Average Number of Real Roots of Random Harmonic Equations

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Let {gk}nk=0be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let ψk(t) (k = 0, 1, 2,…) be the normalized Jacobi polynomials orthogonal with respect to the interval [−1, 1]. Then it is proved that the average number of real roots of the random equations, ∑nk=0gk Ψk(t)=C, where C is a constant, is asymptotically equal to n\√3in the same interval when n is large and even for C → ∞ as long as C=O (n2).

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