Abstract
The goal of this paper is to estimate the rate of convergence of a linear positive operator involving Konhauser polynomials to bounded variation functions on $[0,1]$ . To prove our main result, we have used some methods and techniques of probability theory.
Highlights
In, Konhauser presented the general theory of biorthogonal polynomials [ ]
Afterwards, in [ ], he gave the following pair of biorthogonal polynomials: Yν(n)(x; k) and Zν(n)(x; k) (n >, k ∈ Z+), satisfying
Yν(n)(x; ) = Zν(n)(x; ) = L(νn)(x), where L(νn)(x) are classical Laguerre polynomials and Yν(n)(x; k) Konhauser polynomials given by v
Summary
In , Konhauser presented the general theory of biorthogonal polynomials [ ]. We set mn(x, t; k) = ( – x)(n+ )/k exp t ( – x)– /k – Yυ(n)(t; k)xυ . Our paper concerns the rate of pointwise convergence of the operators given by ( ). By means of the techniques of probability theory, we shall estimate the rate of convergence for the operators (Lnf ) for functions of a bounded variation on [ , ] at points x where f (x+) and f (x–) exist.
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