Search results for "Measurable function"
showing 9 items of 19 documents
Variational Henstock integrability of Banach space valued functions
2016
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…
Kurzweil--Henstock and Kurzweil--Henstock--Pettis integrability of strongly measurable functions
2006
We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by $\sum _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.
Quantitative approximation of certain stochastic integrals
2002
We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.
Lower Semi-frames, Frames, and Metric Operators
2020
AbstractThis paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.
Stability of switched systems: The single input case
2001
We study the stability of the origin for the dynamical system x(t) = u(t)Ax(t) + (1 − u(t))Bx(t), where A and B are two 2×2 real matrices with eigenvalues having strictly negative real part, x ∊ R2 and u(.) : [0, ∞[→ [0,1] is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). This bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation.
Monotonicity and total boundednessin spaces of measurable functions
2017
Abstract We define and study the moduli d(x, 𝓐, D) and i(x, 𝓐,D) related to monotonicity of a given function x of the space L 0(Ω) of real-valued “measurable” functions defined on a linearly ordered set Ω. We extend the definitions to subsets X of L 0(Ω), and we use the obtained quantities, d(X) and i(X), to estimate the Hausdorff measure of noncompactness γ(X) of X. Compactness criteria, in special cases, are obtained.
PIP-Space Valued Reproducing Pairs of Measurable Functions
2019
We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ), where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.
On an approximation problem for stochastic integrals where random time nets do not help
2006
Abstract Given a geometric Brownian motion S = ( S t ) t ∈ [ 0 , T ] and a Borel measurable function g : ( 0 , ∞ ) → R such that g ( S T ) ∈ L 2 , we approximate g ( S T ) - E g ( S T ) by ∑ i = 1 n v i - 1 ( S τ i - S τ i - 1 ) where 0 = τ 0 ⩽ ⋯ ⩽ τ n = T is an increasing sequence of stopping times and the v i - 1 are F τ i - 1 -measurable random variables such that E v i - 1 2 ( S τ i - S τ i - 1 ) 2 ∞ ( ( F t ) t ∈ [ 0 , T ] is the augmentation of the natural filtration of the underlying Brownian motion). In case that g is not almost surely linear, we show that one gets a lower bound for the L 2 -approximation rate of 1 / n if one optimizes over all nets consisting of n + 1 stopping time…
Reproducing pairs of measurable functions
2017
We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several examples, both discrete and continuous, are presented.