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Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Michael D. Perlman · Journal of Multivariate Analysis · 1974

Cited by 76

Conditional Fourier-Feynman transform and conditional convolution product given infinite-dimensional vector-valued conditioning function

Jae Hwi Choi · Filomat · 2024

LetC0[0, T] denoteWiener space. Define an infinite-dimensional random vector XE,? : C0[0, T] ? R? by XE,?(x) = (?e1, x?, ?e2, x?, . . .) where E = {en}? n=1 is an orthonormal sequence in L2[0, T] and ?e, x? denotes the Paley?Wiener?Zygmund (PWZ) stochastic integral. In this paper, we study a conditional Fourier? Feynman transform (CFFT) and a conditional convolution product (CCP) for scale-invariant measurable (SIM) functionals on C0[0, T] with the very general conditioning function XE,?. In particular, we show that the CFFT of the CCP is a product of CFFTs.

Cited by 0Open access

Classifying infinite-dimensional data with unified basis functions: An effective machine learning approach

Jiayu Shang, Xiao‐Jun Zeng · Neurocomputing · 2024

Many real-world applications involve functional data, where each sample is a continuous function rather than a finite-dimensional vector or matrix. This infinite-dimensional nature poses significant challenges for classification tasks. Despite the progress in functional data analysis, existing methods and machine learning algorithms often struggle with the complexities of infinite-dimensional spaces. This paper introduces a novel approach that aims to simplify and improve the accuracy of infinite-dimensional classification tasks within functional data analysis. By addressing the algorithmic and computational complexities of existing high-dimensional classification methods, our approach provides an effective solution for the non-linear classification of infinite-dimensional functional data. The primary innovation lies in identifying a set of unified basis functions to accurately represent all observations originally defined in infinite-dimensional space. Subsequently, these observed functions in an infinite-dimensional space can be represented by their parameter vectors in a finite-dimensional space, and the original functional classification problem is converted to the conventional classification problem in a vector space (parameters) to solve, which leads to the simplicity of the proposed approach. We present the detailed design and implementation of this approach, including an effective framework for identifying functional data representations and B-spline knot placements. This approach not only facilitates the transformation between infinite-dimensional function spaces and finite-dimensional representative parameter spaces but also demonstrates significant accuracy improvements across various datasets in this field, as evidenced by empirical findings and comparative experimental results.

Cited by 0Open access

Automorphic forms and infinite matrices

Tomio Kubota · Nagoya Mathematical Journal · 1992

In the present paper, we show that an infinite dimensional vector whose components are Fourier coefficients of an automorphic form is characterized as an infinite dimensional vector which is annihilated by an infinite matrix constructed by the values of a Bessel function. Results and methods are all simple and concrete. Although the idea in the present paper is applicable to more general cases, our investigation will be restricted to the case of automorphic forms of weight 0, i.e., automorphic functions, with respect to SL(2, Z) on the upper half plane, in order to explain the main idea distinctly.

Cited by 1Open access

Universal harmonic functions

A. Bonilla · Quaestiones Mathematicae · 2002

In this paper, we prove that there exists an infinite dimensional closed vector space M of harmonic functions in R n such that each v ∊ M \{0} is a universal harmonic function.

Cited by 6

Lineability of non-differentiable Pettis primitives

B. Bongiorno, U. B. Darju, Luisa Di Piazza · arXiv (Cornell University) · 2014

Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0; 1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND.

Cited by 0Open access

Strong duality and sensitivity analysis in semi-infinite linear programming

Amitabh Basu, Kipp Martin, Christopher Ryan · arXiv (Cornell University) · 2015

Finite-dimensional linear programs satisfy strong duality (SD) and have the "dual pricing" (DP) property. The (DP) property ensures that, given a sufficiently small perturbation of the right-hand-side vector, there exists a dual solution that correctly "prices" the perturbation by computing the exact change in the optimal objective function value. These properties may fail in semi-infinite linear programming where the constraint vector space is infinite dimensional. Unlike the finite-dimensional case, in semi-infinite linear programs the constraint vector space is a modeling choice. We show that, for a sufficiently restricted vector space, both (SD) and (DP) always hold, at the cost of restricting the perturbations to that space. The main goal of the paper is to extend this restricted space to the largest possible constraint space where (SD) and (DP) hold. Once (SD) or (DP) fail for a given constraint space, then these conditions fail for all larger constraint spaces. We give sufficient conditions for when (SD) and (DP) hold in an extended constraint space. Our results require the use of linear functionals that are singular or purely finitely additive and thus not representable as finite support vectors. The key to understanding these linear functionals is the extension of the Fourier-Motzkin elimination procedure to semi-infinite linear programs.

Cited by 0Open access

Uncertainty Principle for Rational Functions in Hardy Spaces

Dan Xiong, Li Chai, Jingxin Zhang · 2018

Uncertainty principles in finite dimensional vector space have been studied extensively, however they cannot be applied to sparse representation of rational functions. This paper considers the sparse representation for a rational function under a pair of orthonormal rational function bases. We prove the uncertainty principle concerning pairs of compressible representation of rational functions in the infinite dimensional function space. The uniqueness of compressible representation using such pairs is provided as a direct consequence of uncertainty principle.

Cited by 1

Compressible Topological Vector Spaces

Mohammadreza Robaei, Robert Akl · arXiv (Cornell University) · 2022

The optimum subspace decomposition of the infinite-dimensional compressible random processes in the locally convex Hausdorff space has been propose and its dimension has been measured. We conduct topological analysis of finite- and infinite-dimensional compressible vector spaces. We prove that if there are a sufficient number of separating points in compressible topological vector space, the Banach Limit coincides the minimal linear functional. Then, optimum orthogonal decomposition of the compressible topological vector space can be formulated as a limit for which minimal linear functional occurs. It has been shown that the separable space, also referred to as an optimum subspace, is the subset of the compressible vector space that contains its extreme points. The inseparable subspace is characterized using a novel absorbing null space property through Minkowski functional and Lebesgue measure. We prove that the absorbing null space is a connected subspace of the locally convex space. We purpose reflexive homomorphism that establishes a relation between signal space and double dual space. It has been shown that the Banach limit can be determined by the compact convergence of the Cauchy nets on the left and right hands sides of a reflexive homomorphism. Finally, we propose to measure the optimum subspace dimension using the Frechet distance metric. To apply the Frechet distance to the compressible topological vector space, the Kothe sequence sampled from a continuous function has been proposed. Briefly, the proposed approach measures the sufficient number of separating points of the finite- and infinite-dimensional compressible vector space for the given undersampled operator with respect to Hahn-Banach and Daniell-Kolmogorov theorems. The numerical analysis have been presented for finite- and infinite-dimensional signals.

Cited by 2Open access

Localization and the invariant probability measure for photonic band gap structures

Glen J. Kissel · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2007

Optical localization in a randomly disordered infinite length one-dimensional photonic band gap structure is studied using the transfer matrix formalism. Asymptotically, the infinite product of random matrices acting on a nonrandom input vector induces an invariant probability measure on the direction of the propagated vector. This invariant measure is numerically calculated for use in Furstenberg's master formula giving the upper Lyapunov exponent (localization factor) of the infinite random matrix product. A quarter-wave stack model with one of the bilayer thicknesses disordered is used for simulation purposes. In this plane wave model the invariant measure is rarely a uniform probability density function, as is sometimes assumed in the literature. Yet, the assumption of a uniform probability density function for the invariant measure gives surprisingly good results for a highly disordered system in the UV region.

Cited by 5

Strong duality and sensitivity analysis in semi-infinite linear\n programming

Amitabh Basu, Kipp Martin, Christopher Ryan · arXiv (Cornell University) · 2015

Finite-dimensional linear programs satisfy strong duality (SD) and have the\n"dual pricing" (DP) property. The (DP) property ensures that, given a\nsufficiently small perturbation of the right-hand-side vector, there exists a\ndual solution that correctly "prices" the perturbation by computing the exact\nchange in the optimal objective function value. These properties may fail in\nsemi-infinite linear programming where the constraint vector space is infinite\ndimensional. Unlike the finite-dimensional case, in semi-infinite linear\nprograms the constraint vector space is a modeling choice. We show that, for a\nsufficiently restricted vector space, both (SD) and (DP) always hold, at the\ncost of restricting the perturbations to that space. The main goal of the paper\nis to extend this restricted space to the largest possible constraint space\nwhere (SD) and (DP) hold. Once (SD) or (DP) fail for a given constraint space,\nthen these conditions fail for all larger constraint spaces. We give sufficient\nconditions for when (SD) and (DP) hold in an extended constraint space. Our\nresults require the use of linear functionals that are singular or purely\nfinitely additive and thus not representable as finite support vectors. The key\nto understanding these linear functionals is the extension of the\nFourier-Motzkin elimination procedure to semi-infinite linear programs.\n

Cited by 0Open access

Calculus of Variations

Anders Hansson, Martin Andersen · 2023

Here we discuss optimization over infinite-dimensional vector spaces. The most simple case is when the variable is a real-valued function of a real variable. This has applications in optimal control in continuous time, where the optimal control signal is a real-valued function of time. Another important application is the derivation of probability density functions from the principle of maximizing the entropy subject to moment constraints. In this case the variable is the probability density function. For probability density functions the argument is often a vector.

Cited by 0Open access

On the Differentiability of Vector Valued Additive Set Functions

Mangatiana A. Robdera, Dintle N. Kagiso · Advances in Pure Mathematics · 2013

The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function ; that is, for all measurable sets. Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.

Cited by 6Open access

On the team-maxmin equilibria

Takuya Iimura · Economic Theory Bulletin · 2025

Abstract In this note we extend the result of von Stengel and Koller (Games Econ Behav 21:309–321, 1997) to infinite games. Specifically, we show that every infinite zero-sum game between a team and an adversary admits a pure-strategy team-maxmin equilibrium when (i) the strategy sets are compact, convex subsets of (possibly infinite-dimensional) topological vector spaces, and (ii) the payoff function is bounded, upper-semicontinuous on the team’s strategy-profile set and concave in each team member’s strategy, while being lower-semicontinuous and convex in the adversary’s strategy. Because the vector spaces may have arbitrary dimension, we obtain the following corollary: a mixed-strategy team-maxmin equilibrium exists whenever (i) the strategy sets are compact subsets of metric spaces, and (ii) the payoff function is bounded, measurable, upper-semicontinuous on the team’s strategy-profile set, and lower-semicontinuous on the adversary’s strategy set. The proof employs Sion’s minimax theorem.

Cited by 0Open access

Five-dimensional null & time-like supersymmetric geometries

Pasini, Giulio, Shahbazi, C. S. · arXiv (Cornell University) · 2015

We show that there exist supersymmetric solutions of five-dimensional, pure, $\mathcal{N}=1$ Supergravity such that the norm of the supersymmetric Killing vector, built out of the Killing spinor, is a real not-everywhere analytic function such that all its derivatives vanish at a point where the Killing vector field becomes null. The norm of the Killing vector field then is not an analytic function on a neighborhood around this point. We explicitly construct such solutions by using a multi-center Gibbons-Hawking base. Although many of these solutions have infinite charges, we find explicit examples with finite charges that asymptote to $AdS_3\times S_2$ and discuss their physical interpretation.

Cited by 0Open access

Lyapunov characterization of input-to-output stability for infinite-dimensional systems

Patrick Bachmann, Sergey Dashkovskiy, Andrii Mironchenko · 2025

We prove that for control systems defined by semi-linear time-invariant evolution equations, input-to-output stability (IOS) is equivalent to the existence of a vector IOS Lyapunov function. We generalize the IOS Lyapunov theory for ordinary differential equations to a class of infinite-dimensional systems by providing a novel notion of Lyapunov function. Furthermore, we introduce a new notion of robust output stability based on a regularized output, which is equivalent to IOS.

Cited by 0

The failure of the profile likelihood method for a large class of semi-parametric models

Eric Beutner, Laurent Bordes, Laurent Doyen · Bernoulli · 2017

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector $\mathbf{{\theta}}$, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde{\boldsymbol {\theta}}$, say, with probability $p$. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox’s proportional hazards model.

Cited by 6Open access

The infinite Arnoldi exponential integrator for linear inhomogeneous\n ODEs

Antti Koskela, Elias Jarlebring · arXiv (Cornell University) · 2015

Exponential integrators that use Krylov approximations of matrix functions\nhave turned out to be efficient for the time-integration of certain ordinary\ndifferential equations (ODEs). This holds in particular for linear homogeneous\nODEs, where the exponential integrator is equivalent to approximating the\nproduct of the matrix exponential and a vector. In this paper, we consider\nlinear inhomogeneous ODEs, $y'(t)=Ay(t)+g(t)$, where the function $g(t)$ is\nassumed to satisfy certain regularity conditions. We derive an algorithm for\nthis problem which is equivalent to approximating the product of the matrix\nexponential and a vector using Arnoldi's method. The construction is based on\nexpressing the function $g(t)$ as a linear combination of given basis functions\n$[\\phi_i]_{i=0}^\\infty$ with particular properties. The properties are such\nthat the inhomogeneous ODE can be restated as an infinite-dimensional linear\nhomogeneous ODE. Moreover, the linear homogeneous infinite-dimensional ODE has\nproperties that directly allow us to extend a Krylov method for\nfinite-dimensional linear ODEs. Although the construction is based on an\ninfinite-dimensional operator, the algorithm can be carried out with operations\ninvolving matrices and vectors of finite size. This type of construction\nresembles in many ways the infinite Arnoldi method for nonlinear eigenvalue\nproblems. We prove convergence of the algorithm under certain natural\nconditions, and illustrate properties of the algorithm with examples stemming\nfrom the discretization of partial differential equations.\n

Cited by 0Open access

Normed Vector Spaces and Double Duals

Patrick Morandi · 2005

In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces. As an application, we give an example of an infinite-dimensional vector space V for which the natural map η: V → V ∗ ∗ is not an isomorphism. In analysis, duals and double duals of vector spaces are often defined differently than in algebra, by considering continuity. We will be more specific shortly. Let V be an R-vector space. We say that V is a normed vector space if there is a function ‖· ‖ : V → R which satisfies • ‖v ‖ ≥ 0 for all v ∈ V, and ‖v ‖ = 0 if and only if v = 0. • ‖αv ‖ = |α | · ‖v ‖ for all α ∈ R and v ∈ V. • ‖v + w ‖ ≤ ‖v‖+ ‖w ‖ for all v, w ∈ V. The function ‖· ‖ is called a norm on V. If V is a normed vector space, then the function ‖· ‖ allows us to define a metric on V by d(v, w) = ‖v − w‖, just as we do for R or Rn. We can then talk about functions f: V → R being continuous at v0 ∈ V: if for any ε> 0 there exists δ> 0 such that ‖v − v0 ‖ < δ implies ‖f(v) − f(v0) ‖ < ε, then f is continuous at v0. Example 1. If V = Rn, then V is a normed vector space under the usual Euclidean metric ‖(x1,..., xn) ‖ = x21 + · · ·+ x2n. Recall that the set S of all real valued sequences is an R-vector space under pointwise addition and scalar multiplication. The next example gives a collection of subspaces of this sequence space. Example 2. For x = {xn} ∈ S, define, for p ≥ 1, ‖x‖p =

Cited by 0

Plane elastostatic transmission through an interface layer

K. Aderogba · Mathematical Proceedings of the Cambridge Philosophical Society · 2000

A theorem is established for generating by differentiation and integration the Airy stress function due to the presence of any arbitrary two-dimensional singularity near a thick layer separating two other dissimilar semi-infinite solids directly from the corresponding Airy stress function for the unbounded homogeneous solid. A systematic application of the theorem yields the dominant distant effect associated with any arbitrary influencing line singularity in a well-structured physically interpretable form. As an illustration, the problem of a concentrated line force of arbitrary orientation near the interface layer is examined from this standpoint. It is found that a force aligned normal to the interface layer produces a far-field effect of an edge dislocation with Burgers vector parallel to the interface layer, together with a semi-infinite plane of centres of dilatation, whereas a force aligned parallel to the interface layer produces a far-field effect of an edge dislocation with Burgers vector perpendicular to the interface layer together with a semi-infinite plane of concentrated couples.

Cited by 4