ABSTRACT Nazarov Nazarov will investigate exponential polynomials and lacunary series and on weighted norm inequalities. The objectives of the proposed research are: 1) to get a complete version of the Turan Lemma, 2) to investigate the distribution of values of trigonometric polynomials with random spectra, 3) to complete the theory of weighted norm inequalities for singular integral operators with one matrix weight and 4)to start an attack on the scalar two-weight case. The problems described in this proposal are of fundamental character and have numerous connections to other fields some of which in their turn have a direct impact on science and technology. For example, fast computing and data compression based on wavelets. The idea of the wavelet compression is strikingly simple: you just take the wavelet expansion of your image and drop all small terms, memorizing only several "large" ones. The problem is that you should be sure that the large terms are really few and that the small terms together cannot add up to something huge. The formal requirement (which is more or less just the above phrase translated into mathematical language) is that the system of wavelets should form a basis in a certain function space.
