[1] Observations of atomic emissions from the Io plasma torus and previously published laboratory work suggest that proton collisions with SO 2 may be a source of some of the spectral lines. We present an analysis of Balmer series lines seen in the spectra of collisions of protons with SO 2 over the range of 50 to 250 keV. Absolute emission cross sections for these lines are measured to be on the order of 10 À19 cm 2 and are in rough agreement with a parametric model for electron capture which supports charge-transfer ionization of the SO 2 target.
In this note we investigate the existence of frames of exponentials for L 2 (Ω) in the setting of LCA groups. Our main result shows that sub-multitiling properties of Ω ⊂ G with respect to a uniform lattice Γ of G guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of Γ. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.
Abstract. A classical theorem attributed to Naimark states that, given a Parseval frame B in a Hilbert space H, one can embed H in a larger Hilbert space K so that the image of B is the projection of an orthonormal basis for K. In the present work, we revisit the notion of Parseval frame MRA wavelets from [11] and [12] and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provide a different point of view on Parseval frame MRA wavelets than that of [11] and [12].1. Notation and Preliminary Remarks 1.1. Notation. The Fourier transform in this paper will be denoted byWe will identify subsets of the torus, T = R/Z, with subsets of R which are invariant under translations by integers -in particular, functions on T can be considered to be 1-periodic function on R and the Lebesgue measure we associate to T has total mass 1. We will frequently use the operatorsIn particular, D j and T k are unitary maps corresponding to the dyadic dilations and integer translations. We will let ψ jk = D j T k ψ = 2 j/2 ψ(2 j ·−k). Definition 1.1. We will use the notation m • g to denote the function which satisfies m • g = m · g, whenever this function is well-defined.Remark. The above definition makes sense, for instance, if g ∈ L 2 (R) and m ∈ L ∞ (R): then m · g is an L 2 (R) function, and so m • g is well-defined since the Fourier transform is an isometry on L 2 (R). Though this notation is perhaps new, the above operation is actually quite common in a variety of areas of analysis. It comes up repeatedly in the study of shift-invariant spaces, e.g. in [7]. We should take a moment to reference a few of the authors who have made significant contributions to the study of shift invariant spaces upon which much of the present work rests: Helson, [6], de Boor, DeVore, and Ron, [4] and [5], Bownik, [1], as well as many of the references contained in those papers. The operation m • g operation also comes up in the study of singular integrals -for example, the Hilbert transform corresponds to such a map when, say, g ∈ L 2 (R) and m is (a multiple of) the function which is 1 for positive reals and −1 for negative reals.In everything we discuss below -unless otherwise noted -equalities between functions are taken to be equalities in the almost everywhere sense; equalities between sets are taken to be equalities up to sets of measure zero. All functions we will discuss below are measurable.1.2. Dyadic Parseval Frame MRA Wavelets. The term wavelet can mean a variety of things in different contexts. For our purposes, a wavelet will be a function ψ ∈ L 2 (R) so that {ψ jk : j, k ∈ Z} linearly generates L 2 (R) in some way -these are more accurately described as dyadic wavelets since the dilations are dyadic dilations. Classically, wavelets were always taken to be orthonormal wavelets; that is, functions ψ so that {ψ jk : j, k ∈ Z} corresponded to orthonormal bases of L 2 (R) (e.g. the Haar wavelet, Shannon wavelet, Daubechies wavelet...
Abstract. We show partial regularity of bounded positive solutions of some semilinear elliptic equations ∆u = f (u) in domains of R 2 . As a consequence, there exists a large variety of nonnegative singular solutions to these equations. These equations have previously been studied from the point of view of free boundary problems, where solutions additionally are stable for a variational problem, which we do not assume.
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