Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

Nussbaum, Roger D.; Walsh, Bertram

doi:10.1090/s0002-9947-98-01998-9

Cited by 10 publications

(3 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(A cone K in a Banach space X is defined to be a closed convex set for which σu ∈ K whenever u ∈ K and σ ≥ 0, and for which u, −u ∈ K only if u = 0. A linear operator A on X is called positive if A maps K into K. For some general references, see the original paper of Kreȋn and Rutman [27], as well as [4], [9], [38], [39] and [40]. More general results can be found in [32].…”

Section: Introductionmentioning

confidence: 99%

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Mallet‐Paret

Nussbaum

2013

J Dyn Diff Equat

Self Cite

View full text Add to dashboard Cite

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equatioṅwith a single delay, where the delay coefficient is of one sign, say δβ(t) ≥ 0 with δ ∈ {−1, 1}. Positivity properties are studied, with the result that if (−1) k = δ then the k-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients α(t) and β(t) are periodic of the same period, and β(t) satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of u 0 -positivity of the exterior product is investigated when β(t) satisfies a uniform sign condition.Note that equation (1.1) has a single delay. Typically we assume a signed feedback, that is, β(t) is of constant sign, either positive or negative, for almost every t.Delay-differential equations have been studied for at least 200 years. While some of the early work had its origins in certain types of geometric problems and number theory, much of the impetus for the development of the theory came from studies of viscoelasticity and population dynamics (notably by Volterra; see [43], [44], and [45]), and from control theory (see, for example, Bellman [3]). More recent work has involved models from a wide variety of scientific fields, including nonlinear optics [18], [22], [23], economics [2], [29], biology [12], [30], and as well population dynamics [19], [42]. Classic references for much of the fundamental theory are the books of J.K. Hale and S.M. Verduyn Lunel [15] and of O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther [10]. For additional material see [1], [13], [14], [16], [17], and [49]. Equations such as (1 .1) can arise as linearizations around solutions of nonlinear equations. Two such very classic yet still challenging nonlinear equations are Wright's Equationẋ (t) = −αx(t − 1)(1 + x(t)), and the Mackey-Glass Equationẋ (t) = −α 1 x(t) + α 2 f (x(t − 1)), where f (x) = x/(1 + x n ). See [48] and [30], as well as the references in [15] for further information on such equations.Abstractly, a linear (evolutionary) process U (t, τ ) : X → X on a Banach space X is a collection of bounded linear operators U (t, τ ), for t ≥ τ , for which U (τ, τ ) = I and U (t, σ)U (σ, τ ) = U (t, τ ) whenever t ≥ σ ≥ τ , with U (t, τ )x varying continuously in (t, τ ) for each fixed x. Linear processes occur as solution maps of a wide variety of nonautonomous linear equations, including of course the finitedimensional caseẋ = A(t)x of an ordinary differential equation. In the case of the delay-differential equation (1.1) the underlying Banach space is X = C([−1, 0]), and we assume that α : R → R and β : R → R are locally integrable functions.Given an abstract linear process U (t, τ ) as above, and given an integer m ≥ 1, one obtains the so-called compoun...

show abstract

Section: Introductionmentioning

confidence: 99%

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Mallet‐Paret

Nussbaum

2013

J Dyn Diff Equat

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is a classical problem to find the best approximation of a given vector by elements in certain closed convex subset; cf. [5,7,8,10,12,14,16]. The best approximation of functions on an interval by polynomials with nonnegative coefficients is particularly interesting; for instance, it plays a crucial role in the spectral analysis of self-adjoint operators on real Hilbert spaces [12,14].…”

Section: Introduction 1closed Convex Cone Generated By a Sequencementioning

confidence: 99%

The Metric Projections Onto Closed Convex Cones in a Hilbert Space

Qiu¹,

Wang²

2021

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set: $$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$ Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$ . As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.

show abstract

“…It is a classical problem to find the best approximation of a given vector by elements in a given closed convex subset, cf. [5,7,8,10,12,14,16]. The best approximation of functions by polynomials with non-negative coefficients on certain interval is particularly interesting, for instance, it plays a crucial role in the spectral analysis of self-adjoint operators on real Hilbert spaces [12,14].…”

mentioning

confidence: 99%

The metric projections onto closed convex cones in a Hilbert space

Qiu,

Wang

2020

Preprint

View full text Add to dashboard Cite

We study the metric projection onto the closed convex cone in a real Hilbert space H generated by a sequence V = {v n } ∞ n=0 . The first main result of this paper provides a sufficient condition under which we can identify the closed convex cone generated by V with the following set:Then, by adapting classical results on general convex cones, we give a useful description of the metric projection of a vector onto C[[V]]. As applications, we obtain the best approximations of many concrete functions in L 2 ([−1, 1]) by polynomials with non-negative coefficients.

show abstract

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

Cited by 10 publications

References 15 publications

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

The Metric Projections Onto Closed Convex Cones in a Hilbert Space

The metric projections onto closed convex cones in a Hilbert space

Contact Info

Product

Resources

About