Weight functions are used in orthogonal polynomials in an effort to normalize the polynomials in the equation. For example, if you have an absolutely continuous equation, such as Dalpha(x) = W(x)dx, the letter W represents the weight function in the orthogonal polynomial equation. In this specific instance, the weight function will be multiplied by the x variable to get a weighted and more normalized answer to the equation. The x variable in this equation must be a non-negative number between zero and infinity to get an accurate answer to the equation.
A classical weight function refers to the precise way in which the equation utilizing the classical weight function is written. If your orthogonal polynomial equation is written with a classical weight function, it will be written in terms of w. This means that you must rearrange your basic equation to isolate w on one side of the equation. The following orthogonal polynomial equation is referred to as Shepard's Method: F(x,y)= w(i)f(i). To write this equation as a classical weight function it must be rearranged as this: w(i)= hi÷hjf(i), where both hi and hj are known coefficients.
While classical weight functions are easier to solve and are more consistently used with orthogonal polynomials, non-classical weight functions are also utilized in certain types of quadratic and theoretical equations. A non-classical weight function is structured in the same way as a classical weight function, in that the constant w is utilized throughout the equation. Despite this, the coefficients in the equation are unknown. With reference to the previous equation, hi and hj become variables when used as non-classical weight functions.
While weight functions can be consistently infused into a traditional orthogonal polynomial, some equations are more well known and revered then others. Some of the more famous examples of orthogonal polynomials, referred to as classical polynomials, are the Wilson polynomials, Jacobi polynomials, continuous Hahn polynomials and the ultraspherical polynomials and Meixner polynomials. Some types of discrete orthogonal polynomials are a finite sequence, which is a direct result of the weight function that is integrated into the equation.