Average number of real roots of random harmonic equations (Q1272491)

Let \(g_k(\omega)\), \(\omega\in\Omega\), \(k=0,1,\dots, n\), be a sequence of independent, normally distributed random variables defined on a probability space \(( \Omega,\mathcal A,\text{Pr})\) with mean zero and variance one and let \(b_0,b_1, \dots,b_n\) be positive constants. Denote \(f_n(t,\omega)=\sum^n_{k=0}g_k(\omega)b_k\varphi_k(t)\), where \(\varphi_k(t)\) is any function of \(t\). Let \(N_C(a,b)\) be the number of real roots of the equation \(f_n(t,\omega)=C\) in the interval \((a,b)\), with \(EN_C(a,b)\) the expected value of this number. There are many asymptotic estimates for \(EN_C\) for the cases of algebraic equations, \(\varphi_k(t)\equiv t^k\), or trigonometric equations, \(\varphi_k(t)\equiv\cos kt\). Also, \textit{M. Das} [Proc. Am. Math. Soc. 27, 147-153 (1971; Zbl 0212.49401)] considered the case when \(b_k=1\), \(k=0,1,2,\dots,n\), and \(\varphi_k(t)\) are normalized Legendre polynomials orthogonal with respect to the interval \((-1,1)\). He proved that for large \(n\), \(EN_0(-1,1)\) is asymptotically \(n/\sqrt 3\). Further, \textit{M. Das} and \textit{S. S. Bhatt} [Indian J. Pure Appl. Math. 13, 411-420 (1982; Zbl 0481.60067)] considered the case \(C=0\) when \(\varphi_k(t)\) are sequences of classical orthogonal polynomials. In particular, they considered the cases of Jacobi, Hermite and Laguerre polynomials. They showed that the above asymptotic formula \(n/\sqrt 3\) persists for all the above cases. For the case of nonzero \(C\) only the case of Legendre polynomials was studied, under the assumption that \(b_k=1\), \(k=0,1,2,\dots,n \). It was shown by the reviewer [Analysis 16, No. 3, 245-253 (1996; Zbl 0864.60038)] that the asymptotic formula \(EN_C(-1,1)\sim n/\sqrt 3\) remains valid as long as \(C^2/n\) tends to zero as \(n\) tends to infinity. There are also some further, more recently results for the Legendre polynomials by the reviewer [J. Appl. Math. Stochastic Anal. 10, No. 3, 257-264 (1997) and ``Topics in random polynomials'' (Harlow, 1998)]. The present paper considers the case when \(\{\varphi_k(t)\}^n_{ k=0}\) is a sequence of Jacobi polynomials, \(C\) is a nonzero constant, and an attempt is made to obtain an asymptotic formula for \(EN_C(-1,1)\). To this end the interval \((-1,1)\) is divided into two groups of subintervals, and different methods are used for each group. For intervals in a small neighbourhood of 1 and \(-1\), an approach which is based on a generalization of Jensen's theorem is successfully used, and it is shown that in this subinterval the expected number of roots of \(f_n(t,\omega)=C\) is negligible. For the remaining subintervals a formula based on the earlier work of \textit{M. Kac} [Bull. Am. Math. Soc. 49, 314-320 (1943; Zbl 0060.28602)] is used. However, the fact that this formula is obtained for the case of \(C=0\) is overlooked. The generalisation of this formula for the case of nonzero \(C\) can be obtained and, as has been shown, for example by the reviewer [Ann. Probab. 14, 702-709 (1986; Zbl 0609.60074)], would be involved with an exponential function. As this interval provides the main contribution to the expected number of real roots, the additional exponential term is significant and may reduce the expected number of real roots. However, this would be unusual, and, with comparison to the other cases, it is more likely that under mild conditions on \(C\) the author's asymptotic formula would remain valid. However, in the reviewer's view this remains an open and interesting problem worth a definite and thorough investigation.