Runge's Phenomenon refers to the unsettling behavior that occurs when using polynomial interpolation at equidistant nodes, where the interpolation can oscillate wildly and deviate significantly from the expected values, especially at the edges of the interval. This phenomenon highlights the challenges and limitations associated with polynomial interpolation, demonstrating that as the number of interpolation points increases, the error can actually worsen rather than improve. It has important implications in approximation theory and is particularly relevant when considering rational approximations.
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