Orthogonal Systems and Convolution Operators

1. Front Matter

   Pages i-xvi

   PDF

2. Orthogonal Polynomials and Krein’s Theorem

   • Robert L. Ellis, Israel Gohberg

   Pages 1-27

3. Reformulations of Krein’s Theorem

   • Robert L. Ellis, Israel Gohberg

   Pages 29-36

4. Inner Products on Modules and Orthogonalization with Invertible Squares

   • Robert L. Ellis, Israel Gohberg

   Pages 37-51

5. Orthogonal Matrix Polynomials

   • Robert L. Ellis, Israel Gohberg

   Pages 53-70

6. Special Class of Block Toeplitz Matrices

   • Robert L. Ellis, Israel Gohberg

   Pages 71-86

7. Orthogonal Operator-Valued Polynomials: First Generalization

   • Robert L. Ellis, Israel Gohberg

   Pages 87-98

8. Convolution Equations on a Finite Interval

   • Robert L. Ellis, Israel Gohberg

   Pages 99-114

9. Continuous Analogues of Orthogonal Matrix Polynomials

   • Robert L. Ellis, Israel Gohberg

   Pages 115-142

10. Orthogonal Operator-Valued Polynomials

    • Robert L. Ellis, Israel Gohberg

    Pages 143-161

11. Reverse, Left and Right Orthogonalization

    • Robert L. Ellis, Israel Gohberg

    Pages 163-182

12. Discrete Infinite Analogue of Krein’s Theorem

    • Robert L. Ellis, Israel Gohberg

    Pages 183-203

13. Continuous Infinite Analogue of Krein’s Theorem

    • Robert L. Ellis, Israel Gohberg

    Pages 205-226

14. Back Matter

    Pages 227-236

    PDF

In this book we study orthogonal polynomials and their generalizations in spaces with weighted inner products. The impetus for our research was a deep theorem due to M.G. Krein along with subsequent results of Krein and H. Langer. Together with our colleagues, we have worked in this area for nearly fifteen years, and the results of our research are presented here in unified form. We are grateful to the Department of mathematics at the University of Maryland in College Park and to Tel-Aviv University for their support and encouragement. The support of the Silver Family Foundation is also highly appreciated. Introduction The starting point ofthis book is a study ofthe orthogonal polynomials {qn In ?: O} obtained by orthogonalizing the power functions I, Z, z2, ... on the unit circle. The orthogonality is with respect to the scalar product defined by where the weight w is a positive integrable function on the unit circle. These ortho­ gonal polynomials are called the Szego polynomials associated with the weight w.

• C*-algebra
• Operator theory
• convolution
• distribution
• orthogonal polynomials

• Department of Mathematics, University of Maryland, College Park, USA

  Robert L. Ellis

• School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, IL - Ramat Aviv, Israel

  Israel Gohberg

Policies and ethics