The paper presents interesting fast algorithms generalizing the fast Fourier transform to the case of noninteger frequencies and nonequidistant nodes on the interval \([-\pi,\pi]\). The described algorithms are approximate ones, i.e. the calculations are performed with a fixed relative accuracy \(\varepsilon \geq 0\). They are based on a combination of results of the approximation by trigonometric polynomials and fast Fourier transform techniques. The algorithms require \(O(N \log N+N \log(1/ \varepsilon))\) arithmetic operations. Several numerical examples illustrate the efficiency of the approach.