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Number of real roots of a random trigonometric polynomial. (English) Zbl 0770.60055

Summary: We study the expected number of real roots of the random equation \[ g_ 1\cos\theta+g_ 2\cos 2\theta+\cdots+g_ n\cos n\theta=K, \] where the coefficients \(g_ j\)’s are normally distributed, but not necessarily all identical. It is shown that although this expected number is independent of the means of \(g_ j\), \(j=1,2,\dots,n\), it will depend on their variances. The previous works in this direction considered the identical distribution for the coefficients.

MSC:

60G99 Stochastic processes
42B99 Harmonic analysis in several variables
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