Mathematical preprints by Shayne Waldron

1. Putatively optimal projective spherical designs with little apparent symmetry
   
2. An explicit construction of the unitarily invariant quaternionic polynomial spaces on the sphere
   
3. Complex spherical designs from group orbits
   J. Approx. Theory xxx (2024), no. x, xxx-xxx.
4. On Waldron Interpolation on a Simplex in R^d
   
5. Equations for the overlaps of a SIC
   
6. Constructing high order spherical designs as a union of two of lower order
   
7. Testing isomorphism between tuples of subspaces
   
8. A variational characterisation of projective spherical designs over the quaternions
   
9. Multivariate Lagrange interpolation and polynomials of one quaternionic variable
   Dolomites Research Notes on Approximation 17 (2024), 20-27.
10. Tight frames over the quaternions and equiangular lines
    
11. Extremal growth of polynomials
    Anal. Math. 46 no. 2 (2020), 195-224.
12. SICs and the elements of order three in the Clifford group
    J. Phys. A 52, no. 10 (2019), 1-31.
13. Spherical (t,t)-designs with a small number of vectors
    Linear Algebra Appl. 608 (2021), 84-106.
14. The Fourier transform of a projective group frame
    Appl. Comput. Harmon. Anal. 49, no. 1 (2020), 74-98.
15. An introduction to finite tight frames
    Applied and Computational Harmonic Analysis, Birhauser, 2018.
16. Constructing exact symmetric informationally complete measurements from numerical solutions
    J. Phys. A 51, no. 16 (2018), 40 pp.
17. On the number of harmonic frames
    Appl. Comput. Harmon. Anal. 48 (2020), 46-63.
18. A sharpening of the Welch bounds and the existence of real and complex spherical t-designs
    IEEE Trans. Info. Theory 63 (2017), no. 11, 6849-6857.
19. The construction of G-invariant finite tight frames
    J. Fourier Anal. Appl. 22 (2016), no. 5, 1097-1120.
20. Tight frames for cyclotomotic fields and other rational vector spaces
    Linear Algebra Appl. 476 (2015), 98-123.
21. Nice error frames, canonical abstract error groups and the construction of SICs
    Linear Algebra Appl. 516 (2017), 93-117.
22. The projective symmetry group of a finite frame
    New Zealand J. Math. 48 (2018), 55-81.
23. A characterisation of projective unitary equivalence of finite frames
    SIAM J. Discrete Math. 30 (2016), no. 2, 976-994.
24. Multivariate Bernstein operators and redundant systems
    J. Approx. Theory 192 (2015), 215-233.
25. Group frames
    chapter in the book Finite frames (edited by G. Kutyniok and P. Casazza), Springer 2013.
26. On the construction of highly symmetric tight frames and complex polytopes
    Linear Algebra Appl. 439 (2013), no. x, 4135-4151.
27. Frames for vector spaces and affine spaces
    Linear Algebra Appl. 435 (2011), no. 1, 77-94.
28. Affine generalised barycentric coordinates
    Jaen J. Approx. 3 (2011), no. 2, 209-226.
29. A classification of the harmonic frames up to unitary equivalence
    Appl. Comput. Harmon. Anal. 30 (2011), 307-318.
30. The symmetry group of a finite frame
    Linear Algebra Appl. 433 (2010), no. 1, 248-262.
31. Recursive three term recurrence relations for the Jacobi polynomials on a triangle
    Constr. Approx. 33 (2011), no. 3, 405-424.
32. On the construction of equiangular frames from graphs
    Linear Algebra Appl. 431 (2009), no. 11, 2228-2242.
33. On the convergence of optimal measures
    Constr. Approx. 32 (2010), no. 1, 159-179.
34. Increasing the polynomial reproduction of a quasi-interpolation operator
    J. Approx. Theory 161 (2009), 114-126.
35. On the spacing of Fekete points for a sphere, ball or simplex
    Indag. Math. 19 (2008), no. 2, 163-176.
36. Continuous and discrete tight frames of orthogonal polynomials for a radially symmetric weight
    Constr. Approx. 30 (2009), no. 1, 33-52.
37. On the Vandermonde determinant of Padua-like points
    Dolomites Research Notes on Approximation 2 (2009), 1-15.
38. Tight frames generated by finite nonabelian groups
    Numer. Algorithms 48, (2008), 11-28.
39. Scattered data interpolation by box splines
    AMS/IP Studies in Advanced Mathematics 42 (2008), 749-767.
40. Some remarks on Heisenberg frames and sets of equiangular lines
    New Zealand J. Math. 36 (2007), 113-137.
41. Orthogonal polynomials on the disc
    J. Approx. Theory 150 (2008), no. 2, 117-131.
42. Hermite polynomials on the plane
    Numer. Algorithms 45 (2007), 231-238.
43. Multivariate Jacobi polynomials with singular weights
    East J. Approx. 13 (2007), no. 2, 163-183.
44. Computing orthogonal polynomials on a triangle by degree raising
    Numer. Algorithms 42, (2006), 171-179.
45. On computing all harmonic frames of $n$ vectors in $\C^d$
    Appl. Comput. Harmon. Anal. 21 (2006), 168-181.
46. On the Bernstein-Bézier form of Jacobi polynomials on a simplex
    J. Approx. Theory 140 (2006), no. 1, 86-99.
47. Pseudometrics, distances and multivariate polynomial inequalities
    J. Approx. Theory 153 (2008), no. 1, 80--96.
48. Tight frames and their symmetries
    Const. Approx. 21 (2005), no. 1, 83-112.
49. The vertices of the platonic solids are tight frames
    Advances in Constructive Approximation: Vanderbilt 2003, 495-498, (edited by M. Neamtu and E. B. Saff), Nashboro Press, 2004.
50. Metrics associated to multivariate polynomial inequalities
    Advances in Constructive Approximation: Vanderbilt 2003, 133-147, (edited by M. Neamtu and E. B. Saff), Nashboro Press, 2004.
51. A generalised beta integral and the limit of the Bernstein-Durrmeyer operator with Jacobi weights
    J. Approx. Theory 122 (2003), no. 1, 141-150.
52. Generalised Welch Bound Equality sequences are tight frames
    IEEE Trans. Info. Theory 49 (2003), no. 9, 2307-2309.
53. The diagonalisation of the multivariate Bernstein operator
    J. Approx. Theory 117 (2002), no. 1, 103-131.
54. Isometric tight frames
    Electronic Journal of Linear Algebra 9 (2002), 122-128.
55. On the structure of Kergin interpolation for points in general position
    Recent Progress in Multivariate Approximation, 75-88 (edited by W. Haußmann, K. Jetter and M. Reimer), International Series of Numerical Mathematics 137, Birkhauser, Basel, 2001.
56. Signed frames and Hadamard products of Gram matrices
    Linear Algebra Appl. 347 (2002), no. 1-3, 131-157.
57. Extremising the $L_p$-norm of a monic polynomial with roots in a given interval and Hermite interpolation
    East J. Approx. 3 (2001), no. 3, 255-266.
58. The eigenstructure of the Bernstein operator
    J. Approx. Theory 105 (2000), no. 1, 133-165.
59. Mean value interpolation for points in general position
60. Inverse and direct theorems for best uniform approximation by polynomials
61. On Bernstein's comparison theorem, Peano kernels of constant sign and near-minimax approximation
62. Minimally supported error representations and approximation by the constants
    Numer. Math. 85 (2000), no. 3, 469--484.
63. Refinements of the Peano kernel theorem
    Numer. Funct. Anal. Optim. 20 (1999), no. 1-2, 147--161.
64. Sharp error estimates for multivariate positive linear operators which reproduce the linear polynomials
    Approximation Theory IX - Vol. 1, 339--346, (edited by C. K. Chui and L. L. Schumaker), Vanderbilt University Press, 1998.
65. Multipoint Taylor formulae
    Numer. Math. 80 (1998), no. 3, 461-494.
66. The error in linear interpolation at the vertices of a simplex
    SIAM J. Numer. Anal. 35 (1998), no. 3, 1191-1200.
    • and some related open problems
      To appear in East J. Approx.
67. Schmidt's inequality
    East J. Approx. 3 (1997), no. 2, 11-29.
68. $L_p$-error bounds for Hermite interpolation and the associated Wirtinger inequalities
    Constr. Approx. 13 (1997), no. 4, 461-479.
69. Symmetries of Linear Functionals
    Approximation Theory VIII - Vol. 1, 541--550, (edited by C. K. Chui and L. L. Schumaker), World Scientific, 1995.
70. A multivariate form of Hardy's inequality and $L_p$-error bounds for multivariate Lagrange interpolation schemes
    SIAM J. Math. Anal. 28 (1997), no. 1, 233-258. 97h:41020
71. Integral error formulae for the scale of mean value interpolations which includes Kergin and Hakopian interpolation
    Numer. Math. 77 (1997), no. 1, 105--122.
72. Dissertation (University of Wisconsin-Madison, May 1995)

• Error formulae for generalised Taylor interpolation
• Least rules and their numeric and symbolic calculation
• A homothety based argument for computing the best constant in some Hardy's inequalities for the complement of bounded domain
• The eigenstructure of the Bernstein operator