furthermore, relationships between certain Lebesgue constants and M n ( f ) are established.

Local Lipschitz constants

Abstract Let X be a closed subset of I= [− 1, 1], For f ϵ C[X], the local Lipschitz constant is defined to be λ nδ (f) = sup { ∥B n (f) − B n (g)∥ ∥f − g∥: 0 , where Bn(g) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as δ → 0 of the local Lipschitz constant of f, and an explicit expression is given for this common value. The, possibly very different, characterizations of local and global Lipschitz constants are also considered.

Lebesgue constants for certain classes of nodes

Abstract For each f ϵ C(I), let Bn(f) be the best uniform polynomial approximation of degree at most n, and let en(f) = f − Bn(f) be the error function. Denote the set of extreme points of en(f) by En(f), and assume that this set has precisely n + 2 points. If E( F ) is the infinite triangular array of nodes whose nth row consists of the n + 2 points of En(f), then the corresponding Lebesgue constant of order n is designated Λ n (E( F )) . For certain rational and non-rational functions it is shown that Λ n (E( F )) = 0( log n) .

Error bounds for polynomial product approximation

Abstract This paper establishes bounds on the uniform error in the approximation of a continuous function defined on a rectangle by polynomial product approximations. The dependence of product approximations on the basis functions used for the associated polynomial spaces is investigated.

Local Lipschitz and strong unicity constants for certain nonlinear families

Abstract Let X be a compact metric space, and let V = { F ( a , x ): a ϵ A } where A is an open subset of R n , and F ( a , x ) and ∂F ∂a i , 1 ⩽ i ⩽ n , are continuous on A × X . Suppose f ϵ C ( X ) is weakly normal; that is (i) f has a best approximation F(a ∗ , ·) = B v (f) such that N = dim W(a ∗ ) ≡ dim span {( ∂F ∂a i )(a ∗ , ·): 1 ⩽ i

Product Approximation: Error Estimates

n} is maximal, and (ii) certain weakened versions of the local Haar condition, a sign property equivalent to a form of asymptotic convexity, and Property Z hold. For those weakly normal functions f for which { x ϵ X: ¦f(x) − F(a ∗ , x)¦ = ∥f − F(a ∗ , ·)∥ } has exactly N + 1 points, we give constructions of the local Lipschitz and strong unicity constants, as well as show that B v ( f ) is differentiable.