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.
Highlights
This study shows how to derive analytical expressions, called “constrained expressions”, that can be used to represent functions that are subject to a set of linear constraints
This study shows how to derive interpolating expressions subject to a variety of linear constraints, such as functions passing through multiple points with assigned derivatives, or subject to multiple relative constraints, as well as periodic functions subject to multiple point constraints
This study shows how to derive analytical constrained expressions, representing all functions subject to a set of assigned linear constraints
Summary
This study shows how to derive analytical expressions, called “constrained expressions”, that can be used to represent functions that are subject to a set of linear constraints. [6], where least-squares solutions of initial and boundary value problems applied to linear nonhomogeneous differential equations of any order and with nonconstant coefficients are obtained These constrained expressions are provided in terms of a new function, g(x), which is completely free to choose. These interpolating expressions are useful when solving linear differential equations because they can be rewritten using functions with embedded constraints (a.k.a., the “subject to” conditions). Constraints in one point; constraints in two and in n points; multiple linear constraints; relative constraints; constraints on continuous and discontinuous periodic functions
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