The inner kernel theorem for a certain Segal algebra

The Segal algebra ${\textbf{S}}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. Despite the fact that it is a Banach space it is possible to derive a kernel theorem similar to the Schwartz kernel theorem, of course without making use of the Schwartz kernel theorem. First we characterize the bounded linear operators from ${\textbf{S}}_{0}(G_1)$ to ${\textbf{S}}_{0}'(G_2)$ by distributions in ${\textbf{S}}_{0}'(G_1 \times G_2)$. We call this the "outer kernel theorem". The "inner kernel theorem" is concerned with the characterization of those linear operators which have kernels in the subspace ${\textbf{S}}_{0}(G_1 \times G_2)$, the main subject of this manuscript. We provide a description of such operators as regularizing operators in our context, mapping ${\textbf{S}}_{0}'(G_1)$ into test functions in ${\textbf{S}}_{0}(G_2)$, in a $w^{*}$-to norm continuous manner. The presentation provides a detailed functional analytic treatment of the situation and applies to the case of general LCA groups, without recurrence to the use of so-called Wilson bases, which have been used for the case of elementary LCA groups. The approach is then used in order to describe natural laws of composition which imitate the composition of linear mappings via matrix multiplications, now in a continuous setting. We use here that in a suitable (weak) form these operators approximate general operators. We also provide an explanation and mathematical justification used by engineers explaining in which sense pure frequencies "integrate" to a Dirac delta distribution

Compactness Criteria in Function Spaces

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces

Gabor analysis over finite Abelian groups

The topic of this paper are (multi-window) Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane. Our generic approach covers simultaneously multi-dimensional signals as well as non-separable lattices. The main results reduce to well-known fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler-Raz or Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a central role is given to spreading function of linear operators between finite-dimensional Hilbert spaces. Another relevant tool is a symplectic version of Poisson's summation formula over the finite time-frequency plane. It provides the Fundamental Identity of Gabor Analysis.In addition we highlight projective representations of the time-frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe

Asymptotic boundary forms for tight Gabor frames and lattice localization domains

We consider Gabor localization operators $G_{\phi,\Omega}$ defined by two parameters, the generating function $\phi$ of a tight Gabor frame $\{\phi_\lambda\}_{\lambda \in \Lambda}$, parametrized by the elements of a given lattice $\Lambda \subset \Bbb{R}^2$, i.e. a discrete cocompact subgroup of $\Bbb{R}^2$, and a lattice localization domain $\Omega \subset \Bbb{R}^2$ with its boundary consisting of line segments connecting points of $\Lambda$. We find an explicit formula for the boundary form $BF(\phi,\Omega)=\text{A}_\Lambda \lim_{R\rightarrow \infty}\frac{PF(G_{\phi,R\Omega})}{R}$, the normalized limit of the projection functional $PF(G_{\phi,\Omega})=\sum_{i=0}^{\infty}\lambda_i(G_{\phi,\Omega})(1-\lambda_i(G_{\phi,\Omega}))$, where $\lambda_i(G_{\phi,\Omega})$ are the eigenvalues of the localization operators $G_{\phi,\Omega}$ applied to dilated domains $R\Omega$, $R$ is an integer and $\text{A}_\Lambda$ is the area of the fundamental domain of the lattice $\Lambda$.Comment: 35 page

Environmental Policy, the Porter Hypothesis and the Composition of Capital: Effects of Learning and Technological Progress

In this paper the e.ect of environmental policy on the composition of capital is investigated.By allowing for non-linearities it generalizes Xepapadeas and De Zeeuw (Journal of Environmental Economics and Management, 1999) and determines scenarios in which their results do not carry over.In particular, we show that the way acquisition cost of investment decreases with the age of the capital stock is of crucial importance.Also it is obtained that environmental policy has opposite e.ects on the average age of the capital stock in the case of either deterioration or depreciation.We also focus more explicitly on learning and technological progress.Among others we obtain that in the presence of learning, implementing a stricter environmental policy with the aim to reach a certain target of emissions reduction has a stronger negative e.ect on industry pro.ts, which implies quite the opposite as to what is described by the Porter hypothesis.environmental policy;capital;learning;technological change

Dynamic Investment Behavior Taking into Account Ageing of the Capital Good

In standard capital accumulation models all capital goods are equally productive and produce goods of the same quality.However, due to ageing, in reality it holds most of the time that newer capital goods are more productive. Implications of this feature for the firm's investment policies are investigated in an optimal control problem with distributed parameters.It turns out that investing in capital goods of di¤erent age is done such that the net present value of marginal investment equals zero.Comparing the returns of investment in capital goods of different age, the higher productivity of younger capital goods has to be weighed against the lower costs of depreciation, discounting and acquisition of older capital goods.In the steady state it holds that, in the most reasonable scenario, the firm should invest at the highest rate in new capital goods, and dis-investment can only be optimal when costs of acquisition are large and machines are old.investment;capital goods;ageing