Uniform Approximation, Bernstein Polynomials (Real Analysis)
Motivation for Bernstein Polynomials:
A canonical method to approximating a function is by polynomial interpolation at a specified number of points. We study the asymptotic behavior of particular interpolation schemes in Math 128A, Numerical Analysis. However, we note that interpolation can yield wildly inaccurate results near the endpoints of interpolation (Runge Phenomenon), so we may instead want a sequence of polynomials that approximates our desired continuous function uniformly.
As it turns out in common occurrences, the requirement of having our true function continuous is not a difficult hypothesis to meet, and in fact, Bernstein polynomials perform quite well in regards to function approximations. For Math 104, Real Analysis, our textbook Ross develops the construction of Bernstein polynomials as a 'concrete' proof of (example for) the Weierstrass Approximation Theorem.
Read more about Bernstein Polynomials here (.pdf lecture notes).
Math 104 : Real Analysis
Textbook: Ross
Lecturer: Max Wimberley
Lecture notes for this course are provided below after homeworks; however, they are depreciated and incomplete due to the fact this class is structured to be fully encompassed within the Ross textbook. All theorems, in-class discussions and examples are purely out of the textbook. It turned out that notes taken in lecture fell short of textbook contents, completely defeating the purpose of lecture notes, when there already exist numerous valuable resources devoted to the subject of analysis (and calculus).