In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set Ω is periodic, with the period an integral multiple of the measure of Ω. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.

IntroductionLet D denote the unit disc in the complex plane and T the unit circle with normalized HAAR measure do. For 0 < p < co, H P denotes the usual HARDY space of analytic functions in D. Let P be the orthogonal projection from L2( T) onto H 2 . If b is a holomorphic function in D, we define the HANKEL operator with symbol b as a formal FOURIER series by(We are defining Hb as an anti-linear operator only as a matter of convenience). NEHARI (see [S]) proved that Hb extends to a bounded operator on H 2 iff b E BMO. This result is a consequence of duality and RIESZ factorization and also holds for H P if 1 < p < co. JANSON, PEETRE and SEMMES [7] proved that Hb extends to a continuous operator on H P , 0 < p < 1 iff b belongs to the LIPSCHITZ space A'/,-'; and when p = 1, iff b E LMO (Logarithmic Mean Oscillation; see [lo]). This was independently proven by TOLOKONNIKOV [12], who also showed, using methods of functional analysis, that, if b E A",-', then Hb extends to a continuous operator from H P to Hieak and that Hb is bounded from H P (0 < p < 1) to H ' iff b belongs to a generalized LIP~CHITZ Space A, with e(t) = t",-'/log (llt) (for the definition of A, see [9] or [5]). Another proof of these results, which extends to some of the HARDY spaces H, defined by JANSON in [5], was given in [l]. Our aim in this paper is to give a new proof of these results which generalizes easily to all JANSON'S H, spaces and relies on balayage of CARLESON measures. More precisely, we first prove (for the definitions, see section 2) the following theorem, where e denotes a growth function of upper type less than 1. Theorem 1. Let H , be a generalized HARDY space with dual space BMO,. Then Hb extends to a continuous operator from H , into Hkeak iff b E BMO,. Moreover, i f b E BMO,, there exists a continuous operator %from H, into L' such that Hb = P Q T.The operator & is obtained as a balayage with respect to the POISSON kernel for the disc of a generalized CARLESON measure, using a representation theorem for f E BMO, due to SMITH [lo] and Theorem 1 follows easily from the generalized CARLESON property. More generally, we find necessary and sufficient conditions on the symbol b so that the HANKEL operator Hb maps one H,, space to another HV2. This gives Theorem 2 (section 3). The key

We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and symmetric MSF wavelets of L 2 (R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L 2 (R) and H 2 (R). We also enumerate all symmetric wavelets of L 2 (R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H 2 (R). Finally, we construct families of MSF wavelets of L 2 (R) whose Fourier transform does not vanish in any neighbourhood of the origin.Notation. In this article, measure will always mean Lebesgue measure. All subsets of the real line we consider, are assumed to be Lebesgue measurable. With |E| we denote the Lebesgue measure of a measurable set E in R. We say that a relation between measurable sets holds almost everywhere (a.e.) if their characteristic functions are equal a.e. Thus, A = B a.e. means that χ A = χ B a.e., and n A n = A a.e. means that n χ An = χ A a.e., where denotes the disjoint union.