An integral identity for the Rademacher functions (Q1842063)

Denote by \(r_j(t)\) the well-known Rademacher functions with the kernel \[ K_n(s, t):= \sum^n_{j= 1} r_j(s) r_j(t),\quad 0\leq s, t\leq 1\quad\text{and}\quad n= 1, 2,\dots\;. \] The author proves the identity \[ \int^1_0 r_{j_1}(s)\cdots r_{j_m}(s)\;F(K_n(s, t)) ds= C^{m, n}_F r_{j_1}(t)\cdots r_{j_m}(t), \] where \(0\leq m\leq n\), \(1\leq j_1<\cdots< j_m\leq n\), \(F(t)\) is an arbitrary function defined on the set of integers \(\{- n,\dots, n- 1, n\}\); furthermore, he provides an explicit expression for the constant \(C^{m, n}_F\). Then he obtains a limiting version of the above identity when \[ F(t):= n^{m/2} G(t/\sqrt n)\quad\text{and} \quad n\to \infty, \] where \(G\) is a function defined on the real line and satisfies a certain integrability condition.