Abstract:
This paper is concerned with systems of functions on the unit interval which are generated by dyadic dilations and integer translations of a given function. Similar systems have a wide range of applications in the theory of wavelets, in nonlinear, and in particular, in greedy approximations, in the representation of functions by series, in problems in numerical analysis, and so on. Conditions, and in some particular cases, criteria for the generating function are given for the system to be Besselian, to form a Riesz basis or to be an orthonormal system, and separately, to be complete. For this purpose, the concept of the dual function of the generating function of a system is introduced and studied. Some of the conditions given below are easy to verify in practice, as is demonstrated by examples.
Bibliography: 25 titles.
Keywords:Riesz basis, Haar system, affine system of functions, system of dilations and translations.
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