The Almost Orthogonal Polynomial Sequences

Since the orthogonal polynomials have a lot of applications in mathematics (theory of special functions, spectral operators, probability, analytic numbers and approximations) and other sciences (physics, mechanics, electrotechnics), it appeared numerous extensions of the concept of standard orthogonality. We will discuss the concept of almost orthogonality which is less known, but very useful in modeling of electronic systems which generate orthonormal basis. They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials. Like the most important application we will consider approximations of the functions.