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In this chapter, we study approximation of a function \(f\in C[a,\, b]\) by another simpler function \(\phi \in C[a,\, b]\) in a general manner. The function \(\phi \) is mostly considered a polynomial of certain degree n. In some cases, \(\phi \) as a rational function is also considered. Interestingly, this deeply theoretical development originated in Chebyshev’s work on mechanisms of machines (see the text of Steffens cited in the Bibliography); but we confine to a few significant results. These theoretical results have contemporarily been put to good use in computer software for efficiently computing transcendental mathematical functions, and obtaining easy to evaluate functional forms that approximate complicated expressions appearing in theoretical and empirical experimental studies.