The authors consider an integral operator \(A_ t\) with difference kernel of the form \(t^ d{\mathcal A}(t(x-y))\), where \({\mathcal A}:R^ d\to C\) is a functional in the Schwartz class \({\mathcal S}(R^ d)\). The asymptotic behavior of the trace tr\(f(A_ t)\) of a function of \(A_ t\) is given. The case \(d=1\), which is degenerate in certain respects, is written out separately. The obtained results admit a natural generalization to an operator \(\tilde A_ t\) with a kernel of the form \(t^ d{\mathcal A}(t(x- y),x,y)\). The asymptotic behavior of tr\(f(\tilde A_ t)\) was recently got by H. Widom but the technique, which the authors used is fairly elementary. This paper contains simple formulas for the resolvent.