Orthogonal Polynomials And Special Functions

by Marcellan, Francisco; Van Assche, Walter

• ISBN13: 9783540310624
• ISBN10: 3540310622
• Format: Paperback
• Copyright: 2006-08-30
• Publisher: Springer Verlag

Summary

Special functions and orthogonal polynomials in particular have been around for centuries. Can you imagine mathematics without trigonometric functions, the exponential function or polynomials? In the twentieth century the emphasis was on special functions satisfying linear differential equations, but this has now been extended to difference equations, partial differential equations and non-linear differential equations. The present set of lecture notes containes seven chapters about the current state of orthogonal polynomials and special functions and gives a view on open problems and future directions. The topics are: computational methods and software for quadrature and approximation, equilibrium problems in logarithmic potential theory, discrete orthogonal polynomials and convergence of Krylov subspace methods in numerical linear algebra, orthogonal rational functions and matrix orthogonal rational functions, orthogonal polynomials in several variables (Jack polynomials) and separation of variables, a classification of finite families of orthogonal polynomials in Askeya??s scheme using Leonard pairs, and non-linear special functions associated with the Painlev?? equations.

Author Biography

Francisco Marcell+ín is full professor of applied mathematics at the Universidad Carlos III de Madrid in Legan+¬s (Spain) where he served as vice rector of research. In the past he has been teaching at the universities of Zaragoza, Santiago de Compostela and the Polyt+¬cnica de Madrid and was recently a visiting professor at the Georgia Institute of Technology. He was program director of the SIAM activity group on Orthogonal Polynomials and Special Functions from 1999 to 2005. Presently he is the director of the Spanish National Agency for Quality Assurance and Accreditation (ANECA). He has several publications in mathematical analysis (most particularly special functions and approximation) and linear algebra. Walter Van Assche is full professor of mathematics at the Katholieke Universiteit Leuven in Belgium and was a research director of the Belgian National Fund for Scientific Research. He was vice chair of the SIAM activity group on Orthogonal Polynomials and Special Functions from 1999 to 2005 and is on the editorial board of the Journal of Approximation Theory and the Journal of Difference Equations and Applications. He is the author of Asymptotics for Orthogonal Polynomials (Lecture Notes in Mathematics 1265) and wrote two chapters in M.E.H. Ismail's book Classical and Quantum Orthogonal Polynomials in one Variable.

Table of Contents

Orthogonal Polynomials, Quadrature, and Approximation: Computational Methods and Software (in Matlab)
Walter Gautschi	1	(78)
1 Introduction	2	(2)
2 Orthogonal Polynomials	4	(26)
2.1 Recurrence Coefficients	4	(4)
2.2 Modified Chebyshev Algorithm	8	(2)
2.3 Discrete Stieltjes and Lanczos Algorithm	10	(2)
2.4 Discretization Methods	12	(3)
2.5 Cauchy Integrals of Orthogonal Polynomials	15	(3)
2.6 Modification Algorithms	18	(12)
3 Sobolev Orthogonal Polynomials	30	(6)
3.1 Sobolev Inner Product and Recurrence Relation	30	(1)
3.2 Moment-Based Algorithm	31	(1)
3.3 Discretization Algorithm	32	(1)
3.4 Zeros	33	(3)
4 Quadrature	36	(21)
4.1 Gauss-Type Quadrature Formulae	36	(4)
4.2 Gauss—Kronrod Quadrature	40	(2)
4.3 Gauss—Turan Quadrature	42	(1)
4.4 Quadrature Formulae Based on Rational Functions	43	(2)
4.5 Cauchy Principal Value Integrals	45	(2)
4.6 Polynomials Orthogonal on Several Intervals	47	(3)
4.7 Quadrature Estimates of Matrix Functionals	50	(7)
5 Approximation	57	(19)
5.1 Polynomial Least Squares Approximation	57	(6)
5.2 Moment-Preserving Spline Approximation	63	(5)
5.3 Slowly Convergent Series	68	(8)
References	76	(3)
Equilibrium Problems of Potential Theory in the Complex Plane
Andrei Martinez Finkelshtein	79	(40)
1 Background	80	(2)
1.1 Introduction	80	(1)
1.2 Background or What You Should Bring to Class	80	(2)
2 Logarithmic Potentials: Definition and Properties	82	(6)
2.1 Superharmonic Functions	82	(2)
2.2 Definition of the Logarithmic Potential	84	(1)
2.3 Some Principles for Potentials	85	(1)
2.4 Recovering a Measure from its Potential	86	(2)
3 Energy and Equilibrium	88	(20)
3.1 Logarithmic Energy	88	(2)
3.2 Extremal Problem, Equilibrium Measure and Capacity	90	(6)
3.3 Link with Conformal Mapping and Green Function	96	(4)
3.4 Equilibrium in an External Field	100	(6)
3.5 Other Equilibrium Problems. Equilibrium with Constraints	106	(2)
4 Two Applications	108	(8)
4.1 Analytic Properties of Polynomials	108	(4)
4.2 Complex Dynamics	112	(4)
5 Conclusions, or What You Should Take Home	116	(1)
References	116	(3)
Discrete Orthogonal Polynomials and Superlinear Convergence of Krylov Subspace Methods in Numerical Linear Algebra
Bernhard Beckermann	119	(68)
1 Background in Numerical Linear Algebra	120	(18)
1.1 Introduction	120	(4)
1.2 Conjugate Gradients, Lanczos. and Ritz Values	124	(2)
1.3 Krylov Subspace Methods and Discrete Orthogonal Polynomials: Non Symmetric Data	126	(5)
1.4 Krylov Subspace Methods and Discrete Orthogonal Polynomials: Symmetric Data	131	(7)
2 Extremal Problems in Complex Potential Theory and nth Root Asymptotics of OP	138	(18)
2.1 Energy Problems with External Field	138	(7)
2.2 Energy Problems with Constraint and External Field	145	(4)
2.3 Asymptotics for Discrete Orthogonal Polynomials	149	(7)
3 Consequences	156	(26)
3.1 Applications to the Rate of Convergence of CG	156	(9)
3.2 Applications to the Rate of Convergence of Ritz Values	165	(6)
3.3 Circulants, Toeplitz Matrices and their Cousins	171	(5)
3.4 Discretization of Elliptic PDE's	176	(4)
3.5 Conclusions	180	(2)
References	182	(5)
Orthogonal Rational Functions on the Unit Circle: from the Scalar to the Matrix Case
Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, Olav Njåstad	187	(42)
1 Motivation: Why Orthogonal Rational Functions?	188	(3)
1.1 Linear Prediction	189	(1)
1.2 Krylov Subspace Methods	190	(1)
1.3 Numerical Quadrature	191	(1)
2 Orthogonal Rational Functions on the Unit Circle	191	(11)
2.1 Preliminaries	191	(3)
2.2 The Fundamental Spaces	194	(1)
2.3 Reproducing Kernels	195	(2)
2.4 Recurrence Relations	197	(5)
3 Quadrature and Interpolation	202	(4)
3.1 Quadrature	202	(1)
3.2 Interpolation	203	(2)
3.3 Interpolation and Quadrature Using the Kernels	205	(1)
4 Density and the Proof of Favard's Theorem	206	(1)
4.1 Density	206	(1)
4.2 Proof of Favard's Theorem	207	(1)
5 Convergence	207	(4)
5.1 Orthogonal Polynomials w.r.t. Varying Measures	207	(1)
5.2 Szego's Condition and Convergence	208	(3)
6 Szego's Problem	211	(1)
7 Hilbert Modules and Hardy Spaces	212	(2)
7.1 Inner Products and Norms	213	(1)
7.2 Caratheodory Function and Spectral Factor	214	(1)
8 MORF and Reproducing Kernels	214	(5)
8.1 Orthogonal Rational Functions	214	(1)
8.2 Reproducing Kernels	215	(4)
9 Recurrence for the MORF	219	(33)
9.1 The Recursion	219	(3)
9.2 Functions of the Second Kind	222	(1)
10 Interpolation and Quadrature	223	(2)
10.1 The Kernels	223	(1)
10.2 The MORF	224	(1)
11 Minimalisation and Szego's Problem	225	(1)
12 What We did not Discuss	226	(1)
References	227	(2)
Orthogonal Polynomials and Separation of Variables
Vadim B. Kuznetsov	229	(26)
1 Chebyshev Polynomials	230	(5)
1.1 Pafnuty Lvovich Chebyshev	230	(1)
1.2 Notation and Standard Formulae	230	(2)
1.3 Polynomials Tn(x) and Un(x)	232	(2)
1.4 Orthogonality	234	(1)
1.5 Other Results	234	(1)
1.6 Least Possible Deviation from Zero	235	(1)
2 Gegenbauer Polynomials	235	(6)
2.1 Leopold Bernhard Gegenbauer	235	(1)
2.2 Polynomials C(g)n(x)	236	(2)
2.3 Beta Integral and Elliptic Coordinates	238	(1)
2.4 Q-Operator	239	(2)
3 Monomial and Elementary Symmetric Functions	241	(11)
3.1 Definitions	241	(1)
3.2 Factorizing Symmetric Polynomials Eλ1,λ2(χ1, χ2)	242	(2)
3.3 Factorizing the Basis Eλ(χ1,...,χn)	244	(3)
3.4 Lionville Integrable Systems	247	(1)
3.5 Separation of Variables	248	(1)
3.6 Factorizing Symmetric Monomials mλ1,λ2 (χ1, χ2)	249	(1)
3.7 Factorizing the Basis mλ (χ1,...,χx)	250	(2)
References	252	(3)
An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials
Paul Terwilliger	255	(76)
1 Leonard Pairs and Leonard Systems	256	(6)
1.1 Leonard Pairs	256	(1)
1.2 An Example	257	(1)
1.3 Leonard Systems	258	(3)
1.4 The D4 Action	261	(1)
2 The Structure of a Leonard System	262	(9)
2.1 The Antiautomorphism †	265	(1)
2.2 The Scalars ai, χi	266	(2)
2.3 The Polynomials pi	268	(2)
2.4 The scalars v,mi	270	(1)
3 The Standard Basis	271	(8)
3.1 The Scalars bi, ci	272	(2)
3.2 The Scalars ki	274	(1)
3.3 The Polynomials upsiloni	275	(2)
3.4 The Polynomials ui	277	(1)
3.5 A Bilinear Form	277	(2)
4 Askey-Wilson Duality	279	(4)
4.1 The Three-Term Recurrence and the Difference Equation	280	(1)
4.2 The Orthogonality Relations	281	(1)
4.3 The Matrix P	282	(1)
5 The Split Decomposition	283	(9)
5.1 The Split Basis	287	(1)
5.2 The Parameter Array and the Classifying Space	287	(2)
5.3 Everything in Terms of the Parameter Array	289	(3)
6 The Terminating Branch of the Askey Scheme	292	(3)
7 Applications and Related Topics	295	(5)
7.1 A Characterization of Leonard Systems	295	(3)
7.2 Leonard Pairs A, A* with A Lower Bidiagonal and A* Upper Bidiagonal	298	(1)
7.3 Leonard Pairs A, A* with A Tridiagonal and A* Diagonal	298	(1)
7.4 Characterizations of the Parameter Arrays	299	(1)
8 The Askey-Wilson Relations	300	(4)
8.1 Leonard Pairs and the Lie Algebra sl2	301	(1)
8.2 Leonard Pairs and the Quantum Algebra Uq(sl2)	302	(1)
8.3 Leonard Pairs in Combinatorics	303	(1)
9 Tridiagonal Pairs	304	(4)
10 Appendix: List of Parameter Arrays	308	(11)
11 Suggestions for Further Research	319	(7)
References	326	(5)
Painlevé Equations Nonlinear Special Functions
Peter A. Clarkson	331	(82)
1 Introduction	333	(2)
2 Inverse Problems for the Painleve Equations	335	(3)
2.1 Integral Equations	335	(2)
2.2 Isomonodromy Problems	337	(1)
3 Hamiltonian Structure	338	(2)
4 Bäcklund Transformations	340	(7)
4.1 Bäcklund Transformations for PHII	341	(1)
4.2 Bäcklund Transformations for PIII	342	(1)
4.3 Bäcklund Transformations for PIV	343	(1)
4.4 Bäcklund Transformations for PV	344	(1)
4.5 Bäcklund Transformations for PVI	345	(1)
4.6 Affine Weyl Groups	346	(1)
5 Rational Solutions	347	(20)
5.1 Rational Solutions of PII	347	(1)
5.2 The Yablonskii-Vorob'ev Polynomials	347	(7)
5.3 Determinantal Representation of Rational Solutions of PII	354	(4)
5.4 Rational Solutions of PII	358	(2)
5.5 Rational Solutions of PIV	360	(4)
5.6 Rational Solutions of PV	364	(3)
5.7 Rational Solutions of PVI	367	(1)
6 Other Elementary Solutions	367	(4)
6.1 Elementary Solutions of PIII	367	(2)
6.2 Elementary Solutions of PI	369	(1)
6.3 Elementary Solutions of PVI	370	(1)
7 Special Function Solutions	371	(9)
7.1 Special Function Solutions of PII	373	(2)
7.2 Special Function Solutions of PIII	375	(1)
7.3 Special Function Solutions of PIV	376	(2)
7.4 Special Function Solutions of PV	378	(1)
7.5 Special Function Solutions of PVI	379	(1)
8 Other Mathematical Properties	380	(4)
8.1 Hirota Bilinear Forms	380	(1)
8.2 Coalescence Cascade	380	(2)
8.3 The PII Hierarchy	382	(2)
9 Asymptotic Expansions and Connection Formulae	384	(8)
9.1 First Painlevé Equation	384	(3)
9.2 Second Painlevé Equation	387	(1)
9.3 Connection Formulae for PII	388	(4)
10 Applications of Painlevé Equations	392	(8)
10.1 Reductions of Partial Differential Equations	392
10.2 Combinatorics	390	(8)
10.3 Orthogonal Polynomials	398	(2)
11 Discussion	400	(1)
References	400	(13)
Index	413