Bent Fuglede

Commuting self-adjoint partial differential operators and a group theoretic problem

In Rn let Ω denote a Nikodym region (= a connected open set on which every distribution of finite Dirichlet integral is itself in L2(Ω)). The existence of n commuting self-adjoint operators H1,…, Hn in L2(Ω) such that each Hj is a restriction of −i ββxj (acting in the distribution sense) is shown to be equivalent to the existence of a set Λ ⊂Rn such that the restrictions to Ω of the functions exp i ∑ λjxj form a total orthogonal family in L2(Ω). If it is required, in addition, that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω), then this is shown to correspond to the requirement that Λ can be chosen as a subgroup of the additive group Rn. The measurable sets Ω ⊂Rn (of finite Lebesgue measure) for which there exists a subgroup Λ ⊂Rn as stated are precisely those measurable sets which (after a correction by a null set) form a system of representatives for the quotient of Rn by some subgroup Γ (essentially the dual of Λ).

Jensen-Shannon divergence and Hilbert space embedding

This paper describes the Jensen-Shannon divergence (JSD) and Hilbert space embedding. With natural definitions making these considerations precise, one finds that the general Jensen-Shannon divergence related to the mixture is the minimum redundancy, which can be achieved by the observer. The set of distributions with the metric /spl radic/JSD can even be embedded isometrically into Hilbert space and the embedding can be identified.

Orthogonal Exponentials on the Ball

We show that there does not exist an infinite sequence of vectors An in R d, d > 1, such that the corresponding exponentials e i(;~,~,x), x E R d, when considered on the unit ball B in Rd, are pairwise orthogonal in LZ(B) (B being endowed with Lebesgue measure). The weaker result that L2(B) does not have an infinite orthogonal base of exponentials has recently been established by A. Iosevich, N. Katz, and S. Pedersen in (2). For d = 2 the present result was announced in the authors 1974 paper (1).

Localization in Fine Potential Theory and Uniform Approximation by Subharmonic Functions

It is shown how the cone l(U) of superharmonic functions ⩾0 on an open set U in Rn, n ⩾ 3, can be recovered from the cone l of superharmonic functions ⩾0 on the whole of Rn by a process involving the operator of localization associated with U. Actually we treat the more general case where U is open in the Cartan-Brelot fine topology on Rn. As an application we obtain a new proof of a theorem of J. Bliedtner and W. Hansen on uniform approximation by continuous subharmonic functions in open sets containing a given compact set K in Rn.

Some properties of the riesz charge associated with a δ-subharmonic function

The first property is a refinement of earlier results of Ch. de la Vallée Poussin, M. Brelot, and A. F. Grishin. Let w=u−v with u, v superharmonic on a suitable harmonic space Ω (for example an open subset of Rn), and let μ[w]=μ[u]−μ[v] denote the associated Riesz charge. If w≥0, and if E denotes the set of those points of Ω at which the lim inf of w in thefine topology is 0, then the restriction of μ[w] to E is ≤0. Another property states that, if e denotes a polar subset of Ω such that the fine lim inf of |w| at each point of e is finite, then the restriction of μ[w] to e is 0.

The Dirichlet Laplacian on Finely Open Sets

For any decreasing sequence of bounded finely open sets Di ⊂ RN it is shown that, for every n, the nth eigenvalue λn ( Di) of the Dirichlet laplacian A ( Di ) on Di converges to λn ( D ) (the nth eigenvalue of A ( D ) ), where D denotes the fine interior of ∩ Di. Likewise, A ( Di )-1 → A ( D )-1 in operator norm. Similar results are obtained for increasing or just order convergent sequences ( Di ). Furthermore, A ( D )-1 is identified with the integral operator on L2 ( D ) whose kernel is Greens function for D.

No Two Quotients of Normalized Binomial Mid-coefficients Are Equal

The motivation behind the study of Eq. (1) is that, as shown in [2], the numbers pLyp, (q, r = 0, 1, 2, . . . ; q d r) are precisely the eigenvalues of Liouville’s integral operator for the case of a planar circular disc of radius 1 lying in R3. The question thus arose whether it can occur that pPpLs = pLypLI for distinct pairs (p, s) and (q, r) (with p <s, q B r, say). The result of the present note means that this degeneracy cannot occur. As to Liouville’s operator see Berg and Liitzen [l] and references therein to Liouville’s published and unpublished work on this subject. For the binomial mid-coefficients (T) themselves it has been shown by Erdijs [3] and Moser [4] that no single such coefficient can be the 345 0022-3 14X/90 f3.00

On the riesz charge of the lower envelope of δ-subharmonic functions

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more δ-subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.

Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ⊂ X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → [0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1AGω is sufficient for the possibility of the sweeping in question (1A denoting the indicator function of A and Gω the G-potential of ω).

Domains of Existence for Finely Holomorphic Functions

We show that fine domains in ℂ with the property that they are Euclidean Fs and Gd, are in fact fine domains of existence for finely holomorphic functions. Moreover regular fine domains are also fine domains of existence. Next we show that fine domains such as ℂ \ ℚ or ℂ \ (ℚ × iℚ), more specifically fine domains V with the properties that their complement contains a non-empty polar set E that is of the first Baire category in its Euclidean closure K and that (K \ E) ⊂ V, are not fine domains of existence.