On the Location of the Zeros of Certain Orthogonal Functions1
نویسندگان
چکیده
Let R be a finite region of the s-plane and let L2(R) denote the class of functions f(z) each analytic in R with //b|/(8)| 2dS< °°. Let the points ßi, ß2, ß3, • • • be given interior to R and let 0„(z) be that function of class L2(R) for which n(ßi) =n(ß2) = • • • =n(ßn-i) =0, n(ßn) = 1, and which minimizes ffs\ <¡>n(z) \ 2dS over the class L2(R). If the points ßi, ß2, ß3, ■ ■ ■ are not all distinct these requirements of interpolation on „(z) are to be interpreted in the usual way in the theory of interpolation, to refer to the vanishing of suitable derivatives of 4>n(z) in multiple points ßk. The purpose of this note is to establish results on the location of zeros of the functions <£„(z). Our main result is the
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