On approximation of ultraspherical polynomials in the oscillatory region. (English) Zbl 1378.33007
The author studies ultraspherical polynomials given in terms of Jacobi polynomials \(P_k^{(\alpha,\alpha)}(x)\), where the parameter satisfies \(\alpha>-{k+1\over 2}\). As customary, the ‘oscillatory region’ is an interval containing all but maybe a few of the extreme zeros of a polynomial; for background material the reader is referred to the standard work of Szegő (the 1975 reprint of Orthogonal polynomials [4th ed. Providence, R. I.: American Mathematical Society (AMS) (1975; Zbl 0305.42011)]).
Introducing \[ u=(k+\alpha)(k+\alpha+1),\;q=(\alpha^2-1)/u \] and the normal form of the differential equation \[ y''+b^2y=0,\;y=(1-x^2)^{(\alpha+1)/2}P_k^{(\alpha,\alpha)}(x), \] where \[ b=b(x)={\sqrt{(1-q-x^2)u} \over 1-x^2}, \] and define the function \[ g(x)=\sqrt{b(x)}y(x), \] which almost equioscillates on \(|x|<\sqrt{1-q}\), the main result is formulated as:
{Theorem 1.} Let \(k\geq 2\) be even and let \(x\) belong to one of the following intervals, depending on the value of \(\alpha\): \[ \begin{align*}{ &\text{(i)}\quad 0\leq x\leq \sqrt{1-{1\over \alpha}},\ |\alpha|<\sqrt{7\over 6};\cr &\text{(ii)}\quad 0\leq x\leq \sqrt{1-q},\ \alpha\in [-{2k+1\over 4},-{7\over 6}]\cup [{7\over 6},\infty).\cr }\end{align*} \]
Then the following approximation holds \[ g(x)=g(0)(\cos{{\mathcal B}(x)}+r(x)), \] where \[ g(0)=\left(-{1\over 4}\right)^{k/2}{{k+\alpha}\choose{k/2}} (k^2+2k\alpha+\alpha+1)^{1/4}, \]
\[ {\mathcal B}(x)=\begin{cases} \sqrt{u}\left(\arccos{\sqrt{1-q-x^2\over 1-q}}- \sqrt{q}\arccos{\sqrt{1-q-x^2\over (1-q)(1-x^2)}}\right), &q\geq 0,\\ \sqrt{u}\left(\arccos{\sqrt{1-q-x^2\over 1-q}}+ \sqrt{-q} \arccos \text{h}{\sqrt{1-q-x^2\over (1-q)(1-x^2)}}\right), &q<0.\end{cases} \] The error term \(r(x)\) is bounded as follows: \[ r(x)=\begin{cases} {2.72 x\over \sqrt{(1-x^2)u}}, &q<0;\\ {2(1-x^2)x\over (1-q)(1-q-x^2)^{3/2} \sqrt{u}}, &0\leq q<{1\over 2};\\ {(1+q)x\over 4(1-q-x^2)^{3/2}}, &{1\over 2}\leq q<1.\end{cases} \]
As a corollary, the author shows that the ultraspherical polynomials ‘live’ in the interval \((-\sqrt{1-q},\sqrt{1-q})\):
{Theorem 2.} For \(k\geq 10\), even, \(\alpha\geq 3-{k\over 2},\;|\alpha|\geq 1\), \[ \int_{-\eta}^{\eta}\,(1-x^2)( P_k^{(\alpha,\alpha)}(x))^2dx>1-{5\over 3(1-q)^{1/3}u^{1/6}} > 1-{5\over 3}\cdot \left({k+\alpha\over (k+2\alpha)k}\right)^{1/3}, \] where \[ \eta=\sqrt(1-q)\left(1-{4\cdot 2^{1/3}\over 3(1-q)^{2/3} u^{1/3}}\right). \] Here \(P_k^{(\alpha,\alpha)}(x)\) is the orthonormal form of the ultraspherical polynomial.
Introducing \[ u=(k+\alpha)(k+\alpha+1),\;q=(\alpha^2-1)/u \] and the normal form of the differential equation \[ y''+b^2y=0,\;y=(1-x^2)^{(\alpha+1)/2}P_k^{(\alpha,\alpha)}(x), \] where \[ b=b(x)={\sqrt{(1-q-x^2)u} \over 1-x^2}, \] and define the function \[ g(x)=\sqrt{b(x)}y(x), \] which almost equioscillates on \(|x|<\sqrt{1-q}\), the main result is formulated as:
{Theorem 1.} Let \(k\geq 2\) be even and let \(x\) belong to one of the following intervals, depending on the value of \(\alpha\): \[ \begin{align*}{ &\text{(i)}\quad 0\leq x\leq \sqrt{1-{1\over \alpha}},\ |\alpha|<\sqrt{7\over 6};\cr &\text{(ii)}\quad 0\leq x\leq \sqrt{1-q},\ \alpha\in [-{2k+1\over 4},-{7\over 6}]\cup [{7\over 6},\infty).\cr }\end{align*} \]
Then the following approximation holds \[ g(x)=g(0)(\cos{{\mathcal B}(x)}+r(x)), \] where \[ g(0)=\left(-{1\over 4}\right)^{k/2}{{k+\alpha}\choose{k/2}} (k^2+2k\alpha+\alpha+1)^{1/4}, \]
\[ {\mathcal B}(x)=\begin{cases} \sqrt{u}\left(\arccos{\sqrt{1-q-x^2\over 1-q}}- \sqrt{q}\arccos{\sqrt{1-q-x^2\over (1-q)(1-x^2)}}\right), &q\geq 0,\\ \sqrt{u}\left(\arccos{\sqrt{1-q-x^2\over 1-q}}+ \sqrt{-q} \arccos \text{h}{\sqrt{1-q-x^2\over (1-q)(1-x^2)}}\right), &q<0.\end{cases} \] The error term \(r(x)\) is bounded as follows: \[ r(x)=\begin{cases} {2.72 x\over \sqrt{(1-x^2)u}}, &q<0;\\ {2(1-x^2)x\over (1-q)(1-q-x^2)^{3/2} \sqrt{u}}, &0\leq q<{1\over 2};\\ {(1+q)x\over 4(1-q-x^2)^{3/2}}, &{1\over 2}\leq q<1.\end{cases} \]
As a corollary, the author shows that the ultraspherical polynomials ‘live’ in the interval \((-\sqrt{1-q},\sqrt{1-q})\):
{Theorem 2.} For \(k\geq 10\), even, \(\alpha\geq 3-{k\over 2},\;|\alpha|\geq 1\), \[ \int_{-\eta}^{\eta}\,(1-x^2)( P_k^{(\alpha,\alpha)}(x))^2dx>1-{5\over 3(1-q)^{1/3}u^{1/6}} > 1-{5\over 3}\cdot \left({k+\alpha\over (k+2\alpha)k}\right)^{1/3}, \] where \[ \eta=\sqrt(1-q)\left(1-{4\cdot 2^{1/3}\over 3(1-q)^{2/3} u^{1/3}}\right). \] Here \(P_k^{(\alpha,\alpha)}(x)\) is the orthonormal form of the ultraspherical polynomial.
Reviewer: Marcel G. de Bruin (Heemstede)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
orthogonal polynomials; ultraspherical polynomials; Gegenbauer polynomials; uniform approximationCitations:
Zbl 0305.42011References:
| [1] | Anderson, G. D.; Qiu, S.-L., A monotoneity property of the gamma function, Proc. Amer. Math. Soc., 125, 11, 3355-3362 (1997) · Zbl 0881.33001 |
| [2] | Driver, K.; Duren, P., Trajectories of the zeros of hypergeometric polynomials \(F(- n, b; 2 b; z)\) for \(b < - 1 / 2\), Constr. Approx., 17, 169-179 (2001) · Zbl 0987.33001 |
| [3] | Driver, K.; Jordaan, K., Zeros of quasi-orthogonal Jacobi polynomials, SIGMA Symmetry Integrability Geom. Methods Appl., 12, 042 (2016), 13 pages · Zbl 1339.33014 |
| [4] | Driver, K.; Muldoon, M. E., Zeros of pseudo-ultraspherical polynomials, Anal. Appl., 12, 563-581 (2014) · Zbl 1301.33011 |
| [5] | Elbert, A.; Laforgia, A., On some properties of the gamma function, Proc. Amer. Math. Soc., 128, 9, 2667-2673 (2000) · Zbl 0949.33001 |
| [7] | Krasikov, I., On zeros of polynomials and allied functions satisfying second order differential equation, East J. Approx., 9, 51-65 (2003) · Zbl 1110.33002 |
| [8] | Krasikov, I., On extreme zeros of classical orthogonal polynomials, J. Comput. Appl. Math., 193, 168-182 (2006) · Zbl 1099.33008 |
| [9] | Krasikov, I., Approximations for the Bessel and Airy functions with an explicit error term, LMS J. Comput. Math., 17, 1, 209-225 (2014) · Zbl 1294.41024 |
| [10] | Krasikov, I., On the Bessel function \(J_\nu(x)\) in the transition region, LMS J. Comput. Math., 17, 1, 273-281 (2014) · Zbl 1294.41025 |
| [11] | Kuijlaars, A. B.J.; Martínez-Finkelshtein, A., Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, J. Anal. Math., 94, 1, 195-234 (2004) · Zbl 1126.33003 |
| [12] | Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23 (1975) · JFM 65.0278.03 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
