POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS

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2 2 N. ROBIDOUX averages, provided the averages are computed over intervals with disjoint interiors and the points are not located in the interiors of the intervals..2. Discrete Hodge star operators. The discrete Hodge star operators which we study in detail are defined at the end of this section. First, we provide some background. Discrete Hodge star operators have been identified as important components of mixed finite element, finite volume and finite difference methods for the numerical solution of partial differential equations [2, 6, 7, 8, 22, 27, 28, 3, 32, 33, 39, 4, 47, 49, 5, 52]..2.. An important simplification: identifying differential forms with functional proxies. There is a close relationship between the material presented in this article and models of continuum mechanics based on cochains, coboundary operators, discrete constitutive relations and Tonti diagrams [5, 7, 8, 3, 4, 5, 9, 20, 3, 35, 36, 37, 38, 49, 5, 52, 53, 54, 55, 56, 58]. However, no derham cohomology or differential forms theory is needed to understand this article. Because we identify discrete differential forms with polynomial proxies and we restrict ourselves to trivial constitutive relations (continuum) Hodge star operators look like identity mappings, and discrete Hodge star operators can be defined in an elementary way [39] Discrete Hodge star operators: nodal values to cell masses. Given the values of some real function f at n distinct nodes in R d, an estimate for the integral of f over a measurable set C may be obtained by interpolating the nodal values and integrating the interpolant over C, that is, with an interpolatory quadrature or cubature rule [7, 24]. The simultaneous approximation of the integrals of f over n distinct measurable sets is of particular interest: the corresponding mapping, from n nodal values to n integrals, is a discrete analog of the Hodge star operator of differential forms theory which maps 0-forms to d-forms Discrete Hodge star operators: cell masses to nodal values. Conversely, given the values of the integrals of a real function f over n suitable measurable subsets C i of R d, an estimate for the value of f at a node may be obtained by constructing a function which is consistent with this data, and evaluating this function at the node ( analytic substitution ). If d = and the C i s are bounded intervals with disjoint, nonempty interiors, this corresponds to evaluating a histopolant. Of particular interest is the simultaneous approximation of the values of f at n distinct nodes, a discrete analog of the Hodge star operator which maps d-forms to 0-forms. For a given space of differential forms proxies (polynomials, trigonometric polynomials, splines, radial basis functions, etc.), the discrete Hodge star operator which maps cell masses to nodal values is the inverse of the discrete Hodge star operator which maps nodal values to cell masses D Discrete Hodge star operators consistent with boundary conditions. In this article, we study discrete Hodge star operators on the interval (d = ). Hybrid discrete Hodge star operators which map n cell integrals and one boundary point value, or n point values to n 2 cell integrals and two boundary point values, to n nodal values, arise when solving boundary value problems on the interval. Following is a list of the discrete Hodge star operators which, together with their inverses, are most relevant: discrete Hodge star operators which converts n integrals to n interior point values;

4 4 N. ROBIDOUX A likely application of the material contained in this article concerns the placement of geometric degrees of freedom associated with higher order Whitney and edge and face elements [3, 4, 0,, 25, 29]. (Note: Corollary. implies that it is not possible to make the D analogs of face and edge elements superconvergent in the sense of making the corresponding discrete Hodge star operators locally exact on polynomials of one extra order. It is, however, possible to make the discrete Hodge star operators corresponding to nodal (0-forms) and cell (d-forms) Whitney elements superconvergent.) 2. Summary. In Sections 4 to 6, we establish the existence and uniqueness of histopolants for possibly gappy histograms (Theorem 4.). Having defined interhistopolants (inter-histopolants are polynomials with prescribed point values and cell averages), we proceed to establish their existence and uniqueness (Theorem 4.). We also indicate how to construct the corresponding cardinal basis functions (Theorem 5.). In Section 7, we define generic star operators: linear operators which convert arbitrary degrees of freedom to nodal values by way of polynomial proxies. A number of examples are given. In Section 8, we define superconvergence: A star operator which maps n degrees of freedom to n nodal values is superconvergent if it is exact on polynomials of order n +. We then prove the central theorem of this article (Theorem 8.7), which states that a star operator is superconvergent if and only if the nodal degrees of freedom are defined by the roots of a polynomial for which the alternate degrees of freedom vanish. We call such roots optimal nodes. In Section 9, we illustrate the generality of Theorem 8.7 by showing that optimal nodes exist for operators which convert polynomial expansion coefficients to nodal values. This provides another justification of the use of roots grids for pseudospectral collocation. In Section 0, we show that optimal nodes exist for star operators which convert point values, cell averages and/or integrals, to nodal values, provided the corresponding inter-histopolation problem is well-defined (Theorem 0.). We also explain how to compute the locations of the optimal nodes without solving an inter-histopolation problem (Theorem 0.2). In Section, we show that the first three discrete Hodge star operators listed above (Section.2.4) can be made superconvergent (Corollary.). The fourth discrete Hodge star operator cannot be made superconvergent because optimal nodes are located in the interiors of cells when integral degrees of freedom are used, yet the fourth discrete Hodge star operator requires two nodes to be located at the boundary. In Section 2, we establish that the only Jacobi polynomials with roots grids which are optimal for the corresponding extremas and endpoints grids are P (0,0) n (the Legendre polynomial P n ), P (,0) n, P (0,) n and P (,) n (Theorem 2.3). (Note that the extremas and endpoints grids corresponding to other polynomials for example, Chebyshev polynomials do define optimal grids. It is just that the optimal nodes are not given by the corresponding roots grids.) In Section 3, we explain why it is preferable to optimize the nodal degrees of freedom (Theorem 3.3). This is because the dual optimization problem is not well-posed. The remainder of the article is devoted to a numerical study of the discrete star operators which convert n averages to n nodal values (these are rescaled versions of the first discrete Hodge star operator listed in Section.2.4). Most notably, the formal

5 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 5 superconvergence is shown to translate into lower errors and accelerated convergence rates. Discrete star operators based on Legendre, Chebyshev, and uniform grids are seen to yield spectrally small truncation and solution errors for sufficiently smooth test functions. (Only algebraic convergence is observed when a discontinuous test function is used and then, only in RMS norm and for the Legendre and Chebyshev grids.) Finally, discrete star operators based on Legendre and Chebyshev extremas and endpoints grids are seen to be close to the identity matrix and to have condition numbers bounded by 2, the discrete star operators based on optimal grids being closer to the identity and having smaller condition numbers. 3. Terminology and notation. average: The average of the function f over the interval C i is f dx. h i C i average: The average of the function f over the interval C i is the usual average; the average of f over the singleton {x j } is f(x j ). In general, quotes point to the possibility of point values being used in addition to integrals or averages. 6. bins: n bounded intervals which define the bases of the bars of a histogram.., 4. cell: A bounded nonempty interval or a singleton. The cells define the bins of a histogram C i : The ith cell (out of n) extremas and endpoints grid: A grid with points at the endpoints of [, ] together with the extremas of a given polynomial (usually orthogonal). 2.. faces: The boundary points of the cells (endpoints for intervals, singleton points otherwise) degrees of freedom: n numbers which completely determine a polynomial of order n. 7. H: Diagonal matrix of the cell lengths h i gappy histogram: An histogram with gaps (with some bars missing )... histogram: n intervals with non-intersecting interiors together with n corresponding averages (bar heights) or, equivalently, integrals (bar areas). Note: There may be gaps between bins, that is, the intervals defining the histogram may not be contiguous... histogram: n cells together with n corresponding averages. 6. histopolant: A continuous function whose bin averages are the same as the histogram s.., 4. inter-histopolant: A continuous function whose cell averages are the bar heights, that is, a continuous function which interpolates several point values and matches several bar areas h i : The length of the ith cell. 4. interval: (a, b), [a, b) or (a, b] with < a < b <. I: Identity matrix. l i : Cardinal basis polynomial (Lagrange polynomial) with respect to the faces. 5. l i : Cardinal basis polynomial (Lagrange polynomial) with respect to the nodes. 7.. nodes: Points which define the projection operator π 0 and the nodal degrees of freedom. 7.. optimal nodes: For a given π, the unique nodes which make the corresponding and its inverse superconvergent. 8. optimized grid: A dual grid with faces (primal) and the corresponding optimal nodes (dual). 5. polynomials of order n: A polynomial of order n is a polynomial of degree at most n. i : Cardinal basis polynomial with respect to integral projections. 5. i : Cardinal basis polynomial with respect to average projections. 5. P n: Legendre polynomial of degree n (= P (0,0) n ). P (α,β) n : Jacobi (hypergeometric) polynomial of degree n. P n: The space of univariate polynomials of order n (of degree at most n ). regular C i s: Sufficiently disjoint intervals and singletons roots grid: A grid with points at the roots of a given polynomial. 2.. singleton: A set which consists of one point..3, 6. superconvergent: Exact on P n+ even though only n degrees of freedom are used. 8. T n: Chebyshev polynomial of the first kind of degree n ( P ( /2, /2) n ). x i : The ith face (there may be up to 2n faces). 5, 0.2. x i : The ith node (out of n) : Column vector or matrix with zero entries. ι: Inverse of the restriction of π to P n For example, the operator which associates to a histogram the corresponding histopolant ι 0 : Interpolation operator: mapping which reconstructs the unique polynomial of order n which takes n prescribed values at the nodes. 7..

6 6 N. ROBIDOUX π: Linear mapping, from a function space containing P n+ into R n, which is one-to-one on P n. π maps smooth functions to n degrees of freedom For example, mapping which associates to a function its integrals over n invervals π: Mapping which associates to a function its averages over n invervals π 0 : Mapping which associates to a function the column vector of its point values at the n nodes. 7.. ρ: ρ(q) := q ιπ(q). 8. : a b if and only if there exists a nonzero constant λ such that a = λb and b = a/λ. : π 0 ι = ( πι 0). Also, mapping which associates n point values to n cell integrals and point values : Mapping which associates n point values to n cell averages and point values Histopolants for gappy histograms exist and are unique. In this section, we establish the existence and uniqueness of histopolants for histograms with possibly non-contiguous bars. (In Section 6.3, we prove existence and uniqueness in cases in which some the bins are singletons.) Theorem 4.. Given a histogram defined by n bins C i bounded intervals with lengths h i > 0 and disjoint interiors and corresponding bar heights f i R, there exists a unique histopolant in P n, that is, a unique p P n such that p dx = f i, i =,..., n. (4.) h i C i (Equivalently, one can match bin integrals.) Proof. Because P n is an n-dimensional vector space and mapping a function to its values at n points is linear, it is sufficient to show that p 0 if f i = 0 for all i. So, suppose that the bin averages of p vanish. Because p is continuous and C i is an interval with nonempty interior, p must vanish somewhere in the interior of C i by the Intermediate Value Theorem. The interiors of the C i s being disjoint, p must have n distinct (real) roots. The degree of p being less than n, p must be trivial. 5. Cardinal basis polynomials for histograms with contiguous bars. The matrix entries of discrete Hodge star operators which map n integrals or averages to n nodal values can be given in closed form in terms of histopolating cardinal basis functions. In this section, we give formulas for histopolating cardinal basis functions which correspond to histograms without gaps. When the bins are contiguous, an antiderivative of a histopolating cardinal basis function must interpolate a step function with a unit jump in the interior of the corresponding bin by the Fundamental Theorem of Calculus [2, 30, 45]. Theorem 5.. Suppose that C i runs from x i /2 to x i+/2, i =,..., n, where < x /2 < < x n+/2 < (so that the bins are contiguous). Also let l i+/2 be the Lagrange polynomial corresponding to x i+/2, that is, l i+/2 is the polynomial of degree n which vanishes at all x k+/2 s except x i+/2, where it takes the value : l i+/2 := x x k+/2 x k i i+/2 x k+/2 [9]. Finally, let j := n l k+/2 = 2 k=j n l k+/2 2 k=j j l k /2. k=

7 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 7 Then, { i } n i= is a cardinal basis with respect to integral projections, that is, C i j dx = δ j i. Likewise, j := h j, where h j is the length of C j, is a cardinal basis polynomial with respect to average projections, that is, j dx = δ j i h. j C i Proof. n l k+/2 (x i+/2 ) = k=j n k=j δ i k = { 0 if i < j, if i j. Consequently, n k=j l k+/2 interpolates step-function data on the x i /2 s, with the unit jump located in the interior of C j, and C i j dx = xi+/2 x i /2 j dx = n l k+/2 (x i+/2 ) k=j n l k+/2 (x j /2 ) = δ j i. k=j 6. Matching averages and point values. In this section, we extend Theorem 4. to histograms with bins consisting of single values, for which the average matching property boils down to interpolation. At the end of the section, the construction of inter-histopolants is reduced to Hermite interpolation when the bins are contiguous. 6.. Averaging over a singleton. Interpolation/histopolation problems can be stated concisely by redefining averaging so it applies to singletons. Definition 6.. The average of a function f over the singleton {x} is f(x). (Note: The average of a Lebesgue integrable function over a singleton {x} is the limit of averages over intervals which shrink nicely to x [42]. For this reason, bars associated with singletons can be understodd as limits.) 6.2. Regular families of subsets of the real line. We now define families of subsets of the real line for which unique inter-histopolants exist for arbitrary data. Definition 6.2. A cell is a bounded nonempty connected subset of R. Consequently, a cell is either a singleton or a bounded interval with nonempty interior. Definition 6.3. A collection of cells {C i } is regular if if C i and C j have nonempty interiors and i j, then C i and C j have disjoint interiors, if C i and C j are singletons and i j, then C i and C j are disjoint, and if C i is a singleton and C j has nonempty interior, then C i is not contained in the interior of C j. Definition 6.4. A face is a boundary point of a cell (interval endpoint if the cell has positive length, the singleton point itself otherwise). If there are n cells, there are at least n faces (if the cells are singletons) and at most 2n faces (if the cells are separated intervals).

8 8 N. ROBIDOUX Although regularity, as a property, applies to families of sets, we will often abuse language and say that the subsets themselves are regular. Example (Regular subsets of the real line). The cells C = [0, ], C 2 = {}, C 3 = [, 2), C 4 = { 3 }, C 5 = (π, 5), C 6 = (5, 8), C 7 = {8}, C 8 = {7/2}, C 9 = [9, 3], C 0 = [3, 4] and C = {6} are regular. The corresponding faces are 0,, 2, 3, π, 5, 8, 7/2, 9, 3, 4 and 6. J. H. Verner has observed that the key property of regular subsets of the real line, property which allows the forthcoming proofs to carry through, is that they can be linearly ordered [57]. This observation leads to cleaner and more general versions of Theorem 6.5 and its corollaries. Connectedness, for example, is not necessary Inter-histopolants exist and are unique. We are now ready for the first main theorem of this article. Theorem 6.5. Given n numbers f i and n regular C i s, there exists a unique p P n such that the average of p over C i equals f i for all is. Proof. Suppose that the averages of p P n over the C i s vanish. Then, every C i which is not a singleton must have a root of p in its interior. Furthermore, because averages correspond to point values for singletons, p must also vanish at singleton points. Because the C i s are regular, p must have n distinct roots. Definition 6.6. The polynomial p of Theorem 6.5 is called an inter-histopolant. Although elementary, Theorem 6.5 appears to be new. (Maybe some version is found in the statistical literature devoted to density estimation?) Theorem 6.5 bridges interpolation and histopolation: When all C i s are singletons, it states the existence and uniqueness of an interpolant; when all the C i s have positive lengths, the existence and uniqueness of an histopolant; and, when some of the C i s are singletons and some have positive lengths, the existence and uniqueness of an inter-histopolant Construction of the inter-histopolant when the cells are contiguous. In this section we show that the construction of an inter-histopolant can be reduced to the evaluation of the derivative of a Hermite interpolant when the intervals are contiguous and the singletons are located at the endpoints of some or all of the intervals. Consequently, inter-histopolating cardinal functions can be constructed from the Hermite interpolating cardinal functions [48]. Theorem 6.7. Suppose given n numbers f i and n contiguous cells C i, the first m of which are intervals and the remaining n m, singletons (so that the C i s are regular). Suppose that the faces are < x /2 < < x m+/2 <, and that the intervals are ordered in such a way that C i runs between x i /2 and x i+/2 for i =,..., m. Finally, let q P n+ be the Hermite interpolant corresponding to the following data: q(x /2 ) = 0, (6.) i q(x i+/2 ) = h k f k, i =,..., m, (6.2) k= q (x j ) = f j if x j is a singleton, (6.3) where h k is the length of C k. (Note: (6.) (6.2) correspond to interpolating the cumulative integral of a histogram [30].) Then, the inter-histopolant is q. Proof. (6.) (6.3) make up + m + (n m) = n + conditions. Consequently, the corresponding Hermite interpolant q exists and is unique [48]. q = f i for i in

9 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 9 m +,..., n by (6.3). Remains to show that the average of q over C i equal f i for i m; indeed, (6.) and (6.2) imply that q dx = ( q(xi+/2 ) q(x i /2 ) ) = (h i f i ) = f i. h i C i h i h i 7. Conversion between arbitrary degrees of freedom and point values. In this section, we prepare the terrain for the centerpiece of this article, a characterization of superconvergence for linear operators which convert arbitrary degrees of freedom to nodal (point) values by way of polynomial proxies, and their inverses, which convert nodal point values to other degrees of freedom. We begin by defining nodal degrees of freedom; then, we define generic degrees of freedom; finally, we define the linear operators which effect the conversions. We also set notation, list elementary properties, and give examples. 7.. The nodal projection operator π 0 and the interpolation operator ι 0. Definition 7.. The nodes x i are n distinct real numbers. The degrees of freedom associated with nodes, the nodal values, are the point values of functions at the nodes. Definition 7.2. The nodal projection operator π 0 is defined as follows: Given a function f, π 0(f) R n is the column vector with entries the nodal values f(x i ). Definition 7.3. The interpolation operator ι 0 is the inverse of the restriction of π 0 to P n, so that π 0 ι 0 = I, the identity mapping on Rn. ι 0 maps f R n to the unique polynomial p P n such that π 0(p) = f. The greek letters pi (π) and iota (ι) refer to projection and injection (interpolation being a one-to-one mapping). Definition 7.4. l i is the Lagrange polynomial corresponding to the ith node, that is, the polynomial of degree n which vanishes at all nodes except the ith one, where it takes the value : l i := k i x x i x i. x k Because (π 0 l j ) i = δ j i, the Lagrange basis is cardinal with respect to π 0 [9]. Lemma 7.5. ι 0 has a simple formula in terms of the Lagrange polynomials: n ι 0 (f) = f j l j. j= 7.2. Generic degrees of freedom. Degrees of freedom are characterized by a projection operator. Definition 7.6. π : P n+ R n is a linear operator which is one-to-one on P n (actually, the domain of π may be larger than P n+, but no smaller). Definition 7.7. ι is the inverse of the restriction of π to P n, so that πι : R n R n is the identity mapping I. Lemma 7.8. ιπ : P n+ P n is a projector onto P n, that is, ιπ(p) = p for p P n, and (ιπ) (ιπ) = ιπ.

10 0 N. ROBIDOUX Proof. (ιπ) (ιπ) = ι (πι) π = ιiπ = ιπ. Example 2 (Nodal projection). π 0 is a suitable π. Example 3 (Integral projection). A suitable π is the mapping which, given n intervals C i with disjoint and nonempty interiors, maps a function f to the integrals C i f dx, i =,..., n. Example 4 (Weighted integral projection). A suitable π is the mapping which, given an integrable (weight) function w which is positive almost everywhere, and n intervals C i with disjoint interiors, maps a function f to the integrals C i fw dx, i =,..., n. A further generalization concerns integrals C i f dµ with µ a positive Borel measure µ, in which case the proof of Theorem 4. carries through provided the C i s are disjoint open intervals with positive measures [42]. Singletons may be added to the mix as well. Example 5 ( Average projection). A suitable π is the mapping which, given n regular C i s, maps a function to its averages over the C i s. ( Example 6 (Point value plus derivatives at a point). A suitable π is f f(x0 ), f (x 0 ), f (x 0 ),..., f (n ) (x 0 ) ) R n. Example 7 (Expansion coefficients). A suitable π is the mapping which associates to a function the first n of its expansion coefficients with respect to some polynomial basis, for example, the first n coefficients of its expansion in terms of Chebyshev (or Legendre) polynomials, or the first n coefficients its Taylor expansion at some point. Example 8 (Point value plus derivatives at points). A suitable π is the mapping which maps a function to the values of its derivative at n points, together with the value of the function at one point. Example 9. The mapping which maps a function to the values of its derivative at n points is not a suitable π, because constant polynomials are mapped to Generic star operators. ι 0 maps n nodal values to the unique polynomial p P n consistent with these values. On the other hand, π maps p P n to the alternate degrees of freedom. Consequently, πι 0 converts nodal degrees of freedom to the other degrees of freedom. Likewise, π 0ι converts degrees of freedom to nodal values. Lemma 7.9. π 0 ι and πι 0 are invertible, and πι 0 = (π 0 ι). Proof. ιπι 0 = ι 0 because ι 0 maps Rn into P n and ι is the inverse of the restriction of π to P n. Consequently, π 0ιπι 0 = π 0ι 0 = I. Definition 7.0. := π 0ι = (πι 0), and := πι 0 = (π 0ι). and are completely determined by π and π 0. Example 0 (Discrete Hodge star operators). The four discrete Hodge star operators described in Section.2.4 are examples of s. Lemma 7.. For any choice of distinct nodes, and are exact on P n, that is, P n P n = π π 0 R n R n and π 0 P n = R n P n π R n

11 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS commute. Proof. ι 0 (resp. ι) is the inverse of the restriction of π 0 (resp. π) to P n. 8. Optimal nodes are unique if they exist. At issue is whether exactness can be extended to P n+, a form of superconvergence. The following trivial example shows that this is sometimes possible. Example. If π = π 0, then = = I, and both stars convert point values at n nodes to the same degrees of freedom. Consequently, and are not only exact on P n+, they are exact on all functions. We now prepare the terrain for the central result of this article, a general characterization of superconvergent star operators (discrete Hodge or not) on the real line. Definition 8.. Optimal nodes (corresponding to π) are nodes which make and exact on P n+. Example 2. The nodes which define π 0 are optimal if π = π 0 (Example ). Definition 8.2. ρ(q) := q ιπ(q). ρ, like ι, is completely determined by π. Lemma 8.3. If q P n, then ρ(q) = 0. Proof. The restriction of ιπ to P n is the identity (Lemma 7.8). Lemma 8.4. πρ = 0. Proof. π(ρ(q)) = π (q ιπ(q)) = π(q) πιπ(q) = π(q) Iπ(q) = π(q) π(q) = 0. Lemma 8.5. If q is a polynomial of degree n, so is ρ(q). Proof. The range of ι is P n. Consequently, ιπ(q) has degree at most n. Lemma 8.6. Let q be a polynomial of degree n, with leading coefficient c n 0. Then, ρ(q) = c n ρ(x n ); consequently ρ(q) and ρ(x n ) have the same roots. Proof. Because q has degree n, q = c n x n + p, where c n R \ {0} and p P n. Consequently, q ιπ(q) = {c n x n + p} {ιπ(c n x n ) + ιπ(p)} = c n x n ιπ(c n x n ) + p ιπ(p) = c n x n c n ιπ(x n ) + p p = c n {x n ιπ(x n )}. The following theorem establishes that both and are exact on P n+, a gain of one order, provided ρ(x n ) has n simple real roots, and the nodes which define π 0 are these roots. (Recall that the nodes must be distinct, for otherwise π 0 is not one-to-one on P n, in which case ι 0 and are not defined.) Theorem 8.7. The following conditions are equivalent: (a): is superconvergent, that is, commutes. P n+ = P n+ π π 0 R n R n (8.)

12 2 N. ROBIDOUX (b): is superconvergent, that is, P n+ = π 0 P n+ π (8.2) R n commutes. (c): There exists a polynomial p of degree n such that R n π(p) = π 0 (p) = 0. (8.3) (d): The nodes are the (simple) roots of every polynomial q of degree n such that π(q) = 0. (e): The nodes are the (simple) roots of ρ(q) for some polynomial q of degree n. (f): ρ(q) vanishes at the nodes for every polynomial q P n+. (g): The nodes are the roots of ρ(x n ). Proof. (a) (b): If p P n+, π 0(p) = π(p) = π(p). (b) (a): If p P n+, π(p) = π 0(p) = π 0(p). (Note that (8.2) is the mirror image of (8.).) (a) (c): Let q have degree n. Then, p = ρ(q) has degree n by Lemma 8.5, π(p) = 0 by Lemma 8.4, and π 0(p) = π 0(q) π 0ι0 = π 0(q) π 0ιπ(q) = π 0(q) π(q) = π 0(q) π 0(q) = 0. (c) (d): Let p be a polynomial of degree n such that (8.3) holds, and q be a polynomial of degree n such that π(q) = 0. Because π(p) = π(q) = 0, p = ρ(p) and q = ρ(q), so that Lemma 8.6 implies that p and q have the same roots. Consequently, q vanishes at the nodes. Because q has degree n, the nodes are the only roots, and they are simple. (d) (e): Because there are n distinct nodes, it is sufficient to show that there exists a polynomial q of degree n such that π(q) = 0. q = ρ(x n ) has degree n by Lemma 8.5, and π(q) = πρ(x n ) = 0 by Lemma 8.4. (e) (f): If the degree of q is less than n, then ρ(q) vanishes by Lemma 8.3. If q has degree n, ρ(q) and ρ(x n ) have the same roots by Lemma 8.6. Consequently, these roots are independent of q. (f) (g): ρ(x n ) has at most n roots and vanishes at n distinct nodes. (g) (a): Because is linear and exact on P n, it is sufficient to show that π(p) = π 0 (p) for one polynomial p of degree n. Consider p = ρ(xn ). π 0 (p) = 0 because the nodes are located at the roots of ρ(x n ). On the other hand, π(p) = 0 by Lemma 8.4, so that π (p) = 0 as well. Corollary 8.8. For a given π, the optimal nodes are unique if they exist. Proof. The optimal nodes, if they exist, are located at the roots of ρ(x n ), which is completely defined by π. 9. Superconvergent conversion between point values and expansion coefficients. To illustrate the generality of the above results, we make a brief detour through polynomial expansions. Let {p 0, p,..., p n } be a basis of P n, and let p n be a polynomial of degree n, so that {p 0, p,..., p n, p n } is a basis of P n+. Every polynomial in P n+ can be

13 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 3 uniquely written as q = n k=0 c kp k. Consequently, the mapping which maps q P n+ to (c 0, c,..., c n ) R n is a suitable π. Furthermore, n ι ((c 0, c,..., c n )) = c k p k, k=0 ( n ) n ιπ c k p k = c k p k, and ρ(p n ) = p n. k=0 k=0 By Theorem 8.7, the corresponding and are exact on P n+ if and only if p n has n simple real roots and the nodes are located there. Example 3. Suppose that {p 0, p,..., p n } = {, x, x 2,..., x n}, so that π corresponds to evaluation of the first n coefficients of the Taylor polynomial based at the origin. Then, p n = x n has only one zero, and consequently and cannot be made superconvergent for n >. (Likewise, the star operator which corresponds to the evaluations of a function and its first n derivatives at some point cannot be made superconvergent.) Example 4. Consider an operator which converts between n point values and the first n ( spectral ) expansion coefficients [9]. That is, suppose that {p 0, p,..., p n } are polynomials orthogonal with respect to some weight, with p k P n for k < n and p n having degree n, and let be the operator which maps n point values to the coefficients of the interpolating polynomial with respect to this basis. Then, the optimal nodes are the n simple real roots of p n, hence are those which maximize the order of Gaussian integration [9, 7]. For example, the mapping between point values at the roots of the Chebyshev polynomial of degree n and the first n coefficients of the Chebyshev expansion is superconvergent, and so is its inverse. Likewise for transforms based on Legendre and other orthogonal polynomials. 0. Superconvergent conversion between combinations of point values and cell averages and/or integrals, and nodal values. In this section, we establish the existence of optimal nodes when the projection π maps a function to its averages over regular cells (Example 5). We also give a method of computing the optimal nodes without solving the corresponding inter-histopolation problem, in the case when the cells are contiguous. 0.. Histograms with or without gaps. Theorem 0.. Let n regular C i s be given, so that π, the operator which associates to a continuous function its averages over the C i s, is one-to-one on P n, and ι, the inverse of π s restriction to P n, is the operator which maps n cell averages to a polynomial inter-histopolant. Then, ρ(x n ) has n simple real roots, which, chosen as nodes, make and superconvergent. Furthermore, every C i contains a root of ρ(x n ), so that singleton points are optimal nodes, and the other optimal nodes are found in the interior of the C i s with positive lengths. Proof. Let p := ρ(x n ), so that π(p) = 0 by Lemma 8.4. If C i is the singleton x i, p(x i ) = (π(p)) i = 0, and x i must be a root of p. If C i has length h i > 0, C i p dx = h i (π(p)) i = 0, and p must have a root in the interior of C i. Because the C i s are regular, the corresponding roots are distinct. The result follows from Theorem 8.7. What Theorem 0. tells us: if p is a polynomial of degree n which vanishes at every singleton and which averages to zero over every cell which is a nontrivial interval such a polynomial always exists if the cells are regular then the optimal nodes are located at the (automatically simple) roots of p.

14 4 N. ROBIDOUX 0.2. Histograms with contiguous cells. When the cells are contiguous, we can compute the roots of ρ(x n ) = x n ιπ(x n ) without (directly) computing ιπ(x n ). Theorem 0.2. Suppose that the C i s, i =,..., m n, are contiguous bounded intervals with disjoint interiors, ordered so that C i runs from x i /2 to x i+/2, and that the singletons C i, i = m+,..., n, are n m distinct endpoints of these intervals. Then, the optimal nodes are the roots or q, where q = m+ i= ( x xi /2 ) αi /2, (0.) and α j = 2 when {x j } is one of the singletons, and α j = otherwise. Proof. q is a polynomial of degree n + because there are m + endpoints and n m singletons, so that m+ i= α i /2 = n +. Consequently, q has degree n. If C i is an interval, then C i q dx = 0 because q vanishes at the endpoints of C i ; consequently, q has a root in the interior of C i. If C i is the singleton {x j }, then q vanishes because x j is a double root of q. Consequently, π(q ) = 0, so that q = ρ(q ) ρ(x n ) by Lemma 8.6. The result now follows from Theorem 0.. Corollary 0.3. In addition to the hypotheses of Theorem 0.2, suppose that the C i s run between the simple roots of q P m+, and the singletons are located at the simple roots of p P n m+, with p a divisor of q. Then the optimal nodes are the roots of (pq).. Existence of superconvergent discrete Hodge star operators. In this section, we show that the first three discrete Hodge star operators listed in the Introduction can always be made superconvergent, whence the fourth is never superconvergent. We also point out the implications for face elements... Three of the four discrete Hodge star operators can be made superconvergent. Corollary.. Consider the following four discrete Hodge star operators: n to n: the discrete Hodge operator which converts n integrals (over contiguous intervals with positive lengths) to n interior nodal values; (n ) + to (n ) + : the discrete Hodge operator which converts n integrals (over... ) and one boundary value to n interior nodal values and one boundary nodal value (at the same boundary point); (n 2) + 2 to (n 2) + 2: the discrete Hodge operator which converts n 2 integrals (over... ) and two boundary values to n 2 interior nodal values and two boundary nodal values; and n to (n 2) + 2: the discrete Hodge operator which converts n integrals (over... ) to n 2 interior nodal values and two boundary nodal values. For any set of faces, one can choose nodes which make any one of the first three discrete Hodge star operators superconvergent. However, because the boundary nodes are not matched by boundary singletons, the n to (n 2) + 2 discrete Hodge star operator is never superconvergent. Proof. By Theorem 0., given any set of faces, there exists a unique set of optimal nodes. Moreover, every singleton is automatically an optimal node, and the remaining optimal nodes are located in the interiors of the cells with positive lengths. Consequently, one can choose the non-singleton nodes so as to make any of the first three discrete Hodge star operators superconvergent. However, because there are no

15 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 5 singletons in the fourth, n to (n 2) + 2, case, no optimal node is located at the boundary, which clashes with the boundary points being nodes. 2. Superconvergent discrete Hodge star operators on the interval [, ]. In this section, we characterize the optimal nodes when the faces are defined by an extremas and endpoints grid, paying particular attention to dual grids associated with Jacobi polynomials. We discuss discrete Hodge star operators which convert integrals over contiguous cells, cells which cover the interval [, ], possibly with singletons at ±, to nodal values, and leave out n to (n 2) + 2 discrete Hodge star operators, which are never superconvergent in the sense of being exact on P n Optimal nodes for faces located at the zeros of a known polynomial. Definition 2.. The roots grid associated with a polynomial is the grid with points at the roots of the polynomial. The extremas and endpoints grid is the grid with points at ± together with the roots of the derivative of the polynomial [9]. Corollary 2.2. Suppose that the cells are contiguous and run between the n simple real roots of the polynomial q P n+, which are located in the interval (, ). Then, the optimal nodes are located at the roots of q if there are no singletons, the roots of ((x + )q) if the only singleton is located at, the roots of ((x )q) if the only singleton is located at, and the roots of ((x 2 )q) if the only singletons are located at ± Superconvergent discrete Hodge star operators defined by Jacobi extremas and endpoints grids. Theorem 0. and 0.2 and Corollaries 0.3 and 2.2 apply to arbitrary face grids, smooth or not. Consequently, histograms defined by Jacobi polynomial extremas and endpoints grids always define a unique set of optimal nodes. Are these optimal nodes located at the points of the roots grid? Before we answer this question, let us consider, in Examples 5 to 8, discrete Hodge star operators which convert n integrals over contiguous cells which range between the points of an extrema and endpoints grid, to n point values (no singletons); these are the first discrete Hodge star operators of the above list of four. Example 5 (Extremas and endpoints of P 2 ). 0 is the only extrema of P 2 = (3x 2 )/2, the Legendre polynomial of degree 2. The Legendre extremas and endpoints grid is {, 0, }, and the polynomial q of Corollary 2.2 is P 2 = 3x. ( (x 2 )q ) = 3x 2 = 2P 2, and consequently the Legendre roots grid (with nodes at ±/ 3) is optimal. Example 6 (Extremas and endpoints of T 2 ). 0 is the only extrema of T 2 = 2x 2, the Chebyshev polynomial of degree 2. In this case, the Chebyshev extremas and endpoints grid is the same as the Legendre extremas and endpoints grids, and consequently the optimal nodes are located at ±/ 3. The Chebyshev roots grid, on the other hand, has points at ±/ 2, and consequently the Chebyshev roots grid is not optimal for the Chebyshev extremas and endpoints grid. Example 7 (Extremas and endpoints of P 3 ). The extremas of the Legendre polynomial P 3 = (5x 3 3x)/2 are ±/ 5, so that the Legendre extremas and endpoints grid is {, / 5, / 5, }. q = P 3, ( (x 2 )q ) { = 2P3, and the Legendre roots grid 3/5, 0, } 3/5 is optimal. Example 8 (Extremas and endpoints of T 3 ). The extremas of the Chebyshev polynomial 4x 3 3x are ±/2. q = (x 2 )P 3, ( (x 2 )q ) = 48x 3 30x = 6x(8x 2 5),

16 6 N. ROBIDOUX and the optimal nodes are located at 0 and ± 5/8, which are neither the Chebyshev roots (0 and ± 3/2) nor the Legendre roots. In the remainder of this section, we establish that the Jacobi polynomials P (0,0) m = P m, P (,0) m, P (0,) m and P (,) m are the only ones for which the roots grids are optimal for extremas and endpoints grids, under the assumption that the singletons, if any, are located at the endpoints (and that the integrals which define π are unweighted). First, the positive result. Theorem 2.3. Suppose that the C i s, i =,..., m n, are contiguous bounded intervals with disjoint interiors, ordered so that C i runs from x i /2 to x i+/2, with = x /2 < x 3/2 < < x m+/2 =. Suppose also that (i): m = n (an n to n discrete Hodge star operator in the notation of Corollary.), that is, there are no singletons, and the interior faces are located at the extremas of the Legendre polynomial P m, or (ii): m = n (an (n ) + to (n ) + discrete Hodge star operator), the singleton is located at, and the interior faces are located at the extremas of the Jacobi polynomial P (0,) m, or (iii): m = n (an (n ) + to (n ) + discrete Hodge star operator), the singleton is located at, and the interior faces are located at the extremas of the Jacobi polynomial P (,0) m, or (iv): m = n 2 (an (n 2) + 2 to (n 2) + 2 discrete Hodge star operator), the singletons are located at ±, and the interior faces are located at the extremas of the Jacobi polynomial P (,) m. Then, the roots grids are optimal, that is, the optimal nodes are located at the roots of P m in case (i), the roots of P (0,) m together with in case (ii), the roots of P (,0) m together with in case (iii), and the roots of P (,) m together with ± in case (iv). Proof. Let (α, β) equal (0, 0) in case (i), (0, ) in case (ii), (, 0) in case (iii), and (, ) in case (iv). P (α,β) m has m distinct extremas, located in the interval (, ), because P (α,β) m has m simple roots there [, 7]. The polynomial q of Corollary 2.2 is d dx P(α,β) m, and the optimal nodes are located at the roots of { d ( x) α+ ( + x) β+ d } dx dx P(α,β) m. On the other hand, P (α,β) m is the eigenfunction of the Sturm-Liouville problem { } d ( x) α+ β+ dy ( + x) + λ( x) α ( + x) β y = 0 dx dx corresponding to λ = m(m + α + β + ) 0 [6]. Consequently, the optimal nodes are the roots of ( x) α ( + x) β P (α,β) m. We now establish that dual grids based on the roots and extremas of Jacobi polynomials other than the four listed in Theorem 2.3 for example, Chebyshev roots and extremas and endpoints grids are not optimal. Theorem 2.4. Suppose that the C i s have positive lengths and are contiguous. Suppose also that the faces are located at the extremas of P (α,β) m together with ±, and that the singletons, if any, are located at the endpoints ±. Finally, suppose that the

17 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 7 optimal nodes are located at the roots of P (α,β) m. Then, P (α,β) m is one of the four Jacobi polynomials listed in Theorem 2.3. Proof. Let ( α, β) equal (0, 0) if there are no singletons, (0, ) if the singleton is at, (, 0) if it is at, and (, ) if there are two singletons. By Corollary 2.2, we must have that ( x) α ( + x) βp (α,β) m d { ( x) α+ ( + x) β+ d } dx dx P(α,β) m. This implies that P (α,β) m is an eigenfunction of the Sturm-Liouville problem { d ( x) α+ ( + x) β+ dy } + λ( x) α ( + x) βy = 0, dx dx so that α = α and β = β [23]. 3. Nodes are optimal for infinitely many sets of faces. In this section, we indicate why we believe that π 0, not π, should be optimized. Briefly, this is because optimal faces may not exist, and if they exits, they are not unique. Consider discrete Hodge star operators which convert integrals over contiguous intervals (no singletons) to n nodal values. By Theorem 0., given any set of n + (distinct) faces, there exists a unique set of n nodes such that the corresponding discrete Hodge star operator and its inverse are exact on P n+. Does the converse hold? That is, given n (distinct) nodes, does there exist a unique set of faces such that is exact on P n+? The answer is No: If a collection of nodes is optimal for some set of faces, then it is optimal for a one-parameter family of sets of faces. Furthermore, for every n > 3, there exists a set of n nodes which is not optimal for any set of faces. The remainder of this section is devoted to illustrating and establishing these facts. Lemma 3.. Suppose that, which converts n integrals over contiguous cells to nodal values, is exact on P n+. Suppose also that the optimal nodes are located at the roots of the nth degree polynomial p. Then, the faces are located at the roots of one of the antiderivatives of p. Proof. p must be proportional to the derivative of the polynomial q given in (0.). Consequently, q must be proportional to one of the antiderivatives of p. Example 9 (n = : one-parameter family of optimal faces ). Suppose that there is only one node x = 0 (n = ). Because the average of a linear function over a bounded interval with positive length is its value at the center point, is exact on P 2 whenever x /2 = x 3/2. Consequently x = 0 is an optimal node for a one-parameter family of faces. Example 20 (n = 2: one-parameter family of optimal faces ). Suppose that the nodes are ±, the roots of p := x 2 (n = 2). Let q := x 3 /3 + x c be an antiderivative of p. Then, is a local maximum of q and is a local minimum. Because q( ) = 4/3 c and q() = 4/3 c, q has three simple real roots if and only if c ( 4/3, 4/3). Consequently, the nodes ± are optimal for a one-parameter family of sets of faces. Example 2 (n = 3: one-parameter family of optimal faces ). Suppose that the nodes are x < x 2 < x 3, the roots of p := (x x )(x x 2)(x x 3) and let q be an antiderivative of p (n = 3). Then, x and x 3 are local minimas of q, x 2 is a local maximum, and q(x 2 ) > max (q(x ), q(x 3 )). Consequently, q c has three simple

18 8 N. ROBIDOUX real roots if and only if c (max (q(x ), q(x 3 )), q(x 2 )). Consequently, the nodes are optimal for a one-parameter family of sets of faces. Example 22 (n = 3: non-existence of optimal faces ). Suppose that the nodes are evenly spaced and located at ±/3 and ±, the roots of p := (x 2 )(x 2 /9) (n = 4). Let q := x 5 /5 0x 3 /27+x/9 be an antiderivative of p. Then, q c has five simple real roots if and only if c ( 88/3645, 88/3645). Consequently, the nodes are optimal for a one-parameter family of sets of faces. For example, the faces which correspond to c = 0 are 0, ± /(3 3) ±.6, and ± /(3 3).2. For other values of c, the faces are not symmetric about the origin. Example 23. Suppose that the nodes are located at ±/2 and ±, the roots of p := (x 2 )(x 2 /4) (n = 4). Let q := x 5 /5 5x 3 /2 + x/4 be an antiderivative of p. No horizontal line crosses the graph of q more than three times, and q c does not have five real roots for any value of c. Consequently, the nodes are not optimal for any set of faces. Lemma 3.2. Let p be a polynomial of degree n with real simple roots x, x 2,..., x n, and let q be one of its antiderivatives. Then, one of the vertical translates of q has n + simple real roots if and only if a := max {q(x i ) such that q (x i ) > 0} < b := min {q(x i ) such that q (x i ) < 0}. (3.) Furthermore, if (3.) holds, then q c has n + simple real roots if and only if c (a, b). Proof. Because the roots of p are simple, they are either local minimas of q if p (x i ) = q (x i ) > 0 or local maximas of q if p (x i ) = q (x i ) < 0 and they alternate. If q c has n + distinct real roots, they are simple and consequently distinct from the x i s. By Rolle s Theorem, there is exactly one root of p between consecutive roots of q c. Because q c has a constant sign between its roots, we must have that q(x i ) c < 0 if x i is a local minimum, and q(x i ) c > 0 otherwise. Consequently, a < c < b. Conversely, if (3.) holds and a < c < b, q takes the value c between consecutive nodes, as well as between and the leftmost node, and between the rightmost node and. Consequently, q c has n + simple real roots. Theorem 3.3. Consider star operators which convert n 2 integrals over contiguous intervals to n nodal values. Suppose that the n nodes x, x 2,..., x n are the simple roots of the nth degree polynomial p, and let q be an antiderivative of p. Then, there exist n + distinct faces for which the nodes are optimal if and only if the largest local minimum of q is smaller than its smallest local maximum, that is, if (3.) holds. Furthermore, the nodes are optimal if the faces are located at the roots of q c for every c (a, b), and for other values of c, q c does not have n + simple real roots. Proof. Lemmas 3. and 3.2. Corollary 3.4. Consider star operators which convert n integrals over contiguous intervals to n nodal values. For any n > 3, there exists a set of n distinct nodes which is not optimal for any set of n + distinct faces. Proof. For every n > 3, there exists a polynomial q of degree n + with simple roots and a local minimum larger than a local maximum. The extremas of q define a set of nodes which are not optimal for any set of faces.

19 POLYNOMIAL HISTOPOLATION AND DISCRETE HODGE STAR OPERATORS 9 4. Discrete star operators which convert n integrals or averages to n nodal values. In this section, we focus on discrete star operators which convert n integrals or averages, over contiguous intervals with positive lengths (no singletons), to n nodal values, and their inverses. This includes the n to n and n to (n 2) + 2 discrete Hodge star operators. 4.. The cells C i, the faces x j, and the nodes x i. In the remainder of this article, the cells C i are n contiguous intervals with disjoint interiors and lengths 0 < h i <. Consequently, the cells are defined by the locations of n + faces, which we order so that C i runs from x i /2 to x i+/2, where < x /2 < < x n+/2 <. The nodes are n distinct real numbers, ordered so that < x < < x n < The projection operators π (cell masses) and π (cell averages). The degrees of freedom associated with cells, the masses, or integrals, are the integrals of functions over the n cells. Associated with these degrees of freedom are projection operators. The integral projection operator π is defined as follows: Given an integrable function f, π(f) R n is the column vector with entries the masses C i f dx (Example 3). The average projection π is defined similarly: Given an integrable function f, π(f) R n is the column vector with entries the averages h i C i f dx [32, 39, 4, 58] The histopolation operators ι and ῑ. Let ι (resp. ῑ) be the inverse of π s (resp. π s) restriction to P n, so that ι (resp. ῑ) associates a histopolating polynomial in P n to every column vector f R n. ι is a right inverse of π, that is, πι = I; likewise for ῑ: πῑ = I (Theorem 4.). The histopolating cardinal functions (Theorem 5.) provide closed forms for the histopolation operators: ι(f) = n f j j, ῑ(f) = j= n f j j. 4.4., the discrete Hodge star operator from masses to nodal values, and, which maps averages. The discrete Hodge operator is defined as follows: := π 0ι; this corresponds to histopolating (ι), then evaluating the histopolant at the nodes (π 0 ). Equivalently, is the unique linear operator which makes the following diagram commute: P n = P n π π 0 R n R n j= (4.) (Lemma 7.). (The = at the top of diagram (4.) corresponds to the continuum Hodge star operator from - to 0-forms [39].), which maps averages to nodal values, is defined analogously: := π 0 ῑ. Following are closed form formulas for the entries of and in terms of the nodal values of histopolating cardinal basis polynomials: so that [ ] ij = j (x i ), [ ]ij = j(x i ), (4.2) = H, where H is the diagonal matrix of cell lengths h i.

20 20 N. ROBIDOUX 4.5., the discrete Hodge star operator from nodal values to masses, and, which maps to averages. and are invertible. The discrete Hodge operator the inverse of is πι 0 (Lemma 7.9). This corresponds to interpolating (ι 0 ), then integrating the interpolant over the cells (π). Equivalently, is the unique linear operator which makes the following diagram commute: π 0 P n = R n P n π R n. (4.3) (The = in diagram (4.3) corresponds to the continuum Hodge star operator from 0- to -forms.) the inverse of which maps nodal values to averages, can be defined analogously: := πι 0. Following are closed form formulas for the matrix entries of and in terms of cell integrals of nodal Lagrange polynomials (Definition 7.4): [ ] ij = l j dx, [ ] ij = l j dx, (4.4) C i h i C i so that = H. With the help of a computer algebra system, it is not difficult to obtain the entries of the discrete star matrices from formulas (4.2) and (4.4) [9, 40, 47]. Very compact code can be written by calling on polynomial differentiation packages and real polynomial root finders. 5. Numerically tested grids. In the remainder of this article, we present numerical results pertaining to the n averages to n nodal values discrete star operator and its inverse. We consider a number of dual grids with cells which cover the interval [, ]: Uniform grid: The (plain) uniform dual grid has n + equidistant faces so that the cells have lengths h = 2/n and equidistant nodes, located at the centers of the cells [39, 4]. The nodes are not optimal. Optimized uniform grid: The optimized uniform dual grid has the same faces as the uniform grid: x j+/2 = + jh, j = 0,..., n. However, the nodes are located at the extremas of n ( ) j=0 x xj+/2. The nodes are optimal (Theorem 0.2). Chebyshev grid: The (plain) Chebyshev dual grid has faces at the Chebyshev extremas and endpoints grid and nodes at the Chebyshev roots grid. That is, the faces are located at ± together with the extremas of T n, the nodes, at the roots of T n [9]. The nodes are not optimal (Theorem 2.4). Optimized Chebyshev grid: The optimized Chebyshev dual grid has faces at the Chebyshev extremas and endpoints grid, like the (plain) Chebyshev grid. The nodes are located at the extremas of (x 2 )T n. The nodes are optimal (Corollary 2.2). Legendre grid: The (plain) Legendre dual grid has faces at the Legendre extremas and endpoints grid and nodes at the Legendre roots grid [9]. The nodes are optimal (Theorem 2.3). 6. The condition number of. Based on numerical evidence, the assertions made in the remainder of the article should be interpreted as conjectures. Figure 6. shows the condition numbers κ 2 of (and ) as functions of the number n of nodes and cells for the various grids. Most notably, the Legendre and

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