Derivative and approximation theorems on local fields (Q1086452)

The concept of a derivative of functions on local fields K plays a key role in approximation theory. In this note such a concept is given. The formula \(\chi_{\lambda}^{<1>}(x)=| \lambda | \chi_{\lambda}(x)\) for characters \(\chi_{\lambda}\), \(\lambda\in K\) is obtained. With some modification it is applicable to more cases; e.g., to the a-adic group \(\Omega\) a. Let \(f\in L^ r(K)\), \(1\leq r<\infty\), and consider the linear operator \[ L(f,x,\lambda)=\int_{K}f(t)| \lambda | w(\lambda (x-t))dt,\quad \lambda \in K, \] where the kernel w is generated by some \(\omega \in L^ 1(K)\), \(w=\omega\). Then, by means of the above derivative, we prove several lemmas including the Bernstein inequality and establish some inverse approximation theorems for the class \(W[L^ r,| x|^{\alpha}]\) and \(Lip_ r\alpha\). An application to the kernel \(G^{\alpha}\) for the Bessel potential introduced by M. Taibleson is also included.