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Mathematics 2017, 5(4), 57; https://doi.org/10.3390/math5040057

The Theory of Connections: Connecting Points

Aerospace Engineering, Texas A & M University, College Station, TX 77843-3141, USA
This paper is an extended version of our paper published in Mortari, D. The Theory of Connections. Part 1: Connecting Points, AAS 17-255, 2017 AAS/AIAA Space Flight Mechanics Meeting Conference, San Antonio, TX, USA, 5–9 February 2017; (dedicated to John Lee Junkins).
Received: 30 July 2017 / Revised: 17 September 2017 / Accepted: 24 October 2017 / Published: 1 November 2017
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Abstract

This study introduces a procedure to obtain all interpolating functions, y = f ( x ) , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through a single point in three distinct ways: linear, additive, and rational. Then, using the additive formalism, interpolating functions with linear constraints on one, two, and n points are introduced as well as those satisfying relative constraints. In particular, for expressions passing through n points, a generalization of the Waring’s interpolation form is introduced. An alternative approach to derive additive constraint interpolating expressions is introduced requiring the inversion of a matrix with dimensions equally the number of constraints. Finally, continuous and discontinuous interpolating periodic functions passing through a set of points with specified periods are provided. This theory has already been applied to obtain least-squares solutions of initial and boundary value problems applied to nonhomogeneous linear differential equations with nonconstant coefficients. View Full-Text
Keywords: interpolation; linear constraints; embedded constraints interpolation; linear constraints; embedded constraints
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Mortari, D. The Theory of Connections: Connecting Points. Mathematics 2017, 5, 57.

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