Handwriting is one of the most natural ways for a human to record knowledge. In recent years this type of human-computer interaction has received increasing attention due to the rapid evolution of digital ink hardware. This thesis contributes to the art of efficient recognition of handwriting and compact storage of digital ink. In the first part of the thesis, we focus on the development of alg...

A new family of wavelets is introduced, which is associated with Legendre polynomials. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. The method for the wavelet construction is derived from the association of ordinary second order differential equations with multiresolution filters. The low-pass filter associated to Legendre multiresolution analysis is ...

Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal sets that aim to minimise the L∞ error interpolating polynomial. The present work combines comprehensive review known approximation theory results, with new numerical experiments, mo...

The Chebyshev polynomials have many beautiful properties and countless applications, arising in a variety of continuous settings. They are a sequence of orthogonal polynomials appearing in approximation theory, numerical integration, and differential equations. In this paper we approach them instead as discrete objects, counting the sum of weighted tilings. Using this combinatorial approach, on...

in this paper, we introduce a family of fractional-order chebyshev functions based on the classical chebyshev polynomials. we calculate and derive the operational matrix of derivative of fractional order $gamma$ in the caputo sense using the fractional-order chebyshev functions. this matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions P−μ ν and Q −μ ν of degrees 0 ≤ ν ≤ 1, 000, 000 and orders −ν ≤ μ ≤ ν on the interval (−1, 1). Our algorithm, which runs in time independent of ν and μ, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expan...

Abstract— A new polynomial method to solve Volterra–Fredholm Integral equations is presented in this work. The method is based upon Shifted Legendre Polynomials. The properties of Shifted Legendre Polynomials and together with Gaussian integration formula are presented and are utilized to reduce the computation of Volterra–Fredholm Integral equations to a system of algebraic equations. Some num...

In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...

Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the co...

In this paper, a super spectral viscosity method using the Chebyshev differential operator of high order Ds = ( √ 1− x2∂x) is developed for nonlinear conservation laws. The boundary conditions are treated by a penalty method. Compared with the second-order spectral viscosity method, the super one is much weaker while still guaranteeing the convergence of the bounded solution of the Chebyshev–Ga...