Laguerre Transform Research Articles

L-isothermic and L-minimal surfaces

It has recently been shown that a second-order linear nonhomogeneous differential equation is associated with a surface with an isothermic representation of their lines of curvature (L-isothermic surface) (Schief et al 2007 J. Math. Phys. 48 073510). The 6-parameter group acting on linearly independent solutions of the homogeneous version of the latter equation generates a Laguerre transformation of the surface. The Weierstrass representation of the surfaces which are both L-isothermic and L-minimal is presented.

Markerless Human Motion Analysis in Gauss–Laguerre Transform Domain: An Application to Sit-To-Stand in Young and Elderly People

A markerless computer vision technique specifically designed to track natural elements on the human body surface is presented. The method implements the estimate of translation, rotation, and scaling by means of a maximum likelihood approach carried out in the Gauss-Laguerre transform domain. The approach is particularly suitable for human movement analysis in clinical contexts, where kinematics is at present performed by means of marker-based systems. Specific drawbacks of these latter systems, such as the burden of time for marker placement and the intrinsic intrusive nature, would be removed by the proposed method. Experimental results in terms of tracking performance are obtained by analyzing video sequences capturing the execution of the sit-to-stand task in two groups of young and elderly volunteers. The results are compared with clinical studies that used marker-based systems, and are particularly encouraging for a future extension of the approach to other motor tasks and to predict scores obtained from the physical performance batteries that are widely and regularly used by clinicians and physical therapists.

Laguerre Geometry of Surfaces in R 3

Let f : M → R3 be an oriented surface with non–degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L(f) = f (H2 – K)/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R3. And we give a classification theorem of surfaces in R3 with vanishing Laguerre form.

An integral equation method for the numerical analysis of gravity waves in a channel with free boundary

We consider the numerical solution of an evolution equation of second order on a boundary in R 2 that arises in the theory of gravity waves in fluids. First, by Laguerre transformation with respect to time we reduce the non-stationary problem to a sequence of operator equations. Then we handle the cross-section and longitudinal section cases of the channel in different ways. The potential theory for Laplace equation in a bounded domain with corners and the Green's function technique in a semi-infinite domain lead to systems of boundary integral equations of the second kind. For a full discretization we use a Nyström method. In the case of cross-section the trigonometrical quadrature rules in combination with special mesh grading transformation for the accounting of densities singularities in corners are applied. In the longitudinal section case the quadratures are based on sinc-approximation. The results of numerical experiments are presented.

The numerical solution of an evolution problem of second order in time on a closed smooth boundary

We consider an initial value problem for the second-order differential equation with a Dirichlet-to-Neumann operator coefficient. For the numerical solution we carry out semi-discretization by the Laguerre transformation with respect to the time variable. Then an infinite system of the stationary operator equations is obtained. By potential theory, the operator equations are reduced to boundary integral equations of the second kind with logarithmic or hypersingular kernels. The full discretization is realized by Nyström's method which is based on the trigonometric quadrature rules. Numerical tests confirm the ability of the method to solve these types of nonstationary problems.

On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.

Audio Effects Based on Biorthogonal Time-Varying Frequency Warping

We illustrate the mathematical background and musical use of a class of audio effects based on frequency warping. These effects alter the frequency content of a signal via spectral mapping. They can be implemented in dispersive tapped delay lines based on a chain of all-pass filters. In a homogeneous line with first-order all-pass sections, the signal formed by the output samples at a given time is related to the input via the Laguerre transform. However, most musical signals require a time-varying frequency modification in order to be properly processed. Vibrato in musical instruments or voice intonation in the case of vocal sounds may be modeled as small and slow pitch variations. Simulation of these effects requires techniques for time-varying pitch and/or brightness modification that are very useful for sound processing. The basis for time-varying frequency warping is a time-varying version of the Laguerre transformation. The corresponding implementation structure is obtained as a dispersive tapped delay line, where each of the frequency dependent delay element has its own phase response. Thus, time-varying warping results in a space-varying, inhomogeneous, propagation structure. We show that time-varying frequency warping is associated to an expansion over biorthogonal sets generalizing the discrete Laguerre basis. Slow time-varying characteristics lead to slowly varying parameter sequences. The corresponding sound transformation does not suffer from discontinuities typical of delay lines based on unit delays.

On the Numerical Solution of the Hyperbolic Evolution Problem on Closed Curve

AbstractWe consider on evolution problem for a hyperbolic equation with the Dirichlet to Neumann operator. For the numerical solution we combine the Laguerre transformation with respect to the time variable and the boundary integral equation method. Then an infinite sequence of the integral equations of the second kind with a logarithmic singularity is obtained. The full discretization is realized by Nyström's method which is based on trigonometric quadrature rules.

A variational problem for surfaces in Laguerre geometry

We consider the variational problem defined by the functional ∫ H 2 − K K d A \int {\frac {{H^{2}-K}}{{K}}}dA on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.

New euclidean theorems by the use of Laguerre transformations ? Some geometry of Minkowski (2+1)-space

Examples of the use of Laguerre transformations to discover theorems in the Euclidean and Minkowski planes.

Nonsteady temperature field in a thin plate between two tubes interacting with flows of heat carriers

The article states and solves the problem of a nonsteady temperature field in a thin plate between two tubes interacting with flows of heat carrier in the case of unidirectional, heterodirectional, loop, and circulating flow by the method of integral Laguerre transformation with respect to a time variable.

The Laguerre Transform and a Family of Functions with Nonnegative Laguerre Coefficients

The Laguerre transform induced in Keilson and Nunn (Keilson, J., W. R. Nunn. 1979. Laguerre transformation as a tool for the numerical solution of integral equations of convolution type. Appl. Math. Comp. 5 313–359.) and Keilson, Nunn and Sumita (Keilson, J., W. R. Nunn, U. Sumita. 1981. The bilateral Laguerre transform. Appl. Math. Comp. 8 137–174.) maps functions in L2 into discrete sequences and permits rapid numerical calculation of basic continuum operations such as differentiation, integration and convolution. In this paper we study a family of functions 𝒫 with nonnegative Fourier–Laguerre coefficients. A uniform bound for truncation error can be easily found for this family. It will be shown that the family 𝒫 is closed under positive scalar multiplication, convex mixing, and multiplication by completely monotone functions.

The Laguerre Transform of some Elementary Functions

This paper deals with the calculation of the generalized Laguerre transformation of a class of functions, which often appear in applications. The result is used for the derivation of the solution of an ordinary differential equation.

Laguerre transformation as a tool for the numerical solution of integral equations of convolution type

A novel transform is presented which maps continuum functions (such as probability densities) into discrete sequences and permits rapid numerical calculation of convolutions, multiple convolutions, and Neumann expansions for Volterra integral equations. The transform is based on the Laguerre polynomials, associated Laguerre functions and their simple convolution properties. A second transform employs Erlang functions as elements of the basis. The limitations and advantages of the two transforms are discussed. Numerical inversion of Laplace transforms relates simply to the Erlang transform. The deconvolution of two functions, i.e., the solution of a( t) = x( t)* b( t), may also be obtained quickly in this way.

A new inversion method for a class of Z transforms with application to Laguerre transform signal analysis of transients on uniformly distributed RGC lines

A new efficient method for signal analysis of transients on uniformly distributed RGC lines, based on Laguerre transformation, is presented. To facilitate implementation of this technique a new inversion method for a class of Z transforms has been derived.

The Dual Poisson–Laguerre Transform of a Class of Generalized Functions

The inversion formula of F. M. Cholewinski and D. T. Haimo for the dual Poisson–Laguerre transform is extended to a class of generalized functions interpreting convergence in the weak distributional sense. It is proved that this class of generalized functions contains tempered distributions defined over $(0,\infty )$. A structure formula for a class of generalized functions whose dual Poisson–Laguerre transform exists is also determined.Further, it is shown that the dual Poisson–Laguerre transform of the class of generalized functions satisfies the heat equation $[xD^2 + (\alpha + 1 - x)D]u = {{\partial u} / {\partial t}}$, $\alpha > - 1$.

Application of Laguerre Transform on heat conduction problem

In this paper the author has utilised the Laguerre transform introduced by him in an earlier paper (1960), to solve the heat equation for one-dimensional linear flow of heat in a very long non-homogenous bar of prescribed thermal conductivity under certain boundary conditions, the source of heat is being taken within the medium.

On the Differential Invariants of the Laguerre Group

1. In a paper “Beiträge zur Inversionsgeometric,” which will appear in the forthcoming volume of theScience Reports of the Tôhoku Imperial University, I have treated the problem of determining the necessary and sufficient condition in order that two plane curves which are given by natural equations should be transformable into each other by inversion. In connection with that paper I propose here to treat the analogous problem for Laguerre transformations:Having given two plane curves, by natural equations, in which the functional relations are all supposed to be analytic, it is required to determine the necessary and sufficient condition in order that the two plane curves should be transformable into each other by a Laguerre transformation.