THE FOURIER-TRANSFORM OF ORDER-STATISTICS WITH APPLICATIONS TO LORENTZ SPACES

We present a formula for the Fourier transforms of order statistics in R(n) showing that all these Fourier transforms are equal up to a cons tant multiple outside the coordinate planes in R(n). For a(1) greater than or equal to ... greater than or equal to a(n) greater than or equ al to 0 and q > 0, denote by l(w,q)(n) the n-dimensional Lorentz space with the norm \\(x(1),...,x(n))\\ = (a(1)(x(1))(q) +...+a(n)(x(n)*)( q))(1/q),(a(1)(x(1))(q)+... a(n)(x(n)*)(q))(1/q), where (x(1)*,..x(n) ) is the non-increasing permutation of the numbers \x(1)\,...,\x(n)\. We use the above mentioned formula and the Fourier transform criterio n of isometric embeddability of Banach spaces into L(q) [10] to prove that, for n greater than or equal to 3 and q less than or equal to 1, the space l(w,q)(n) is isometric to a subspace of L(q) if and only if the numbers a(1),...,a(n) form an arithmetic progression. For q > 1, a ll the numbers a(z) must be equal so that lw,qn = l(q)(n). Consequentl y, the Lorentz function space L(w,q)(0, 1) is isometric to a subspace of L(q) is and only if either 0 < q < infinity and the weight w is a c onstant function (so that L(w,q) = L(q)), or q less than or equal to 1 and w(t) is a decreasing linear function. Finally, we relate our resu lts to the theory of positive definite functions.