Lp Bos

Lp Bos
The University of Calgary | HBI · Department of Mathematics and Statistics

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144
Publications
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Publications

Publications (144)
Article
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We give an exposition of some simple but applicable cases of worst-case growth of a polynomial in terms of its uniform norm on a given compact set K ⊂ C d. Included is a direct verification of the formula for the pluripotential extremal function for a real simplex.
Preprint
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We model the power applied by a cyclist on a velodrome -- for individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The dissipative forces to be overcome by power are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, the power is used to increase the mechani...
Article
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We show that the problem of finding the measure supported on a compact set K⊂C such that the variance of the least squares predictor by polynomials of degree at most n at a point z0∈Cd\K is a minimum is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K, with extremal growth at z0. We use this to find the pol...
Preprint
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We model the instantaneous power applied by a cyclist on a velodrome -- for individual pursuits and other individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the cons...
Article
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We discuss a generalization of Berrut’s first and second rational interpolants to the case of equally spaced points on a triangle in R2.
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We prove the equivalence—under rotations of distinct terms—of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to \({{\mathbb {R}}}^3\), nor is it limited to the elasticity tensor wi...
Preprint
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We model the instantaneous power on a velodrome for individual pursuits, taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their re...
Article
In this article we study the probability that the maximum over a symmetric interval [−a, a] of a univariate polynomial of degree at most n is attained at an endpoint. We give explicit formulas for the degree n = 1, 2,3 cases. The formula for the degree 3 case leads to a lower bound for the true probability. Numerical experiments indicate that this...
Preprint
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For a constant power output, the mean ascent speed (VAM) increases monotonically with the slope. This property constitutes a physical background upon which various strategies for the VAM maximization can be examined in the context of the maximum sustainable power as a function of both the gear ratio and cadence.
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We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measur...
Preprint
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We correct the calculation of the Monge-Amp\`ere measure of a certain extremal plurisubharmonic function for the complex Euclidean ball in C^2.
Preprint
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We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K with extremal growth at thi...
Article
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We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do c...
Article
The classical Markov polynomialinequality bounds the norm of the derivative of a polynomial on an interval in terms of its degree squared and the norm of the polynomial itself with the factor of degree squared being the optimal (worst case) upper bound. Here we study what this factor should be on average, for random polynomials with independent, id...
Article
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We provide a MATLAB package for the computation of near-optimal sampling sets and weights (designs) for nth degree polynomial regression on discretizations of planar, surface and solid domains. This topic has strong connections with computational statistics and approximation theory. Optimality has two aspects that are here treated together: the car...
Article
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The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP s...
Article
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As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a...
Article
The classical Markov polynomial inequality bounds the uniform norm of the derivative of a polynomial on an interval in terms of its degree squared and the norm of the polynomial itself, with the factor of degree squared being the optimal upper bound. Here we study what this factor should be on average, for random polynomials with independent N(0,1)...
Preprint
Full-text available
The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the...
Article
For over a decade, there has been intensive work on the numerical and analytic construction of SICs (d ² equiangular lines in C d ) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vect...
Preprint
The purpose of this paper is to prove the equivalence$-$under rotations of distinct terms$-$of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to ${\mathbb R}^3$, nor is it limited...
Article
Full-text available
We prove a version of the Bernstein-Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex body P. As a consequence, we validate and clarify some observations of Trefethen in multivariate approximation th...
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In this paper, we examine the applicability of the approximation, fg‾≈f‾g‾\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{f g}\approx \overline{f}\,\overline{g...
Article
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As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average...
Preprint
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Everyone learns to find the maximum of a function over an interval by finding the critical points and then checking the endpoints. Often, in examples, the endpoints are treated almost as an afterthought. It cannot be denied that critical points are more sophisticated in comparison to the rather pedestrian idea of an endpoint. Indeed many students e...
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We discuss the Siciak-Zaharjuta extremal function of pluripotential theory for the unit ball in C^d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated.
Article
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In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers. In ove...
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We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval [-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[...
Article
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For a∈Z>0d we let ℓa(t):=(cos (a1t),cos (a2t),⋯,cos (adt)) denote an associated Lissajous curve. We study such Lissajous curves which have the quadrature property for the cube [-1,1]d that. ∫[-1,1]dp(x)dμd(x)=1π∫0πp(ℓa(t))dt for all polynomials p(x)∈V where V is either the space of d-variate polynomials of degree at most m or else the d-fold tensor...
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We give an explicit example for the selection of the shape parameter for a certain univariate radial basis function (RBF) interpolation problem.
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We give a simple recipe for Lissajous curves that for (certain) numerical purposes can serve as a proxy for the cube [−1, 1] d. 2010 AMS subject classification: 41A05, 41A10, 41A63, 65D05. For a ∈ Z d >0 we let a (t) := (cos(a 1 t), cos(a 2 t), · · · , cos(a d t)), t ∈ R. (1) denote the associated Lissajous curve with frequencies a 1 , · · · , a d....
Article
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Recently [1] gave a remarkable orthogonality property of the classical Legendre polynomials on the real interval [−1, 1]: polynomials up to degree n from this family are mutually orthogonal under the arcsine measure weighted by the degree-n normalized Christoffel function. We show that the Legendre polynomials are (essentially) the only orthogonal...
Article
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A new method to interpolate vertex data in IR n is developed with an emphasis on relative simplicity and ease of implementation. The vertices are first (simplicially) triangulated and then interpolating curves along edges in the triangulation are defined. Edge curves are then transfinitely extended, at each vertex vi, to the interior of surrounding...
Article
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In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical obj...
Article
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Let $K={\bf R}^n\subset {\bf C}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2 = x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in ${\bf R}^n\subset {\bf C}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of ${\bf C}^n$ we prove the following explicit formula for t...
Research
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Christoffel orthogonality of the Legendre Polynomials.
Article
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We study Lissajous curves in the three-dimensional cube that generate algebraic cubature formulas on a special family of rank-1 Chebyshev lattices. These formulas are used to construct trivariate hyperinterpolation polynomials via a single one-dimensional Fast Chebyshev Transform (by the Chebfun package) and to compute discrete extremal sets of Fek...
Data
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We apply Hyperbolic Potential Theory to the study of the asymp-totics of Fekete type points for univariate ridge function interpolation.
Article
It has recently been shown that the Lebesgue constant for Berrut's rational interpolant at equidistant nodes grows logarithmically in the number of interpolation nodes. In this paper we show that the same holds for a very general class of well-spaced nodes and essentially any distribution of nodes that satisfies a certain regularity condition, incl...
Article
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In the conclusion of their seminal paper Dahlen et al.(Geophys. J. Int., 141, 157–174, 2000) state that their results extend ray theory to finite-frequency waves. In view of the importance of that paper, which led to developments in theoretical and computational seismology, we feel that it is important to clarify certain mathematical statements the...
Article
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We discuss a polynomial interpolation problem where the data are of the form of a set of algebraic curves in ℝ2 on each of which is prescribed a polynomial. The object is then to construct a global bivariate polynomial that agrees with the given polynomials when restricted to the corresponding curves.
Article
We study theoretically and numerically trigonometric interpolation on symmetric subintervals of [-π,π][-π,π], based on a family of Chebyshev-like angular nodes (subperiodic interpolation). Their Lebesgue constant increases logarithmically in the degree, and the associated Fejér-like trigonometric quadrature formula has positive weights. Application...
Article
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Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than th...
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We calculate the transfinite diameter for the real unit ball Bd:= {x ∈ ℝd: |x| ≤ 1} and the real unit simplex.
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We give the asymptotics for D-optimal (equivalently G-optimal) designs on a compact (possibly complex) design space.
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We update the state of the subject approximately 20 years after the publication of a previous article on this topic. This report is mostly a survey, with a sprinkling of assorted new results throughout.
Article
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In the seminal paper by F. A: Dahlen et al. [“Fréchet kernels for finite-frequency traveltimes. I: Theory”, Geophys. J. Int. 141, No. 1, 157–174 (2000; doi:10.1046/j.1365-246X.2000.00070.x)], the authors formulated an important expression as a first-order estimate of the signal-traveltime delay for seismic studies. The authors left out a term in a...
Article
It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation...
Article
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Using the concept of Geometric Weakly Admissible Meshes together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.
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We present a brief survey on (Weakly) Admissible Meshes and correspond- ing Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points. These provide new computational tools for polynomial least squares and interpolation on multidimensional compact sets, with different applications such as numerical cuba- ture, digital filter...
Article
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We discuss sampling (interpolation) by translates of sinc functions for data restricted to a finite interval. We indicate how the Floater–Hormann (cf. [8]) of the Berrut normalization (cf. [2]), in the case of equally spaced nodes, can be regarded as a sampling operator with improved approximation properties that remains numerically stable. We prov...
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Using classical univariate polynomial inequalities (Ehlich and Zeller, 1964), we show that there exist admissible meshes with O(n 2) points for total degree bivariate polynomials of degree n on convex quadran-gles, triangles and disks. Higher-dimensional extensions are also briefly discussed. 2000 AMS subject classification: 41A10, 41A63, 65D10.
Article
We construct symmetric polar WAMs (weakly admissible meshes) with low cardinality for least-squares polynomial approximation on the disk. These are then mapped to an arbitrary triangle. Numerical tests show that the growth of the least-squares projection uniform norm is much slower than the theoretical bound, and even slower than that of the Lebesg...
Article
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Ray-centred coordinates are used in investigating seismic theory and in computing its results. It is commonly stated that they are local in nature. In this paper, we discuss the region of invalidity of such coordinates. Notably, the existence of such regions limits the generality of theoretical conclusions based on a proof that makes use of these c...
Article
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In the seminal paper by Dahlen et al. the authors formulate an important expression as a first-order estimate of traveltime delay. The authors left out a term which would at first glance seem nontrivial, on the basis that their intention was to derive the Fr\'echet derivative linking the observed delay to the model perturbation (Nolet 2009, pers. c...
Article
We discuss Bernstein–Walsh type inequalities for holomorphic polynomials restricted to curves of the form $$\bigl(z,e^{P_{1}(z)},e^{P_{2}(z)},\ldots,e^{P_{d}(z)}\bigr)\in\mathbb{C}^{d+1},$$ where P 1,P 2,…,P d are fixed polynomials on ℂ (such that the functions z and \(e^{P_{k}(z)}\) are algebraically independent). The existence of such inequalitie...
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We derive the characteristic equations, and the so-called Christoffel equation, for the vector elastodynamic equations in terms of both hypersurfaces of nonuniqueness and as wavefronts based on a physical definition. We follow Courant and Hilbert in defining a wavefront as a surface for which a solution may be zero on one side but nonzero on the ot...
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We discuss and compare two greedy algorithms, that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called "Approximate Fekete Points" by QR factorization with column pivoting of Vandermonde-like matrices. The second computes Discrete Leja Points by LU f...
Article
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Recently [1] gave a simple, geometric and explicit construction of bivariate interpolation at points in a square (the so-called Padua points), and showed that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log(n)) 2). One may observe that these points have the structure of the un...
Article
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Using recent results of Berman and Boucksom (arXiv: 0807. 0035), we show that for a nonpluripolar compact set K⊂ℂd and an admissible weight function w=e −φ , any sequence of optimal measures converges weak-star to the equilibrium measure μ K,φ of (weighted) pluripotential theory for K,φ.
Article
We discuss three natural pseudodistances and pseudomet- rics on a bounded domain in IRN based on polynomial inequalities.
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Suppose that K ⊂ ℝd is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c1, c2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, Fn ⊂ K, for all a ∈ Fn. Here dist(a, b) is a natural distance on K that will be described in the text.
Article
We construct new multivariate polynomial interpolation schemes of Hermite type. The interpolant of a function is obtained by specifying suitable discrete differential conditions on the restrictions of the function to algebraic hypersurfaces. The least space of a finite-dimensional space of analytic functions plays an essential role in the definitio...
Article
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We discuss the class of univariate Radial Basis Functions for which the ith cardinal function ui for interpolation at x1 < x2 < · · · < xn has support [x i−1 , x i+1 ]. We also give an explicit example where it can be proven that the points in an interval [a, b] for which the associated Lebesgue constant is minimal, are equally spaced.
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We discuss some theoretical aspects of the univariate case of the method recently introduced by Sommariva and Vianello [Comput. Math. Appl., to appear] for the calculation of approximate Fekete points for polynomial interpolation.
Article
We show that on the curves gamma := (x, e(t(x))), x is an element of [a, b], where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials P-N (x, y) of degree at most N in each variable x, y : \\(P-N(x,e(t(x))))'\\([a, b]) less than or equal to CN4\\P-N\\gamma, and the exponent four is shar...
Article
We introduce and study the notion of Taylorian points of algebraic curves in ℂ2, which enables us to define intrinsic Taylor interpolation polynomials on curves. These polynomials in turn lead to the construction of a well-behaved Hermitian scheme on curves, of which we give several examples. We show that such Hermitian schemes can be collected to...
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We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights. Symmetry conditions are often use...
Article
We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the Lebesgue constant has minimal order of growth, i.e. log square of the degree. To the best of our knowledge this is the first comple...
Article
In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220-238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [-1, 1](2) , and derived a compact form of the corresponding Lagrange interpolation formula. In...
Article
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The Padua points are a family of points on the square [−1, 1]2 given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpola...
Article
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We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., finding a set of d 2 unit vectors (φ j) in C d with ||φ j , φ k | = 1 √ d + 1 , j = k. Such 'equally spaced configurations' have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tight frames, isometric embeddings 2 (d) → 4 (d 2),...