Search results for "Hermite polynomials"
showing 10 items of 23 documents
Multiwavelet Frames Originated From Hermite Splines
2015
The chapter presents a method for the construction of multiwavelet frame transform for manipulation of discrete-time signals. The frames are generated by three-channel perfect reconstruction oversampled multifilter banks. The design of the multifilter bank starts from a pair of interpolating multifilters, which originate from the cubic Hermite splines. The remaining multifilters are designed by factoring polyphase matrices. Input to the oversampled analysis multifilter bank is a vector-signal, which is derived from an initial scalar signal by one out of three pre-processing algorithms. The post-processing algorithms convert the vector output from the synthesis multifilter banks into a scala…
Products of Bessel functions and associated polynomials
2013
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
The discretized harmonic oscillator: Mathieu functions and a new class of generalized Hermite polynomials
2003
We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansa…
Indefinite integrals for some orthogonal polynomials obtained using integrating factors
2020
A method has been presented recently for deriving integrals of special functions using two kinds of integrating factor for the homogeneous second-order linear differential equations which many spec...
Error analysis of the orthogonal series solution of linear time-invariant systems
1989
Similarities in the error analysis of the polynomial series solution of linear time-invariant systems are pointed out.
Biorthogonal Multiwavelets Originated from Hermite Splines
2015
This chapter presents multiwavelet transforms that manipulate discrete-time signals. The transforms are implemented in two phases: 1. Pre (post)-processing, which transforms a scalar signal into a vector signal (and back). 2. Wavelet transforms of the vector signal. Both phases are performed in a lifting way. The cubic interpolating Hermite splines are used as a predicting aggregate in the vector wavelet transform. Pre(post)-processing algorithms which do not degrade the approximation accuracy of the vector wavelet transforms are presented. A scheme of vector wavelet transforms and three pre(post)-processing algorithms are described. As a result, we get fast biorthogonal algorithms to trans…
Construction of a fundamental set of solutions of an arbitrary homogeneous linear difference equation
2002
Abstract The detailed construction of a prefixed fundamental set of solutions of a linear homogeneous difference equation of any order with arbitrarily variable coefficients is reported. The usefulness of the resulting resolutive formula is illustrated by simple applications to the Hermite polynomials and to the Fibonacci sequence.
Optimized Hermite-Gaussian ansatz functions for dispersion-managed solitons
2001
Abstract By theoretical analysis, we show that the usual procedure of simply projecting the dispersion-managed (DM) soliton profile onto the basis of an arbitrary number of Hermite-gaussian (HG) polynomials leads to relatively accurate ansatz functions, but does not correspond to the best representation of DM solitons. Based on the minimization of the soliton dressing, we present a simple procedure, which provides highly accurate representation of DM solitons on the basis of a few HG polynomials only.
Data analysis procedures for pulse ELDOR measurements of broad distance distributions
2004
The reliability of procedures for extracting the distance distribution between spins from the dipolar evolution function is studied with particular emphasis on broad distributions. A new numerically stable procedure for fitting distance distributions with polynomial interpolation between sampling points is introduced and compared to Tikhonov regularization in the dipolar frequency and distance domains and to approximate Pake transformation. Distance distributions with only narrow peaks are most reliably extracted by distance-domain Tikhonov regularization, while frequency-domain Tikhonov regularization is favorable for distributions with only broad peaks. For the quantification of distribut…
Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions
2019
Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \Omega_2 \subset \Omega_1 \subset \mathbb{R}^n$: $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Ome…