Differential and Falsified Sampling Expansions

Differential and falsified sampling expansions ∑ k ∈ Z d c k φ ( M j x + k ) , where M is a matrix dilation, are studied. In the case of differential expansions, c k = L f ( M - j · ) ( - k ) , where L is an appropriate differential operator. For a large class of functions φ , the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of φ from this class is required. In the present paper, we obtain similar results for a class of band-limited functions φ with the discontinuous Fourier transform. In the case of falsified expansions, c k is the mathematical expectation of random integral average of a signal f near the point M - j k . To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in L p -norm are given in terms of the Fourier transform of f .