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.

In this paper we prove the bilinear analogue of de Leeuw's result for periodic bilinear multipliers and some Jodeit type extension results for bilinear multipliers.

Abstract. In [2] it was shown that "Tiling implies Spectral" holds for a union of three intervals and the reverse implication was studied under certain restrictive hypotheses on the associated spectrum. In this paper, we reinvestigate the "Spectral implies Tiling" part of Fuglede's conjecture for the three interval case. We first show that the "Spectral implies Tiling" for two intervals follows from the simple fact that two distinct circles have at most two points of intersections. We then attempt this for the case of three intervals and except for one situation are able to prove "Spectral implies Tiling". Finally, for the exceptional case, we show a connection to a problem of generalized Vandermonde varieties.

Let Ω ⊂ R be a compact set with measure 1. If there exists a subset Λ ⊂ R such that the set of exponential functions E Λ := {e λ (x) = e 2πiλx | Ω : λ ∈ Λ} is an orthonormal basis for L 2 (Ω), then Λ is called a spectrum for the set Ω. A set Ω is said to tile R if there exists a set T such that Ω + T = R. A conjecture of Fuglede suggests that Spectra and Tiling sets are related. Lagarias and Wang [14] proved that Tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in[3], [8]. In this paper, we give some partial results to support the rationality of the spectrum.