Mikhail SODIN

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Part IV. Examples
1. Trigonometric ensemble X = T m (m-dim torus)
Hn,m ⊂ L 2 (T m ), subspace of trigonometric polynomials in m variables of
degree ≤ n in each of the variables.
The repro-kernel (= covariance): the Dirichlet kernel
m sin [π(2n + 1)(xj − yj)]
Kn,m(x, y) =
(2n + 1) sin [π(xj − yj)]
✫
j=1
scaling parameter L = n (the degree).
After scaling, covariance converges together with partial derivatives of
m sin 2πuj
any order to the limiting kernel k(u − v), k(u) =
.
2πuj
Hence, the limiting spectral measure is the normalized Lebesgue measure
on the cube [−1, 1] m ⊂ R m , which meets all our assumptions.
17
j=1
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