Recovering orthogonality from quasi‐type kernel polynomials using specific spectral transformations

Abstract

In this work, the concept of quasi‐type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi‐type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This leads to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited. Moreover, examples are discussed for obtaining the orthogonality of quasi‐type kernel polynomials of order 1.