We prove two-sided inequalities between the integral moduli of smoothness of a function on R d /T d and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.

Abstract. We study the two-parameter family of unitary operatorswhich are called (k, a)-generalized Fourier transforms and defined by the adeformed Dunkl harmonic oscillator ∆ k,a = |x| 2−a ∆ k − |x| a , a > 0, where ∆ k is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of F k,a to radial functions is given by the a-deformed Hankel transform H λ,a .We obtain necessary and sufficient conditions for the weighted (L p , L q ) Pitt inequalities to hold for the a-deformed Hankel transform. Moreover, we prove two-sided Boas-Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for F k,a transform in L 2 (R d ) with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for F k,a .

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Weighted L p (R n ) → L q (R n ) Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function. Extensions to general weights are also given.

We consider an extremum problem posed by Turan. The aim of this problem is to find a maximum mean value of 1-periodic continuous even function such that sum of Fourier coefficient modules for this function is equal to 1 and support of this function lies in [−h, h], 0 < h ≤ 1/2. We show that this extremum problem for rational h = p/q is equivalent two finitedimensional linear programming problems. Here there are exact results for rational h = 2/q, h = p/(2p + 1), h = 3/q, and asymptotic equalities.