Related papers: A General Integral
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make…
We define the notion of distribution on an infinite dimensional space motivated by the notion of Feynman path integral and by construction of probability measures for generalized random fields. This notion of distribution turns out to be…
A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with…
In this paper, we discuss a similar functional to that of a standard integral. The main difference is in its definition: instead of taking a sum, we are taking a product. It turns out this new "star-integral" may be written in terms of the…
For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…
For general thinning procedures, its inverse operation, the condensing, is studied and a link to integration-by-parts formulas is established. This extends the recent results on that link for independent thinnings of point processes to…
Integration at a point is a new kind of integration derived from integration over an interval in infinitesimal and infinity domains which are spaces larger than the reals. Consider a continuous monotonic divergent function that is…
In this paper we define a type of generalized Riemann-Lebesgue (decomposition) integral for non-negative real functions with respect to two non-additive set functions. For this integral we present some classical properties.
The core of this article is a general theorem with a large number of specializations. Given a manifold $N$ and a finite number of one-parameter groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots, X_{(d)} $, we…
For non-decreasing real functions $f$ and $g$, we consider the functional $ T(f,g ; I,J)=\int_{I} f(x)\di g(x) + \int_J g(x)\di f(x)$, where $I$ and $J$ are intervals with $J\subseteq I$. In particular case with $I=[a,t]$, $J=[a,s]$, $s\leq…
We investigate Bochner integrabilities of generalized Wiener functionals. We further formulate an It\^o formula for a diffusion in a distributional setting, and apply to investigate differentiability-index $s$ and integrability-index $p…
By discussing several examples, the theory of generalized functional models is shown to be very natural for modeling some situations of reasoning under uncertainty. A generalized functional model is a pair (f, P) where f is a function…
In general, while obtaining the probability density function of sums and products of shifted random variables, ordinary analytical methods such as Fourier and Mellin transforms tend to provide integrals which cannot be expressed in terms of…
A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of…
In the probability theory \emph{selfdecomposable, or class $L_0$ distributions} play an important role as they are limiting distributions of normalized partial sums of sequences of independent, not necessarily identically distributed,…
In this paper we study a broad class of distribution functions which is defined by means of reflected generalized beta distribution. This class includes that of Beta-generated distribution as a special case. In particular, we use this class…
Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as…
We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…