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Zeros of classical orthogonal polynomials in two variables Lidia Fern andez Joint work with Antonia M. Delgado, Teresa E. P erez & Miguel A. Pi nar Optimal points configuration and orthogonal polynomials April 2017, Castro Urdiales

  1. Background Zeros of orthogonal polynomials in two variables Common zeros of orthogonal polynomials in two variables (Y. Xu) Let P n = { P n , k : 0 ≤ k ≤ n } denotes a basis of V n . A common zero of P n is a zero for every polynomial P n , k , 0 ≤ k ≤ n . It can be considered as a zero of the subspace V n . Properties: All common zeros of P n are points in R 2 . All zeros of P n are distinct and simple. P n and P n − 1 do not have common zeros. They are related with the joint eigenvalues of some truncated block Jacobi matrices.

  2. Background Zeros of orthogonal polynomials in two variables Common zeros of orthogonal polynomials in two variables (Y. Xu) Let P n = { P n , k : 0 ≤ k ≤ n } denotes a basis of V n . A common zero of P n is a zero for every polynomial P n , k , 0 ≤ k ≤ n . It can be considered as a zero of the subspace V n . Properties: All common zeros of P n are points in R 2 . All zeros of P n are distinct and simple. P n and P n − 1 do not have common zeros. They are related with the joint eigenvalues of some truncated block Jacobi matrices. Not every inner product has orthogonal polynomials with common zeros. In fact, there are not common zeros in centrally symmetric inner products such as on the unit disk.

  3. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk: Hermite 1865 Ch. Hermite, Sur quelques dev´ eloppments en s´ erie de fonctions de plusieurs variables , Comptes Rendus de l’Acad´ emie des Sciences, 1865.

  4. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk: Hermite 1865

  5. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk: Hermite 1865

  6. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk: Hermite 1865

  7. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Ch. Hermite, Sur quelques dev´ eloppments en s´ erie de fonctions de plusieurs variables , Comptes Rendus de l’Acad´ emie des Sciences, 1865.

  8. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Ch. Hermite, Sur quelques dev´ eloppments en s´ erie de fonctions de plusieurs variables , Comptes Rendus de l’Acad´ emie des Sciences, 1865. W. Tramm, Geometrische Diskussion des Hermite’schen Polynoms , Inaugural disertation, Zurich, 1908. Ch. Willigens, Sur les polynˆ omes U m , n , Nouvelles Annales de Math´ ematiques , 1911.

  9. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Ch. Hermite, Sur quelques dev´ eloppments en s´ erie de fonctions de plusieurs variables , Comptes Rendus de l’Acad´ emie des Sciences, 1865. W. Tramm, Geometrische Diskussion des Hermite’schen Polynoms , Inaugural disertation, Zurich, 1908. Ch. Willigens, Sur les polynˆ omes U m , n , Nouvelles Annales de Math´ ematiques , 1911. P . Appell and J. Kamp´ e de F´ eriet, Fonctions Hyperg´ eom´ etriques et HyperSph´ eriques. Polynomes d’Hermite , Gauthier-Villars, Paris, 1926.

  10. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ

  11. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ Four different bases:

  12. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ Four different bases: U m , n Orthogonal basis obtained by the Rodrigues formula.

  13. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ Four different bases: U m , n Orthogonal basis obtained by the Rodrigues formula. P n , k Orthogonal basis obtained by Koornwinder construction.

  14. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ Four different bases: U m , n Orthogonal basis obtained by the Rodrigues formula. P n , k Orthogonal basis obtained by Koornwinder construction. P n Orthogonal basis in polar coordinates. j , i

  15. Zeros of orthogonal polynomial on the disk Orthogonal polynomials on the disk Inner product on B 2 := { ( x , y ) ∈ R 2 : x 2 + y 2 � 1 } � f , g � µ = 1 � B 2 f ( x , y ) g ( x , y ) ( 1 − x 2 − y 2 ) µ dx , µ > − 1 ω µ Four different bases: U m , n Orthogonal basis obtained by the Rodrigues formula. P n , k Orthogonal basis obtained by Koornwinder construction. P n Orthogonal basis in polar coordinates. j , i V m , n Orthogonal basis biorthogonal with U m , n .

  16. Zeros of orthogonal polynomial on the disk Rodrigues formula Basis U m , n For m ≥ 0, n ≥ 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n

  17. Zeros of orthogonal polynomial on the disk Rodrigues formula Basis U m , n For m ≥ 0, n ≥ 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n deg ( U m , n ) = m + n

  18. Zeros of orthogonal polynomial on the disk Rodrigues formula Basis U m , n For m ≥ 0, n ≥ 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n deg ( U m , n ) = m + n { U n − k , k ( x , y ) : 0 ≤ k ≤ n } basis not mutually orthogonal of V n .

  19. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 ∂ m + n 1 ( 1 − x 2 − y 2 ) m + n + µ � � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n 1.5 1.5 1.0 1.0 0.5 0.5 0.0 , 0.0   - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 1

  20. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 , 0.0 , 0.0   - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 2

  21. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n 1.5 1.5 1.0 1.0 0.5 0.5 0.0 , 0.0 ,  - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 ,  - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 3

  22. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 , ,  - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 , 0.0 , 0.0  - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 4

  23. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 ∂ m + n 1 � ( 1 − x 2 − y 2 ) m + n + µ � U m , n ( x , y ) = ( 1 − x 2 − y 2 ) µ ∂ x m ∂ y n 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 , 0.0 , 0.0 ,  - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 , 0.0 , 0.0  - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 5

  24. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n , for µ = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 , 0.0 ,  - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 1.5 1.5 1.0 1.0 0.5 0.5 0.0 , 0.0  - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 Degree 15 (Some of them)

  25. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 is generally composed of multiple curves.

  26. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of U m , n = 0 when m or n are odd.

  27. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of U m , n = 0 when m or n are odd. The curves are inside the unit disk (except the axes).

  28. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of U m , n = 0 when m or n are odd. The curves are inside the unit disk (except the axes). Every variety U m , n = 0 is composed of closed curves symmetric with respect to the axes, centered on the origin and concentric.

  29. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of U m , n = 0 when m or n are odd. The curves are inside the unit disk (except the axes). Every variety U m , n = 0 is composed of closed curves symmetric with respect to the axes, centered on the origin and concentric. The only multiple points (except for the axes) coming from the intersection of several branches are ( 0 , − 1 ) , ( 0 , 1 ) , ( − 1 , 0 ) y ( 1 , 0 ) and they only appear for n ≥ 4 or m ≥ 4.

  30. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U 5 , 4 = 0 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  31. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , 0 = 0 is decomposed in [ m 2 ] ellipses whose axis of symmetry are the coordinates axes.

  32. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , 0 = 0 is decomposed in [ m 2 ] ellipses whose axis of symmetry are the coordinates axes. The semi-major axis is of size 1 and it is on the axis OY.

  33. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , 0 = 0 is decomposed in [ m 2 ] ellipses whose axis of symmetry are the coordinates axes. The semi-major axis is of size 1 and it is on the axis OY. The semi-minor axis is on OX and its size is given by one of the roots of the Gegenbauer polynomial P ( µ,µ ) . m

  34. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , 0 = 0 is decomposed in [ m 2 ] ellipses whose axis of symmetry are the coordinates axes. The semi-major axis is of size 1 and it is on the axis OY. The semi-minor axis is on OX and its size is given by one of the roots of the Gegenbauer polynomial P ( µ,µ ) . m U 8 , 0 = 0 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  35. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n U m , n = 0 intersects a line parallel to OX in m points and a line parallel to OY in n points. U 5 , 4 = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  36. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n Every straight line (not the axes) passing through the origin intersects U m , n = 0 in n + m real points. U 5 , 4 = 0 U 5 , 5 = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  37. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n : Interlacing U 5,0 = 0 U 4,0 = 0 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  38. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n : Interlacing U 5,5 = 0 U 5,4 = 0 U 5,5 = 0 U 4,5 = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  39. Zeros of orthogonal polynomial on the disk Rodrigues formula Zeros of U m , n : Density 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  40. Zeros of orthogonal polynomial on the disk Koornwinder basis Basis P n , k For 0 ≤ k ≤ n � � y ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) √ n − k k 1 − x 2

  41. Zeros of orthogonal polynomial on the disk Koornwinder basis Basis P n , k For 0 ≤ k ≤ n � � y ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) √ n − k k 1 − x 2

  42. Zeros of orthogonal polynomial on the disk Koornwinder basis Basis P n , k For 0 ≤ k ≤ n � � y ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) √ n − k k 1 − x 2

  43. Zeros of orthogonal polynomial on the disk Koornwinder basis Basis P n , k For 0 ≤ k ≤ n � � y ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) √ n − k k 1 − x 2 deg ( P n , k ) = n

  44. Zeros of orthogonal polynomial on the disk Koornwinder basis Basis P n , k For 0 ≤ k ≤ n � � y ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) √ n − k k 1 − x 2 deg ( P n , k ) = n { P n , k ( x , y ) : 0 ≤ k ≤ n } basis mutually orthogonal of V n .

  45. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k For 0 ≤ k ≤ n � � y P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) √ n − k k 1 − x 2

  46. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k For 0 ≤ k ≤ n � � y P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) √ n − k k 1 − x 2 n − k lines x = a where a is a zero of P ( k + µ + 1 , k + µ + 1 ) n − k

  47. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k For 0 ≤ k ≤ n � � y P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) √ n − k k 1 − x 2 n − k lines x = a where a is a zero of P ( k + µ + 1 , k + µ + 1 ) n − k k ellipses (for k ≥ 2) a 2 x 2 + y 2 = a 2 where a is a positive zero of P ( µ + 1 / 2 ,µ + 1 / 2 ) . k For k = 1 we obtain y = 0.

  48. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k For 0 ≤ k ≤ n � � y P n , k ( x , y ) = P ( k + µ + 1 , k + µ + 1 ) ( x ) ( 1 − x 2 ) k / 2 P ( µ + 1 / 2 ,µ + 1 / 2 ) √ n − k k 1 − x 2 n − k lines x = a where a is a zero of P ( k + µ + 1 , k + µ + 1 ) n − k k ellipses (for k ≥ 2) a 2 x 2 + y 2 = a 2 where a is a positive zero of P ( µ + 1 / 2 ,µ + 1 / 2 ) . k For k = 1 we obtain y = 0. The ellipses are centered at the origin, oriented on OX , with semi-major axis equal to 1 and semi-minor axis equal to a .

  49. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k Degree 5 0 1 2 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 , , ,  0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 3 4 5 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 , ,  0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  50. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k Degree 6 0 1 1.5 1.5 1.0 1.0 0.5 0.5 , ,  0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 3 1.5 1.5 1.0 1.0 0.5 0.5 , , 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 4 5 6 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 , ,  0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  51. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines).

  52. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of P n , k = 0 when n − k or k are odd.

  53. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of P n , k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines.

  54. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of P n , k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in P n , k = 0 are symmetric with respect to the axes, centered on the origin and concentric.

  55. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of P n , k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in P n , k = 0 are symmetric with respect to the axes, centered on the origin and concentric. The straight lines in P n , k = 0 are symmetric with respect to the axis OY and they are inside ( − 1 , 1 ) × R .

  56. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of P n , k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in P n , k = 0 are symmetric with respect to the axes, centered on the origin and concentric. The straight lines in P n , k = 0 are symmetric with respect to the axis OY and they are inside ( − 1 , 1 ) × R . There are multiple points coming from the intersection of several branches.

  57. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , 0 = 0 is decomposed in n straight lines. P 8 , 0 = 0 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  58. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , n = 0 is decomposed in [ n 2 ] ellipses whose axis of symmetry are the coordinates axes. The semi-major axis is of size 1 and it is on the axis OX The semi-minor axis is on OY and its size is given by one of the roots of the Gegenbauer polynomial P ( µ + 1 / 2 ,µ + 1 / 2 ) . n P 8 , 8 = 0 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  59. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k P n , k = 0 intersects a line parallel to OX in n − k points and a line parallel to OY in k points. P 9 , 4 = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  60. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k Every straight line passing through the origin intersects P n , k = 0 in n real points. P 9 , 4 = 0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 - 0.5 - 0.5 - 1.0 - 1.0 - 1.5 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  61. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k : Interlacing P 9,4 P 9,5 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  62. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k : Interlacing P 9,4 P 8,4 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  63. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k : Density 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  64. Zeros of orthogonal polynomial on the disk Koornwinder basis Zeros of P n , k : Density For n = k 1.5 1.0 0.5 0.0 - 0.5 - 1.0 - 1.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5

  65. Zeros of orthogonal polynomial on the disk Polar coordinates Basis P n j , 1 , P n j , 2 Polar coordinates ( x , y ) = ( r cos ( θ ) , r sin ( θ )) , with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2 π

  66. Zeros of orthogonal polynomial on the disk Polar coordinates Basis P n j , 1 , P n j , 2 Polar coordinates ( x , y ) = ( r cos ( θ ) , r sin ( θ )) , with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2 π For 0 ≤ j ≤ n / 2, ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j ( 2 r 2 − 1 ) r n − 2 j sin (( n − 2 j ) θ ) j , 2 ( x , y ) = P ( µ, n − 2 j ) P n j

  67. Zeros of orthogonal polynomial on the disk Polar coordinates Basis P n j , 1 , P n j , 2 Polar coordinates ( x , y ) = ( r cos ( θ ) , r sin ( θ )) , with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2 π For 0 ≤ j ≤ n / 2, ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j ( 2 r 2 − 1 ) r n − 2 j sin (( n − 2 j ) θ ) j , 2 ( x , y ) = P ( µ, n − 2 j ) P n j deg ( P n j , i ) = n for i = 1 , 2

  68. Zeros of orthogonal polynomial on the disk Polar coordinates Basis P n j , 1 , P n j , 2 Polar coordinates ( x , y ) = ( r cos ( θ ) , r sin ( θ )) , with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2 π For 0 ≤ j ≤ n / 2, ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j ( 2 r 2 − 1 ) r n − 2 j sin (( n − 2 j ) θ ) j , 2 ( x , y ) = P ( µ, n − 2 j ) P n j deg ( P n j , i ) = n for i = 1 , 2 { P n j , i ( x , y ) : 0 ≤ j ≤ n / 2 , i = 1 , 2 } basis mutually orthogonal of V n .

  69. Zeros of orthogonal polynomial on the disk Polar coordinates Zeros of P n j , 1 , P n j , 2 ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) = 0 j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j

  70. Zeros of orthogonal polynomial on the disk Polar coordinates Zeros of P n j , 1 , P n j , 2 ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) = 0 j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j

  71. Zeros of orthogonal polynomial on the disk Polar coordinates Zeros of P n j , 1 , P n j , 2 ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) = 0 j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j j circles centered on ( 0 , 0 ) with radio r = a + 1 < 1 where a is a 2 zero of P ( µ, n − 2 j ) j

  72. Zeros of orthogonal polynomial on the disk Polar coordinates Zeros of P n j , 1 , P n j , 2 ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) = 0 j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j j circles centered on ( 0 , 0 ) with radio r = a + 1 < 1 where a is a 2 zero of P ( µ, n − 2 j ) j 2 k + 1 n − 2 j lines passing through the origin with angle θ = 2 ( n − 2 j ) π , k ∈ Z .

  73. Zeros of orthogonal polynomial on the disk Polar coordinates Zeros of P n j , 1 , P n j , 2 ( 2 r 2 − 1 ) r n − 2 j cos (( n − 2 j ) θ ) = 0 j , 1 ( x , y ) = P ( µ, n − 2 j ) P n j j circles centered on ( 0 , 0 ) with radio r = a + 1 < 1 2 n − 2 j lines passing through the origin. 6  0 P 2,1 1.0 0.5 - 1.0 - 0.5 0.5 1.0 - 0.5 - 1.0

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