A formula for the Fourier transform of certain odd differentiable functions.
The author proves a Fourier transform formula for a certain class of differentiable functions \(K(x)=B(x)/x\) defined on the real line, relating the global behaviour of the Fourier transform of \(K\) to the function \(B\). Let \(\widehat{f}(z)=\int{f(x)e^{-izx} \,dx}\), \(z\) real, denote the Fourier transform of \(f\), and let \(A\approx B\) mean the existence of positive constants \(C_1\) and \(C_2\) such that \(C_1B\leq A\leq C_2B\). The following theorem is proved. Theorem: Let \(K\) be positive for \(x>0\) and odd. Suppose there exist constants \(c_1\) and \(c_2\), \(0< c_1\leq c_2<2\), such that \[ c_1\frac{K(x)}{x}\leq -K'(x)\leq c_2\frac{K(x)}{x} \] for \(x> 0\). Then, \(\widehat{K}\) exists as a tempered distribution and can be identified with a function (which we also call \(\widehat{K}\)) such that for \(z>0\), \[ i\widehat{K}(z)\approx B(\pi/z). \]
scientific article; zbMATH DE number 1992550