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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Real zeros of a random sum of orthogonal polynomials


Author: Minaketan Das
Journal: Proc. Amer. Math. Soc. 27 (1971), 147-153
MSC: Primary 60.20
MathSciNet review: 0268933
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Abstract: Let $ {c_0},{c_1},{c_2}, \cdots $ be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let $ P_k^ \ast (x)(k = 0,1,2, \cdots )$ be the normalised Legendre polynomials orthogonal with respect to the interval $ ( - 1,1)$. It is proved that the average number of the zeros of $ {c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x)$ in the same interval is asymptotically equal to $ {(3)^{ - 1/2}}n$ when $ n$ is large.


References [Enhancements On Off] (What's this?)

  • [1] Minaketan Das, The average number of real zeros of a random trigonometric polynomial., Proc. Cambridge Philos. Soc. 64 (1968), 721–729. MR 0233398
  • [2] G. Sansone, Orthogonal functions, Revised English ed. Translated from the Italian by A. H. Diamond; with a foreword by E. Hille. Pure and Applied Mathematics, Vol. IX, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1959. MR 0103368
  • [3] Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0268933-9
Keywords: Normally distributed random variables, mathematical expectation, variance, orthogonal polynomials
Article copyright: © Copyright 1971 American Mathematical Society