This report examines the time windows used for linear prediction (LP) analysis of speech. The goal of windowing is to create frames of data each of which will be used to calculate an autocorrelation sequence. Several factors enter into the choice of window. The time and spectral properties of Hamming and Hann windows are examined. We also consider windows based on Discrete Prolate Spherical Seq...

This paper develops a family of preconditioners for pseudospectral approximations of pth-order linear differential operators subject to various types of boundary conditions. The approximations are based on ultraspherical polynomials with special attention being paid to Legendre and Chebyshev polynomial methods based on Gauss–Lobatto quadrature points. The eigenvalue spectrum of the precondition...

Let q ≥ 1 be an integer, Sq be the unit sphere embedded in the Euclidean space Rq+1. A Zonal Function (ZF) network with an activation function φ : [−1, 1] → R and n neurons is a function on Sq of the form x 7→ nk=1 akφ(x · ξk), where ak’s are real numbers, ξk’s are points on Sq. We consider the activation functions φ for which the coefficients {φ̂(`)} in the appropriate ultraspherical polynomial...

The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of non-trivial metric on manifold. applies to some classical Gagliardo-Nirenberg-Sobolev the sphere, optimal constants. Very nonlinear regimes close critical Sobolev exponent can be covered using parabolic flows porous medium or fast diffusion type. Considering power law ...

In this paper we extend complex uniform convexity estimates for $\mathbb{C}$ to $\mathbb{R}^n$ and determine best constants. Furthermore, provide the link log-Sobolev inequalities hypercontractivity ultraspherical measures.

We introduce a novel spectral element method based on the ultraspherical and hierarchical Poincar\'{e}-Steklov scheme for solving second-order linear partial differential equations polygonal domains with unstructured quadrilateral or triangular meshes. Properties of lead to almost banded systems, allowing be competitive in high-polynomial regime ($p > 5$). The enables precomputed solution opera...