%0 Journal Article %1 barnard1998polynomials %A Barnard, R.W %A Dahlquist, G %A Pearce, K %A Reichel, L %A Richards, K.C %D 1998 %J Journal of Approximation Theory %K 33c15-confluent-hypergeometric-functions-whittaker-functions 33c45-orthogonal-polynomials-and-functions-of-hypergeometric-type 41a10-approximation-by-polynomials 42c10-fourier-series-in-special-orthogonal-functions %N 1 %P 128 - 143 %R https://doi.org/10.1006/jath.1998.3181 %T Gram Polynomials and the Kummer Function %U http://www.sciencedirect.com/science/article/pii/S0021904598931811 %V 94 %X Let φknk=0,n<m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form(g, h)d:=1m∑j=1mg(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 ofkandmfor fixedx∈−1, 1. We show that whenn⩽2.5m1/2, the polynomials in the family φknk=0are 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 degreekis close tom, andm⩾10, thenφk(x) oscillates with large amplitude for values ofxnear the endpoints of −1, 1, and this behavior makesφkpoorly 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 long-standing conjectured inequality for the confluent hypergeometric function1F1, also known as Kummer's function, i.e., that1F1((1−a)/2, 1, t2)⩽1F1(1/2, 1, t2) for alla⩾0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants.

@article{barnard1998polynomials, abstract = {Let {φk}nk=0,n<m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form(g, h)d:=1m∑j=1mg(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 ofkandmfor fixedx∈[−1, 1]. We show that whenn⩽2.5m1/2, the polynomials in the family {φk}nk=0are 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 degreekis close tom, andm⩾10, thenφk(x) oscillates with large amplitude for values ofxnear the endpoints of [−1, 1], and this behavior makesφkpoorly 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 long-standing conjectured inequality for the confluent hypergeometric function1F1, also known as Kummer's function, i.e., that1F1((1−a)/2, 1, t2)⩽1F1(1/2, 1, t2) for alla⩾0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants.}, added-at = {2020-01-15T05:14:56.000+0100}, author = {Barnard, R.W and Dahlquist, G and Pearce, K and Reichel, L and Richards, K.C}, biburl = {https://www.bibsonomy.org/bibtex/27d3354b894f923a22ecbfc91f54c0929/gdmcbain}, doi = {https://doi.org/10.1006/jath.1998.3181}, interhash = {a3fd2e99a9f9d89bce9f8640ef8810ce}, intrahash = {7d3354b894f923a22ecbfc91f54c0929}, issn = {0021-9045}, journal = {Journal of Approximation Theory}, keywords = {33c15-confluent-hypergeometric-functions-whittaker-functions 33c45-orthogonal-polynomials-and-functions-of-hypergeometric-type 41a10-approximation-by-polynomials 42c10-fourier-series-in-special-orthogonal-functions}, number = 1, pages = {128 - 143}, timestamp = {2020-01-15T05:14:56.000+0100}, title = {Gram Polynomials and the Kummer Function}, url = {http://www.sciencedirect.com/science/article/pii/S0021904598931811}, volume = 94, year = 1998 }