# The Theory of Connections. Connecting Points

## TL;DR

This paper presents a mathematical framework for constructing functions that inherently satisfy specified linear constraints, enabling more efficient solutions to differential equations and interpolation problems.

## Contribution

It introduces a general method to derive explicit functions with embedded linear constraints, including point and derivative conditions, applicable to differential equations and interpolation.

## Key findings

- Derived general expressions for constrained functions passing through points.
- Introduced a generalized form of Waring's interpolation for constrained points.
- Provided a procedure to compute coefficients using matrix inversion.

## Abstract

This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with embedded specific constraints. The paper first shows how to express the most general explicit function passing through a single point in three distinct ways: linear, additive, and rational. Then, functions with constraints on single, two, or multiple points are introduced as well as those satisfying relative constraints. This capability allows to obtain general expressions to solve linear differential equations with no need to satisfy constraints (the "subject to:" conditions) as the constraints are already embedded in the constrained expression. In particular, for expressions passing through a set of points, a generalization of the Waring's interpolation form, is introduced. The general form of additive constrained expressions is introduced as well as a procedure to derive its coefficient functions, requiring the inversion of a matrix with dimensions as the number of constraints.