This morning I posted some notes on orthogonal polynomials and Gaussian quadrature.
“Orthogonal” just means perpendicular. So how can two polynomials be perpendicular to each other? In geometry, two vectors are perpendicular if and only if their dot product of their coordinates is zero. In more general settings, two things are said to be orthogonal if their inner product (generalization of dot product) is zero. So what was a theorem in basic geometry is taken as a definition in other settings. Typically mathematicians say “orthogonal” rather than “perpendicular.” The basic idea of lines meeting at right angles acts as a reliable guide to intuition in more general settings.
Two polynomials are orthogonal if their inner product is zero. You can define an inner product for two functions by integrating their product, sometimes with a weighting function.
Orthogonal polynomials have remarkable properties that are easy to prove. Last week I posted some notes on Chebyshev polynomials. The notes posted today include Chebyshev polynomials as a special case and focus on the application of orthogonal polynomials to quadrature. (“Quadrature” is just an old-fashioned word for integration, usually applied to numerical integration in one dimension.) It turns out that every class of orthogonal polynomials corresponds to an integration rule.