Abstract
Let R =( -∞, ∞), and let Q ∈ C 1 (R ): R → R + := (0, ∞ )b e an even function. We consider the exponential-type weights w(x )= e -Q(x) , x ∈ R. In this paper, we obtain a mean and uniform convergence theorem for the Lagrange interpolation polynomials Ln(f )i nLp ,1< p ≤∞ with the weight w. MSC: 41A05
Highlights
Introduction and preliminariesLet R = (–∞, ∞), and let Q ∈ C (R) : R → R+ := [, ∞) be an even function, and w(x) =exp(–Q(x)) be the weight such that ∞ xnw (x) dx < for all n = we can construct the orthonormal polynomials pn(x) = pn(w ; x) of degree n with respect to w (x).That is, pn(x)pm(x)w (x) dx = δmn (Kronecker’s delta) –∞and pn(x) = γnxn + · · ·, γn >
We investigate the convergence of the Lagrange interpolation polynomials with respect to the weight w ∈ F (C +)
Author details 1Department of Mathematics Education, Sungkyunkwan University, Seoul, 110-745, Republic of Korea. 2Department of Mathematics, Meijo University, Nagoya, 468-8502, Japan
Summary
The following example is the Freud-type weight: Q(x) = |x|α, α >. The following examples give the Erdős-type weights w = exp(–Q). When we consider the Erdős-type weights, the following definition follows from Damelin and Lubinsky [ ]. Damelin and Lubinsky [ ] got the following results with the Erdős-type weights w = exp(–Q) ∈ E. For any nonzero real-valued functions f (x) and g(x), we write f (x) ∼ g(x) if there exist constants C , C > independent of x such that C g(x) ≤ f (x) ≤ C g(x) for all x. For a fixed constant β > , we define φ(x) := + x –β/ Using this function, we have the following theorem. If wl,α,m is an Erdős-type weight, that is, T(x) := Tl,α,m(x) is unbounded, it is easy to show lim λ(b) =.
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