( g {\displaystyle w(x)={\sqrt {1-x^{2}}}} whenever = , eigenfunctions), leading to generalized Fourier series. x f [ As well as the different function notation, note that the parentheses have been replaced by angled brackets < >. {\displaystyle \{1,x,x^{2},\dots \}} − It’s technically correct to use them, but I’m leaving them out here so that the following example is accessible to calculus students without any background in set theory or linear algebra. It is worth noting that because of the weight function ˆbeing the Jacobian of the change of variable to polar coordinates, Bessel functions that are scaled as in the above orthogonality relation are also orthogonal with respect to the unweighted scalar product over a circle of radius a. Orthogonal Function Sequences. m is of functions of L2-norm one, forming an orthonormal sequence. Orthogonal Functions -Orthogonal Functions -DDefinitionefinition ... another Example ... f (x) =x2, 0