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# Polynomial Approximation

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Polynomial Approximation book

# Polynomial Approximation

DOI link for Polynomial Approximation

Polynomial Approximation book

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## ABSTRACT

Basis functions are the fundamental building blocks for most approximation processes. There are inﬁnitely many choices for basis functions such as polynomials, trigonometric functions, radial basis functions, etc. A central diﬃculty in learning input-output relationships directly from measurements lies in choosing appropriate basis functions and the choice of basis functions unfortunately depends on the characteristics of an unknown input-output map. The problem of choosing an appropriate basis function is diﬃcult since one usually cannot say in advance how complex the input-output map will be, or specify its characteristics. Furthermore, one typically would not have the time or patience to search some handbook of known functions for a set that best represents what we want to study. Hence, we would like to choose building blocks which allow the adaptive construction of input-output maps eﬃciently and quickly. Fortunately, there is frequently a lack of uniqueness, in that many feasible choices exist for basis functions. So the challenge is not an impossible quest. Consider the Stone-Weierstrass theorem which gives one of the most remarkable results in the ﬁeld of approximation theory: Qualitatively, there exists a sequence of polynomials that converge uniformly to any prescribed continuous function on a compact interval. This theorem was ﬁrst stated by Weierstrass for polynomial approximations in 1-D spaces [7] and was later modiﬁed by Stone to generalize it for polynomial approximation in compact 2-D spaces [8-10]. For a general compact space, this theorem can be generalized to N dimensions as follows [11].