In this paper we treat cohomology groups H1(G, C2q-1, M) of meromorphic Eichler integrals for a finitely generated Fuchsian group G of the first kind. According to L. V. Ahlfors [2] and L. Bers [4], H1(G, C2q-1, M) is the space of periods of meromorphic Eichler integrals for G. In the previous paper [8], we had period relations and inequalities of holomorphic Eichler integrals for a certain Kleinian groups.

One of the interesting problems in the potential theory for a function-kernel is to investigate the relation between the domination principle and the balayage principle.

If X is a complex space, the Kobayashi pseudo-distance dx is an intrinsic pseudometric on X defined as follows. If p and q are points of X, a chain α from p to q consists of intermediate points p0, …, pr with p0 = p and pr = q together with maps fi of the unit disc D = {z ∈ Cl‖z| < 1} into X and points ai and bi in D such that fi(ai) = Pi−1 and fi(bi) = pi for i = 1, …, r.

This paper deals with cluster sets. While cluster sets can be considered in a more abstract setting, we shall limit ourselves to the study of functions f defined in the open unit disc D of the complex plane and taking their values on the Riemann sphere C̅. For p a point of the unit circle C, we denote by C(f,p) the cluster set of f at p, i.e., the set of all values w ∈ C̅ for which there is a sequence {zn}, zn ∈ D, such that zn → p and f(zn) → w. The point p is called a point of determination for f if C(f, p) is a singleton. In Section 1 we characterize the set of points of determination of a function f defined in D.

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

We discuss some peculiar features of the diffusion process whose characterization is given below. Let D be a bounded domain in the d-dimensional Euclidean space Ed with a smooth boundary ∂D. The domain D contains open balls (i = 1, · · ·, n) which are mutually disjoint. Our process is a diffusion process on the state space D ∪ ∂D which is locally equivalent to the Brownian motion except on the spheres ∂ and the boundary ∂D. By a diffusion process we mean a continuous strong Markov process. As to the terminology about Markov processes we refer to [2].

A basic problem in the representation theory of a finite group G is the determination of all indecomposable G-modules. Thus, for G = C(n) = a cyclic group of order n over an arbitrary field, the indecomposable representations, finite in number, are known from the theory of a single linear transformation. In 1954 Higman [9] showed that, in sharp contrast to the classical case of characteristic zero, an arbitrary finite group G has indecomposables of arbitrarily high dimension over any field of prime characteristic p iff the p-Sylow subgroup of G is non-cyclic (cf. unbounded representation type [3, p. 431]). Examples published by Heller and Reiner [8] in 1961 indicated that this phenomenon is even more extensive; reinterpreting a result of Dieudonné [4] as classifying the indecomposable modules for a square zero algebra on two generators, they showed that G = C(p) × C(p) (and therefore many other groups) has infinitely many non-isomorphic indecomposables in every even dimension over an infinite field of characteristic p (cf. strong unbounded representation type).

Dans toute la suite Rn désigne l’espace euclidien à dimension n (≧1). On désigne par Δ l’opérateur de Laplace sur Rn. Dans la théorie du potentiel, un noyau de convolution sur Rn signifie une mesure de Radon positive dans Rn. Rappelons qu’un noyau de convolution de Dirichlet N sur Rn est un noyau de convolution sur Rn tel que 1/N̂ soit égal à une fonction définie-négative dans Rn à valeurs réelles, où N̂ désigne la transformée de Fourier de N (cf. [1]). Pour un nombre p > 0, Gp désigne le noyau de convolution de Dirichlet sur Rn vérifiant (Δ — p)Gp = — ε (au sens des distributions), où ε est la mesure de Dirac à l’origine. Si n ≧ 3, on note G = G0 le noyau newtonien avec ΔG = −ε.

Dans toute la suite Rn désignera l’espace euclidien à dimension n (≧ 1). Pour un point x = (x1, x2, · · ·, xn) de Rn, on notera |x| = .La coordonnée sphérique dans Rn s’écrira (r, σ). R+ désignera l’ensemble {t ∈ R1; t ≧ 0}.

A complex Finsler structure F on a complex manifold M is a function on the tangent bundle T(M) with the following properties. (We denote a point of T(M) symbolically by (z, ζ), where z represents the base coordinate and ζ the fibre coordinate.)

Dans toute la suite X désignera un groupe abélien localement compact, séparé et dénombrable à l’infini; ξ sera sa mesure de Haar. Rappelons qu’un noyau de convolution N sur X signifie une mesure de Radon positive dans X et que, pour une mesure de Radon réelle μ dans X, N*μ s’appelle le N-potentiel de μ dès que cette convolution est définie au sens des mesures. Dans cette note, pour simplifier la discussion, nous supposerons qu’il n’existe aucun sous-groupe compact de X excepté {0}.

This work grew out of an attempt to prove the false result that an n-ple Markov process in the sense of Hida (1960) or Lévy (1956a) has multiplicity one. Instead we proved the representation theorem (Theorem III. 1.) that a centered Gaussian process x(t) is n-ple Markov iff it can be written in the form

(I.1)

where is a Gaussian martingale with

(I.2) sp {x(s): s ≤ t} ≡ sp {ai(s): s ≤ t and 1 ≤ i ≤ n}

and A(t) and {ei(t)} satisfy some non-degeneracy condition. We also show (Corollary IV. 13.) that for any Gaussian martingale A(t) with simple left innovation spectrum, continuous ei(t) may be found so that the process x(t) given in (I.1) will satisfy (I.2).