Some orthogonality preserving operators
W. A. Al‐Salam, Anil Kumar Verma · Proceedings of the American Mathematical Society · 1969
where ao, a,, . . . are constants independent of x. Thus LES. With no loss of generality we shall take in the following ao= 1. Since L is orthogonality preserving, i.e. if {pn(x) } is s.o.p. then {Lpn(x) } is also s.o.p., it is of interest to see what operators in S preserve the orthogonality property of all s.o.p. We are also interested in elements of S which preserve orthogonality of particular s.o.p. In this connection let an s.o.p. {pn(x) } be given. We say that { pn (x) } possesses an orthogonality preserving operator if there is JE S such that { pn(x) } and { JPn(x) } are sets of orthogonal polynomials. We shall prove below the following theorems: