We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theo...

The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors) into an orthonormal basis (a set of orthogonal, unit-length vectors). The process consists of taking each vector and then subtracting the elements in common with the previous vectors. This paper introduces an Enhanced version of the Gram-Schmidt Process (EGSP) with inverse, which is ...

In the present paper, we discuss a method to design a linear phase 1-dimensional Infinite Impulse Response (IIR) filter using orthogonal polynomials. The filter is designed using a set of object functions. These object functions are realized using a set of orthogonal polynomials. The method includes placement of zeros and poles in such a way that the amplitude characteristics are not changed wh...

Here we propose an algorithm, named generalized orthogonal components regression (GOCRE), to explore the relationship between a categorical outcome and a set of massive variables. A set of orthogonal components are sequentially constructed to account for the variation of the categorical outcome, and together build up a generalized linear model (GLM). This algorithm can be considered as an exten...

Abstract. An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges’ sense [Numer. Math. 67 (1994), 271–288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonome...