Lagrange polynomials over Clifford numbers

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ0,2, or to the real Clifford algebra ℝ0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ0,0 ≃ ℝ, ℝ0,1 ≃ ℂ and the trivial case ℝ1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝp,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

Solution of Nonlinear Volterra-Fredholm Integrodifferential Equations via Hybrid of Block-Pulse Functions and Lagrange Interpolating Polynomials

An efficient hybrid method is developed to approximate the solution of the high-order nonlinear Volterra-Fredholm integro-differential equations. The properties of hybrid functions consisting of block-pulse functions and Lagrange interpolating polynomials are first presented. These properties are then used to reduce the solution of the nonlinear Volterra-Fredholm integro-differential equations to the solution of algebraic equations whose solution is much more easier than the original one. The validity and applicability of the proposed method are demonstrated through illustrative examples. The method is simple, easy to implement and yields very accurate results.

On approximation by trigonometric Lagrange interpolating polynomials II

We show that trigonometric Lagrange interpolating approximation with arbitrary real distinct nodes in Lp space for 1 ≤ p < ∞, as that with equally spaced nodes in Lp space for 1 < p < ∞ in an earlier paper by T.F. Xie and S.P. Zhou, may also be arbitrarily “bad”. This paper is a continuation of this earlier work by Xie and Zhou, but uses a different method.

On approximation by trigonometric Lagrange interpolating polynomials

It is well-known that the approximation to f(x) ∈ C2π, by nth trigonometric Lagrange interpolating polynomials with equally spaced nodes in C2π, has an upper bound In(n)En(f), where En(f) is the nth best approximation of f(x). For various natural reasons, one can ask what might happen in Lp space? The present paper indicates that the result about the trigonometric Lagrange interoplating approximation in Lp space for 1 < p < ∞ may be “bad” to an arbitrary degree.