Exponentially-Convergent Strategies for Defeating the Runge Phenomenon for the Approximation of Non-Periodic Functions, Part I: Single-Interval Schemes

Approximating a function from its values f(x_i) at a set of evenly spaced points x_i through (N+1)-point polynomial interpolation often fails because of divergence near the endpoints, the "Runge Phenomenon". Here we briefly describe seven strategies, each employing a single polynomial over the entire interval, to wholly or partially defeat the Runge Phenomenon such that the error decreases exponentially fast with N. Each is successful in obtaining high accuracy for Runge's original example. Unfortunately, each of these single-interval strategies also has liabilities including, depending on the method, various permutations of inefficiency, ill-conditioning and a lack of theory. Even so, the Fourier Extension and Gaussian RBF methods are worthy of further development.