Fluctuations of the Number of Critical Points of Random Trigonometric Polynomials*

Statistics of Linear Families of Smooth Functions on Knots

Given a knot K in an Euclidean space R, and a finite dimensional subspace V ⊂ C∞(K), we express the expected number of critical points of a random function in V in terms of an integral-geometric invariant of K and V . When V consists of the restrictions to K of homogeneous polynomials of degree ` on R, this invariant takes the form of total curvature of a certain immersion of K. In particular, ...

Random Sampling of Multivariate Trigonometric Polynomials

We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for the associated Vandermonde-type and Toeplitz-like matrices. The results provide a solid theoretical...

On the average number of sharp crossings of certain Gaussian random polynomials

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

Probability against Condition Number and Sampling of Multivariate Trigonometric Random Polynomials

PROBABILITY AGAINST CONDITION NUMBER AND SAMPLING OF MULTIVARIATE TRIGONOMETRIC RANDOM POLYNOMIALS ALBRECHT BÖTTCHER AND DANIEL POTTS Abstract. The difficult factor in the condition number of a large linear system is the spectral norm of . To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To be more precise, we randomly take from a ball with the uniform ...