Gram Polynomials and the Kummer Function
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Gram Polynomials and the Kummer Function

10.1006/jath.1998.3181

Richards, Kendall C.

20180227

20180227

1998

Barnard, R. W., Dahlquist, G., Pearce, K., Reichel, L., & Richards, K. C. (January 01, 1998). Gram Polynomials and the Kummer Function. Journal of Approximation Theory, 94, 1, 128143.

https://collections.southwestern.edu/s/suscholar/item/257

Let {φk}n
k=0, n<m, be a family of polynomials orthogonal with respect to the positive semidefinite bilinear form
(g, h)d := 1
m Xm
j=1
g(xj )h(xj ), xj := −1 + (2j − 1)/m.
These polynomials are known as Gram polynomials. The present paper investigates the growth of φk(x) as a function of k and m for fixed x ∈ [−1, 1]. We show that when n ≤ 2.5m1/2, the polynomials in the family {φk}n
k=0 are of modest size on [−1, 1], and they are therefore well suited
for the approximation of functions on this interval. We also demonstrate that if the degree k is close to m, and m ≥ 10, then φk(x) oscillates with large amplitude for values of x near the endpoints of [−1, 1], and this behavior makes φk poorly suited for the approximation of functions on [−1, 1]. We study the growth properties of φk(x) by deriving a second order differential equation, one solution of which exposes the growth. The connection between Gram polynomials and this solution to the differential equation suggested what became a longstanding conjectured inequality for the confluent hypergeometric function 1F1, also known as Kummer’s function, i.e., that 1F1((1 − a)/2, 1, t2) ≤ 1F1(1/2, 1, t2) for all a ≥ 0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants.

English

Journal of Approximation Theory

Confluent hypergeometric function

Polynomial approximation

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