Laguerre Function Expansions Research Articles

Besov and Triebel–Lizorkin spaces associated with Laguerre expansions of Hermite type

Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $\alpha \in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calder\'on reproducing formula.

Sharp Estimates for Potential Operators Associated with Laguerre and Dunkl-Laguerre Expansions

We study potential operators associated with Laguerre function expansions of convolution and Hermite types, and with Dunkl-Laguerre expansions. We prove qualita- tively sharp estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q ≤∞ , for which the potential operators are L p − L q bounded. These results are sharp analogues of the classical Hardy-Littlewood-Sobolev fractional integration theorem in the Laguerre and Dunkl-Laguerre settings.

A weighted transplantation theorem for Laguerre function expansions

We establish a generalized weighted transplantation theorem for Laguerre function expansions, which extends the corresponding result by G. Garrigos et al. “A sharp weighted transplantation theorem for Laguerre function expansions” (J. Funct. Anal. 244 (2007), pp. 247–276).

Laplace Type Multipliers for Laguerre Expansions of Hermite Type

We investigate Laplace transform type and Laplace-Stieltjes type multipliers associated to the multi–dimensional Laguerre function expansions of Hermite type. We prove that, under the assumption α i ≥ −1/2, α i ∉ (−1/2, 1/2), these operators are Calderón-Zygmund operators. Consequently, their mapping properties follow by the general theory.

DIMENSION FREE $L^P$ ESTIMATES FOR RIESZ TRANSFORMS ASSOCIATED WITH LAGUERRE FUNCTION EXPANSIONS OF HERMITE TYPE

We prove dimension free $L^p$ estimates for Riesz transforms associated with multi-dimensional Laguerre function expansions of Hermite type. The range of the admissible Laguerre type multi-index $\alpha$ in these estimates depends on $p \in (1,\infty)$; for $1 \lt p \le 2$ this range is almost optimal. The proof is based on suitably defined square functions with Poisson and modified Poisson semigroups involved.

Negative Powers of Laguerre Operators

AbstractWe study negative powers of Laguerre differential operators in ℝd,d≥ 1. For these operators we prove two-weightLp−Lqestimates with ranges ofqdepending onp. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices. we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call theconvexity principleto cover the continuous range of ϵ 2 [−1/2, ∞)d. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on ℝdand isomorphic to ℤd2. The two weightLp−Lqestimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

MULTIPLIERS OF LAPLACE TRANSFORM TYPE IN CERTAIN DUNKL AND LAGUERRE SETTINGS

AbstractWe investigate Laplace type and Laplace–Stieltjes type multipliers in the d-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to ℤd2 and in the related context of Laguerre function expansions of convolution type. We use Calderón–Zygmund theory to prove that these multiplier operators are bounded on weighted Lp, 1<p<∞, and from L1 to weak L1.

Calderón–Zygmund operators related to Laguerre function expansions of convolution type

We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices α in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of α. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood–Paley–Stein type square functions and multipliers of Laplace and Laplace–Stieltjes transforms type, are (vector-valued) Calderón–Zygmund operators in the sense of the associated space of homogeneous type.

On g-functions for Laguerre function expansions of Hermite type

We examine weighted Lp boundedness of g-functions based on semigroups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderon–Zygmund operators is used.

Littlewood–Paley–Stein type square functions based on Laguerre semigroups

We investigate g-functions based on semigroups related to multi-dimensional Laguerre function expansions of convolution type. We prove that these operators can be viewed as Calderon–Zygmund operators in the sense of the underlying space of homogeneous type, hence their mapping properties follow from the general theory.

Incremental Modeling of Nonlinear Distributed Parameter Processes via Spatiotemporal Kernel Series Expansion

In this article, an incremental modeling approach is proposed to model nonlinear distributed parameter systems, with the help of the newly constructed spatiotemporal Volterra kernels. The complex spatiotemporal process is first decomposed into a series of spatiotemporal kernels, upon which the time−space separation can be further conducted with the spatial Karhunen−Loève and temporal Laguerre basis function expansions. These two decompositions can gradually separate the nonlinear time/space coupled dynamics. Finally, the kernels in the spatiotemporal model are estimated from the experimental data incrementally, which can easily achieve satisfactory modeling performance. Simulations of two transport−reaction processes demonstrate the effectiveness of the proposed modeling approach.

Riesz transforms for multi-dimensional Laguerre function expansions

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , the appropriately defined Riesz–Laguerre transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz–Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy–Riemann type and to recover the Riesz–Laguerre transforms on the boundary.

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α = ( α 1 , … , α d ) such that α i ⩾ − 1 / 2 , α i ∉ ( − 1 / 2 , 1 / 2 ) , the appropriately defined Riesz transforms R j α , j = 1 , 2 , … , d , are Calderón–Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α ∈ ( − 1 / 2 , 1 / 2 ) , by means of a local Calderón–Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy–Riemann type equations and to recover the Riesz–Laguerre transforms on the boundary. The two specific values of α, ( − 1 / 2 , … , − 1 / 2 ) and ( 1 / 2 , … , 1 / 2 ) , are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.

A sharp weighted transplantation theorem for Laguerre function expansions

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in L p spaces with power weights. Namely, the operators interchanging { L k α } and { L k β } are bounded in L p ( y δ p ) if and only if − ρ 2 − 1 p < δ < 1 − 1 p + ρ 2 , where ρ = min { α , β } . This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ ⩽ 0 . Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in L p ( y δ p ) .

Nonlinear neuronal mode analysis of action potential encoding in the cockroach tactile spine neuron.

Neuronal mode analysis is a recently developed technique for modelling the behavior of nonlinear systems whose outputs consist of action potentials. The system is modelled as a set of parallel linear filters, or modes, which feed into a multi-input threshold. The characteristics of the principal modes and the multi-input threshold device can be derived from Laguerre function expansions of the computed first- and second-order Volterra kernels when the system is stimulated with a randomly varying input. Neuronal mode analysis was used to model the encoder properties of the cockroach tactile spine neuron, a nonlinear, rapidly adapting, sensory neuron with reliable behavior. The analysis found two principal modes, one rapid and excitatory, the other slower and inhibitory. The two modes have analogies to two of the pathways in a block-structured model of the encoder that was developed from previous physiological investigations of the neuron. These results support the block-structured model and offer a new approach to identifying the components responsible for the nonlinear dynamic properties of this neuronal encoder.

Analysis of Linear Systems by Means of Laguerre Functions

The use of Laguerre functions is proposed for the analysis of heavily damped linear systems where the input is not subject to experimental control. An equation is derived which links the corresponding coefficients in the Laguerre function expansions of the input, the impulse response and the output of a linear system. This equation enables the third set of Laguerre coefficients to be calculated when the other two sets of coefficients are known. The connection between the Laguerre function expansion and the representation of the system response by a series of gamma distributions is noted and the latter series identified as defining an analog system composed entirely of branches of linear storage elements.