Approximation theory is a branch of mathematics that focuses on how functions can be approximated by simpler or more manageable functions. It seeks to find the best way to represent complex functions using polynomials, trigonometric functions, or other simpler forms, making calculations easier and more efficient. This field is important in various applications, including numerical analysis, computer graphics, and data fitting. Techniques from approximation theory help in optimizing algorithms and improving the accuracy of models used in fields like engineering, physics, and statistics.