Approximation of Functions

For a given function u(x) on an interval [a,b] we want to find an approximation p(x) where • either p is a polynomial: p ∈Pn = span{1,x, . . . ,xn} • or p is a trigonometric polynomial: p ∈Tn = span{1,cosx,sinx, . . . ,cos(nx),sin(nx)} We can measure the error in two ways: • (weighted) mean square error ‖u− p‖L2 w 2 = ́ b a (u(x)− p(x)) 2 w(x)dx with a weight function w(x) (e.g., w(x) = 1) • maximum error‖u− p‖∞ = sup x∈[a,b] |u(x)− p(x)|