Fourier-Laguerre Series Research Articles

Pointwise summability of Fourier-Laguerre series of integrable functions

We present an approximation version of the results of D. P. Gupta [ J. of Approx. Theory, 7 (1973), 226-238] A. N. S. Singroura [Proc. Japan Acad., 39 (4) (1963), 208-210] and G. Szeg? [Math. Z., 25 (1926), 87-115]. Some corollaries and examples will also be given.

Efficient numerical calculation of frequency-domain matching method based on an analytical control surface

The matching method based on an analytical control surface has been developed to predict wave-induced motions and loads of ships with forward speed. Taking advantage of several computational techniques, this study proposes an efficient scheme for the numerical calculation of this novel method. Different from the conventional domain decomposition strategy, the control surface is analytical, and the physical quantities on it are expanded into the Fourier-Laguerre series. Therefore, the integration over the control surface associated with the forward speed Green function is performed analytically, avoiding the convergence problem caused by the singular and oscillatory behaviors of the forward speed Green function. Because the introduction of the analytical control surface may cause large computational burdens, an efficient scheme is developed by combing threaded parallelism, Chebyshev series approximation, and shared B-spline data. The excellent agreement between numerical results and experimental measurements shows that the present numerical method is reliable and practical in several applications. Furthermore, the numerical validation reveals that the present numerical scheme is affordable on the standard computing platform.

On the Degree of Approximation of a Function by Nörlund Means of its Fourier Laguerre Series

In this paper, we have proved the degree of approximation of function belonging to L[0, ∞) by Nörlund Summability of Fourier-Laguerre series at the end point x = 0. The purpose of this paper is to concentrate on the approximation relations of the function in L[0, ∞) by Nörlund Summability of Fourier- Laguerre series associate with the given function motivated by the works [3], [9] and [13].

Problems in the Calculus of Variations on Unbounded Intervals—Fourier–Laguerre Analysis and Approximations

In this paper, we consider a class of variational problems on an unbounded interval of the real axis. This type of problems arises, e.g., in quantum mechanics and asymptotic controllability. The problem is treated in a Hilbert space setting with uniformly and non-uniformly weighted Sobolev spaces as state spaces. We provide sufficient conditions that a function from a weighted Sobolev space can be expanded into a Fourier–Laguerre series converging together with its distributional derivative pointwisely and uniformly. With this result, the considered variational problem is transformed into a problem in the sequence space of Fourier–Laguerre coefficients. We develop a Fourier–Laguerre method in these spaces in order to construct a polynomial approximation scheme for the solution of this problem.

Pointwise Convergence of Fourier-Laguerre Series of Integrable Functions

Abstract We extend and improve the some results of Xh. Z. Krasniqi [Int. J. of Anal. and Appl. Vol. 1, 33-39 (2013)], M. L. Mittal and M. V. Singh [Operators, Int. J. of Analysis, Vol. 2015, Article ID 478345, 4 pages] and from many other papers on summability of Fourier-Laguerre series to strong summability proving the estimate of the deviation of the partial sums from considered functions. There also is a remark on summability methods used in cited papers.

Approximation of functions belonging toL[0, ∞) by product summability means of its Fourier-Laguerre series

In this paper, we have proved the degree of approximation of functions belonging to L[0,∞) by Harmonic-Euler means of its Fourier-Laguerre series at x=0. The aim of this paper is to concentrate on the approximation properties of the functions in L[0,∞) by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f.

Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.

Some frequency-domain approaches to the model reduction of delay systems

We review a selection of techniques which have been found effective in the H2 and H∞ model reduction of delay systems, modeled using transfer functions. These include Fourier–Laguerre series, Padé approximants, shift-based approximations, Malmquist bases, partial fractions, wavelet-based techniques, Hankel-norm approximants and truncated state-space realizations. Some discussion of achievable error bounds is included.

Some frequency-domain approaches to the model reduction of delay systems

We review a selection of techniques which have been found effective in the H2 and H∞ model reduction of delay systems, modelled using transfer functions. These include Fourier-Laguerre series, Padé approximants, shift-based approximations, Szegó bases, partial fractions, wavelet-based techniques, Hankel-norm approximants and truncated state-space realizations. Some discussion of achievable error bounds is included.

Application of Laguerre polynomials to invert a semi-infinite convolution

We propose a spectral method of inverting one- and two-dimensional semi-infinite convolutions using the Laguerre polynomials. We construct systems of linear algebraic equations to determine the coefficients of the corresponding Fourier-Laguerre series. We analyze the computational error in the coefficients of the orthogonal series that arise as a result of errors in the data. Applying the Laguerre polynomials makes it possible to eliminate the discretization procedure, which may take the solution of the perturbed equation outside the region of well-posedness.

Online system identification using Laguerre series

An online system identification scheme is proposed based on a Fourier-Laguerre series representation of the unknown impulse response. The unknown parameters are determined using a gradient estimator. Noise effects are considered. The proposed identification scheme is applied to a system with time delay.

Approximation of delay systems by fourier-laguerre series

Error estimates are given for the approximation of stable recorded delay systems in the L 2 and H ∞ norms, using two recently advocated techniques based on Laguerre series. In addition, some theoretical results on L ∞(0, ∞) approximation are derived.

A quasi-equivalence between Borel summability and convergence for Fourier-Laguerre series at the end-point

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.

A further note on certain dual equations involving Fourier-Laguerre series 1)

The present note aims at providing a systematic investigation of the solution of a certain pair of dual equations involving Fourier-Laguerre series. It exhibits equivalence of the solutions obtained earlier (see [5] and [6]) by using a multiplying factor technique as well as by considering separately the equations when ( i) g( x)≡0, (ii) f( x)≡0, and reducing the problem in each case to that of solving an Abel integral equation. It is also shown how the solution of these dual series equations are affected when the parameters involved are constrained differently.