Linear Functional Operations

Let us consider the functional operation \(I_n(f) = \int ^b_a f(\alpha )\varphi (\alpha ,n)\,{\text {d}}\alpha \) , which associates a number with each function \(f(\alpha )\) belonging to a certain family \(\mathscr {F}\) of functions. We aim to investigate the conditions on the kernel \(\varphi (\alpha ,n)\) under which \(I_n\) converges to zero as \(n \rightarrow \infty \) , for any f.