We take three contemporary problems arising from random matrices and show how approximation theory is crucial in solving them. However this is not merely an "off-the-shelf" application of known results from within approximation theory, but require extensions beyond the realm of its current boundaries.
The first example comes from wireless communication systems with random noise and requires the calculation of low order moments of say, the von Neumann entropy, amongst other relevant statistics. Classical Laguerre orthogonal polynomials appear here but novel integrals of these are required.
The second example comes from the analytic number theory of the Riemann zeta function and the modelling of it by the characteristic polynomial of a random unitary matrix. Here we examine the zeros of the derivative of the characteristic polynomial, and a novel system of bi-orthogonal polynomials and associated functions is identified here.
The third example, in contrast to the second, is the evaluation of moment determinants generalising Toeplitz determinants and which arise from another number theoretic source - the matrix integrals over the classical groups other than the unitary group.