Let C([`R]) be the set of all real-valued functions continuous on the extended reals [`R] = [-¥,¥]; thus F Î C([`R]) if and only if F Î C(-¥,¥) and limu®±¥F(u) are finite. Let U = {{uin}i = 1n}n = 1¥ and V = {{vin}i = 1n}n = 1¥, where u1n ³ u2n ³ ¼ ³ unn and v1n ³ v2n ³ ¼ ³ vnn, n ³ n0. We say that U and V are absolutely equally distributed if (A) åi = 1n|F(uin)-F(vin)| = o(n) for every F in C([`R]). If P is a set of pairs (U,V), we say that U and V are uniformly absolutely equally distributed for (U,V) in P if, for each F in C([`R]), (A) holds uniformly for (U,V) in P.
We give simple necessary and sufficient conditions for
U and V to be absolutely equally distributed, or
uniformly absolutely equally distributed for
(U,V) in P. We use these conditions to extend and
strengthen some known results on Weyl-Tyrtyshnikov equal
distribution of the singular values of sequences
{An}n = n0¥ and {Bn}n = n0¥, where
An, Bn Î Chn×kn with min(hn,kn) = n,
and of the eigenvalues of Hermitian {An}n = n0¥ and
{Bn}n = n0¥. We apply these results to some
structured matrices.
If (X,A,m) is a measure space, S Î A,
0 < m(S) < ¥, and g is measurable and real-valued
on S, we say that
U is
distributed like the values of g if
n-1åi = 1nF(uin) = (m(S))-1òS F°g dm
for every F in C([`R]). We show that if U and
V are both distributed like the values of g, then
U and V are absolutely equally distributed. A
special case of this result provides additional insight into the
conclusions of the Szegö and Avram-Parter distribution theorems
if the generating functions are Riemann integrable.