                             Technical Report
                              4 November 1999

The eigenstructure of the Bernstein operator

Shaun Cooper and Shayne Waldron

Mathematics, Massey University, Albany Campus, Private Bag 102904, Auckland, New Zealand email: S.Cooper@massey.ac.nz

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: waldron@math.auckland.ac.nz (http://www.math.auckland.ac.nz/˜ waldron)

ABSTRACT

The Bernstein operator B_{n} reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of B_{n}. Its eigenvalues are

λ_{k}^{(n)} := \frac{n!}{(n − k)!} \frac{1}{n^{k}} , k = 0, 1, …, n,

and the corresponding monic eigenfunctions p_{k}^{(n)} are polynomials of degree k, which have k simple zeros in [0, 1]. By using an explicit formula, it is shown that p_{k}^{(n)} converges as n → ∞ to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p_{k}^{(n)} converge as n → ∞ to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at n + 1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein–Durrmeyer operator, which is self adjoint.

Key Words: (multivariate) Bernstein operator, diagonalisation, eigenvalues, eigenfunctions, total positivity, Stirling numbers, Jacobi polynomials, semigroup, quasi–interpolant

AMS (MOS) Subject Classifications: primary 41A10, 15A18, 38B42, secondary 33C45, 41A36

1. Introduction

It is well known that the Bernstein operator B_{n} : C[0, 1] → C[0, 1], n = 1, 2, …, defined by

B_{n}f(x) := ∑_{k=0}^{n} \binom{n}{k}x^{k}(1 − x)^{n−k}f(\frac{k}{n}),  (1.1)

reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. There are a number of other forms for B_{n}f from which the remaining eigenstructure of B_{n} is more apparent. The simplest of these to deal with is the expansion in terms of the monomials (see, e.g., Widder [Wi41:p.155])

B_{n}f(x) = ∑_{j=0}^{n} \binom{n}{j}x^{j}Δ_{1/n}^{j}f(0).  (1.2)

Here Δ_{h}^{j} is the j–th order forward difference operator

Δ_{h}^{j}f(x) := ∑_{i=0}^{j}(−1)^{j−i} ( \begin{matrix}j \\ i \end{matrix} ) f(x + ih),

which annihilates polynomials of degree < j. Let e_{k} be the monomial x ↦ x^{k}. It follows from (1.2) that B_{n} maps polynomials of degree k = 0, 1, …, n to polynomials of degree k (is degree reducing), and so has an eigenfunction of degree k corresponding to the eigenvalue

λ_{k}^{(n)} = ( \begin{matrix}n \\ k \end{matrix} ) Δ_{1/n}^{k}e_{k}(0) = \frac{n!}{(n − k)!} \frac{1}{n^{k}} ,  k = 0, 1, …, n.  (1.3)

This observation can be found in Berens and DeVore [BD80]. Let p_{k}^{(n)} denote the corresponding monic eigenfunction of degree k, and take

p_{0}^{(n)}(x) := 1,  p_{1}^{(n)}(x) := x − 1/2.  (1.4)

The paper is set out as follows.

In Section 2, we give an explicit description of this diagonal form of the Bernstein operator. This includes a formula for the eigenfunctions p_{k}^{(n)} in terms of the monomial basis, and a description of the dual functionals to them, together with some symmetry properties. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein–Durrmeyer operator, which is self adjoint.

In Section 3, we use the theory of totally positive matrices to show that p_{k}^{(n)} has k distinct real roots in the interval [0, 1], and to describe their location. It is observed numerically that the zeros of successive eigenfunctions of B_{n} interlace, but a proof of this fact using the oscillatory properties of the Bernstein kernel has yet to be given.

In Section 4, we show that p_{k}^{(n)} converges as n → ∞ to a polynomial related to a Jacobi polynomial. Limits of the dual functionals are also obtained. These results are compared with the eigenstructure of the Bernstein–Durrmeyer operator.

Sections 5, 6 and 7 contain applications of the previous sections. Simple and illuminating proofs of results for iterated Boolean sums of B_{n}, limits of iterates of B_{n}, and representations of an associated C_{0}–semigroup are given. The eigenstructure is used to compare the approximation properties of B_{n} with L_{n}, the operator of Lagrange interpolation at the same n + 1 equally spaced points. This allows the possibility of defining a family Bernstein quasi–interpolants which vary from B_{n} to L_{n}. The eigenstructure of the Kantorovich operator is deduced from the eigenstructure of the Bernstein operator.

We conclude with some comments about the multivariate Bernstein operator. To simplify the presentation, a number of examples, including an alternative method for computing the dual functionals, are arranged in the appendix.

2. The diagonalisation and description of the eigenfunctions

Since B_{n} is degree reducing, writing the eigenfunction equation

B_{n}p_{k}^{(n)} = λ_{k}^{(n)}p_{k}^{(n)}  (2.1)

relative to a basis {b_{0}, b_{1}, ⋯, b_{n}} of Π_{n}, with the degree of b_{j} equal to j, leads to an upper triangular system. We now solve this system when the b_{j} are the monomials e_{j}.

The shifted factorial function is defined by

(x)_{j} := x(x + 1) ⋯ (x + j − 1), j = 1, 2, …, (x)_{0} := 1,

and the Stirling numbers of the second kind S(k, j) are defined by

x^{k} = ∑_{j=0}^{k} S(k, j) x(x − 1) ⋯ (x − j + 1).

Note the well known identity

S(k, j) = \frac{1}{j!} ∑_{i=0}^{j} ( \begin{matrix}j \\ i \end{matrix} ) (−1)^{j−i}i^{k}, 0 ≤ j ≤ k.  (2.2)

Theorem 2.3 (Diagonalisation). The Bernstein operator B_{n} can be represented in the diagonal form

B_{n}f = ∑_{k=0}^{n} λ_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f),  ∀f ∈ C[0, 1],  (2.4)

with λ_{k}^{(n)} and p_{k}^{(n)} its eigenvalues and eigenfunctions, and μ_{k}^{(n)} the dual functionals to p_{k}^{(n)}.

The eigenvalues are given by

λ_{k}^{(n)} := \frac{n!}{(n − k)!} \frac{1}{n^{k}} = 1 (1 − \frac{1}{n}) (1 − \frac{2}{n}) ⋯ (1 − \frac{k − 1}{n}) ,  k = 0, …, n, (2.5)

and they satisfy

1 = λ_{0}^{(n)} = λ_{1}^{(n)} > λ_{2}^{(n)} > λ_{3}^{(n)} > ⋯ > λ_{n}^{(n)} > 0.

The eigenfunction for λ_{k}^{(n)} is a polynomial of degree k given by

p_{k}^{(n)}(x) = ∑_{j=0}^{k} c(j, k, n) x^{j} = x^{k} − \frac{k}{2} x^{k−1} + lower order terms,  (2.6)

where the coefficients can be computed using the recurrence formula

\begin{matrix}c(k, k, n) := 1, c(k − 1, k, n) := −k/2, \\ c(k − j, k, n) := \frac{1}{(n−k+1)_{j} − n^{j}} ∑_{i=0}^{j−1} n^{i} S(k − i, k − j) c(k − i, k, n), j = 2, …, k. \end{matrix} (2.7)

The dual functional μ_{k}^{(n)} ∈ span{f ↦ f(j/n) : j = 0, 1, …, n}, defined on C[0, 1], satisfies

μ_{k}^{(n)}(p_{i}^{(n)}) = δ_{ik},  ∀i, k,  (2.8)

and is given by

μ_{k}^{(n)}(f) = ∑_{j=0}^{n} v(j, k, n)f(\frac{j}{n}),  k = 0, …, n,  (2.9)

where the (n + 1) × (n + 1) matrix of coefficients V := [v(j, k, n)]_{j,k=0}^{n} is the inverse of

P := [p_{i}^{(n)}(j/n)]_{i,j=0}^{n}.

The eigenfunctions and dual functionals have the symmetries

p_{k}^{(n)}(x) = (−1)^{k}p_{k}^{(n)}(1 − x),  μ_{k}^{(n)}(f) = (−1)^{k}μ_{k}^{(n)}(f ∘ R),  (2.10)

where R : x ↦ 1 − x is reflection about the point 1/2. The eigenfunctions of degree ≥ 2 can be factored as follows

\begin{matrix}p_{2j}^{(n)}(x) = x(x − 1)q(x − 1/2), \\ p_{2j+1}^{(n)}(x) = x(x − 1/2)(x − 1)q(x − 1/2),  j = 1, 2, …, \end{matrix}  (2.11)

where in each case q is an even monic polynomial.

Proof: We have already seen that the eigenvalues of B_{n} are given by (2.5), and the linear polynomials are eigenfunctions for eigenvalue λ_{0}^{(n)} = λ_{1}^{(n)} = 1, for which the p_{0}^{(n)}, p_{1}^{(n)} of (1.4) are clearly a basis which satisfies (2.6) and (2.7). It remains only to consider the 1–dimensional λ_{k}^{(n)}–eigenspace of polynomials of exact degree k = 2, 3, …, n.

By (1.2) and (2.2),

B_{n}e_{k}(x) = ∑_{j=0}^{k} a(j, k, n) x^{j},  (2.12)

where

a(j, k, n) = ( \begin{matrix}n \\ j \end{matrix} ) Δ_{1/n}^{j}e_{k}(0) = \frac{S(k, j)n!}{n^{k}(n − j)!},  0 ≤ j ≤ k ≤ n.  (2.13)

Note that

a(k, k, n) = λ_{k}^{(n)}, 0 ≤ k ≤ n. (2.14)

Express the eigenfunctions in the form

p_{k}^{(n)}(x) = ∑_{r=0}^{k} c(r, k, n) x^{r}, c(k, k, n) := 1.

Then the eigenfunction equation (2.1) gives

\begin{aligned}λ_{k}^{(n)} ∑_{s=0}^{k} c(s, k, n)x^{s} & = ∑_{r=0}^{k} c(r, k, n) ∑_{s=0}^{r} a(s, r, n)x^{s} \\ & = ∑_{s=0}^{k} ∑_{r=s}^{k} c(r, k, n)a(s, r, n)x^{s}. \end{aligned}

Equating the coefficients of x^{s} above gives

λ_{k}^{(n)}c(s, k, n) = ∑_{r=s}^{k} c(r, k, n)a(s, r, n).

Into this substitute s = k − j and r = k − i, to obtain

λ_{k}^{(n)}c(k − j, k, n) = ∑_{i=0}^{j} c(k − i, k, n)a(k − j, k − i, n),

which, for k > 1, can be solved for c(k − j, k, n) to give

c(k − j, k, n) = \frac{1}{λ_{k}^{(n)} − a(k − j, k − j, n)} ∑_{i=0}^{j−1} c(k − i, k, n) a(k − j, k − i, n), j = 1, …, k.

Equation (2.7) now follows from this using (2.5) and (2.13). Taking j = 1 in (2.7), gives

c(k − 1, k, n) = \frac{S(k, k − 1)}{−k + 1} = −\frac{k}{2} ,  (2.15)

which is (2.6). Using (2.9), the biorthogonality condition μ_{k}(p_{i}) = δ_{ik} can be written as

∑_{j=0}^{n} p_{i}^{(n)}(\frac{j}{n})v(j, k, n) = δ_{ik},

i.e., P V = I, and so V = P^{−1}.

Let R be x ↦ 1 − x, i.e., reflection about the point 1/2. From (1.1), it follows that

B_{n}(f ∘ R) = (B_{n}f) ∘ R,  (2.16)

so that

B_{n}(p_{k}^{(n)} ∘ R) = (B_{n}p_{k}^{(n)}) ∘ R = λ_{k}^{(n)} (p_{k}^{(n)} ∘ R),

and p_{k}^{(n)} ∘R is a λ_{k}^{(n)}–eigenfunction. For k = 0, 1 the symmetry property of p_{k}^{(n)} is obvious, and for k ≥ 2 the eigenfunction p_{k}^{(n)} ∘ R must be a scalar multiple of p_{k}^{(n)} (the eigenspace is 1–dimensional), which by equating powers of x^{k} must be

p_{k}^{(n)} = (−1)^{k}p_{k}^{(n)} ∘ R.  (2.17)

In other words, p_{k}^{(n)} is even (resp. odd) about the point 1/2 when k is even (resp. odd). In particular, the zeros of p_{k}^{(n)} are symmetric about 1/2. Similarly, (2.16) implies that

∑_{k=0}^{n} λ_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f ∘ R) = ∑_{k=0}^{n} λ_{k}^{(n)}(p_{k}^{(n)} ∘ R)μ_{k}^{(n)}(f) = ∑_{k=0}^{n} λ_{k}^{(n)}(−1)^{k}p_{k}^{(n)}μ_{k}^{(n)}(f),

and equating coefficients of p_{k}^{(n)} in the above gives

μ_{k}^{(n)}(f) = (−1)^{k}μ_{k}^{(n)}(f ∘ R).  (2.18)

Taking j = k in (2.7) and using S(m, 0) = 0, m ≥ 1, gives

c(0, k, n) = 0, k ≥ 2.

Hence, for k ≥ 2, x = 0 is a zero of p_{k}^{(n)}, and by the symmetry property (2.17) so is x = 1. Further, when k is odd the symmetry property of the zeros implies that x = 1/2 must be a zero of p_{k}^{(n)}, which proves the factorisations (2.11). This completes the proof.

A list of the first few eigenfunctions and their dual functionals, and an explicit formula for V = P^{−1} is given in the appendix.

0  p_{0}^{(6)}, p_{1}^{(6)}, …, p_{6}^{(6)}  1

Fig. 2.1. The eigenfunctions p_{k}^{(6)}, k=0,...,6, for the Bernstein operator B_{6} (scaled to have the same absolute maxima). Note the symmetries, and interlacing of the zeros.

3. Zeros of the eigenfunctions

Next we determine the distribution of roots the eigenfunctions by using the theory of oscillating kernels (total positivity). The kernel defining the Bernstein operator

K_{n}(k, x) := ( \begin{matrix}n \\ k \end{matrix} ) x^{k}(1 − x)^{n−k}  (3.1)

is extended totally positive ETP(x) in k = 0, 1, …, n and 0 < x < 1 (see Karlin [K68:p.298]). The current theory of totally positive kernels (see the survey of Pinkus [P96]) cannot be applied directly, since this kernel is discrete in the first variable and continuous in the second (a case not yet considered), and it is not totally positive if the values x = 0, 1 are allowed. We circumvent these difficulties by considering the truncated Bernstein operator B_{n}^{×} defined by

B_{n}^{×}f(x) := ∑_{k=1}^{n−1} ( \begin{matrix}n \\ k \end{matrix} ) x^{k}(1 − x)^{n−k}f(\frac{k}{n}),  n = 2, 3, …,  (3.2)

as a matrix operator.

Theorem 3.3 (Zeros of the eigenfunctions). The eigenfunction p_{k}^{(n)}, k = 0, 1, …, n has k simple real roots contained in [0, 1], which we denote by

ξ_{1,k}^{(n)} < ξ_{2,k}^{(n)} < ⋯ < ξ_{k,k}^{(n)}.

More generally, for any nontrivial (a_{m_{1}}, …, a_{m_{2}}), 2 ≤ m_{1} ≤ m_{2} ≤ n,

m_{1} ≤ Z ( ∑_{k=m_{1}}^{m_{2}} a_{k} p_{k}^{(n)}) ≤ m_{2},

where Z counts the number of zeros in [0, 1]. The zeros are symmetric about 1/2, i.e.,

ξ_{i,k}^{(n)} + ξ_{k+1−i,k}^{(n)} = 1,  ∀i.  (3.4)

For k ≥ 2, there are common roots of 0 and 1, i.e.,

ξ_{1,k}^{(n)} = 0, ξ_{k,k}^{(n)} = 1,  k ≥ 2,  (3.5)

and the roots inside (0, 1) satisfy the inclusions

\frac{i − 1}{n} < ξ_{i,k}^{(n)} < 1 − \frac{(k − i)}{n}, 2 ≤ i ≤ k − 1.  (3.6)

Proof: The result is clearly true for k = 0, 1, and (3.4), (3.5) follow from (2.11). We now consider the case k ≥ 2. Since p_{k}^{(n)} vanishes at 0 and 1,

B_{n}^{×}p_{k}^{(n)} = B_{n}p_{k}^{(n)} = λ_{k}^{(n)}p_{k}^{(n)}, k = 2, 3, …, n,

and so the eigenvalues of B_{n}^{×} are λ_{k}^{(n)}, k = 2, …, n, with p_{k}^{(n)} a basis for the corresponding 1–dimensional eigenspace. Consider the matrix representation of B_{n}^{×}

A = [a_{ij}] : IR^{n−1} → IR^{n−1} : (f(i/n))_{i=1}^{n−1} ↦ (B_{n}^{×}f(i/n))_{i=1}^{n−1}  (3.7)

which is given by

a_{ij} := ( \begin{matrix}n \\ j \end{matrix} ) (i/n)^{j}(1 − i/n)^{n−j}.

A is an oscillation matrix, i.e., is totally positive and invertible, with a_{i,i+1}, a_{i+1,i} > 0 (see [P96:Prop.5.1]). Hence by the (Perron–Frobenius) spectral theory of such matrices (see [P96:Th.5.2]) it follows that its eigenvectors (p_{k}^{(n)}(i/n))_{i=1}^{n−1}, k = 2, 3, …, n satisfy

r − 2 ≤ S^{−} (∑_{k=r}^{s} c_{k} p_{k}^{(n)}) ≤ S^{+} (∑_{k=r}^{s} c_{k} p_{k}^{(n)}) ≤ s − 2, 2 ≤ r ≤ s ≤ n (some c_{k} ≠ 0),

where S^{−}(f), S^{+}(f) count the number of sign changes in the sequence

f(1/n), f(2/n), …, f((n − 1)/n)

with zero terms discarded (respectively assigned arbitrary values ±1). In particular, the eigenvector (p_{k}^{(n)}(i/n))_{i=1}^{n−1} has k − 2 strong sign changes, and so in addition to 0, 1, the eigenfunction p_{k}^{(n)} has k − 2 real roots inside (0, 1). Clearly, ξ_{2,k}^{(n)} > \frac{1}{n}, ξ_{2,k}^{(n)} > \frac{2}{n}, …, i.e.,

ξ_{i,k}^{(n)} > \frac{i − 1}{n}, 2 ≤ i ≤ k − 1,

and similarly (or by symmetry) we obtain the other half of (3.6).

Remark. When k is large with respect to n, (3.6) implies the roots of p_{k}^{(n)} are nearly evenly spaced. For example, when k = n we have \frac{i−1}{n} < ξ_{i,n}^{(n)} < \frac{i}{n}, for 2 ≤ i ≤ n − 1.

Interlacing of zeros

Numerical evidence suggests that the zeros of the eigenfunctions interlace, that is,

ξ_{j,k+1}^{(n)} < ξ_{j,k}^{(n)} < ξ_{j+1,k+1}^{(n)}, 1 < j < k.  (3.8)

The classical theorems for interlacing of eigenvectors and eigenfunctions of totally positive matrices and kernels can not be applied here, where the kernel (3.1) is discrete in the first variable and continuous in the second (a case not yet considered), and is not totally positive if the values x = 0, 1 are allowed. By Ando [A87:Th.6.3], it follows that the nodes of consecutive eigenvectors (p_{k}^{(n)}(i/n))_{i=1}^{n−1} are interlacing, in the sense that zeros of the piecewise interpolants indicated in Fig. 3.1 are. Unfortunately, this is not enough to conclude that the roots of consecutive p_{k}^{(n)} are interlacing. This is the subject of further investigation.

0  1

Fig. 3.1. The interlacing of the zeros of the piecewise linear interpolants to p_{5}^{(6)}, p_{6}^{(6)}.

4. Asymptotics of the eigenfunctions and their dual functionals

The limiting eigenfunctions We now show that the sequence of eigenfunctions p_{k}^{(n)} converges as n → ∞.

Theorem 4.1 (Limits of the eigenfunctions). For 0 ≤ j ≤ k,

lim_{n→∞} c(j, k, n) = c^{∗}(j, k),  (4.2)

where

c^{∗}(0, 1) := −\frac{1}{2}, c^{∗}(j, k) := ∏_{i=1}^{k−j} \frac{(k + 1 − i)(k − i)}{i(i − 2k + 1)} j ≠ 0, k ≠ 1. (4.3)

In other words, p_{k}^{(n)} converges uniformly on [0, 1] to p_{k}^{∗} ∈ Π_{k} as n → ∞, where

p_{k}^{∗}(x) := ∑_{j=0}^{k} c^{∗}(j, k) x^{j} = x^{k} − \frac{k}{2} x^{k−1} + \frac{k(k − 1)(k − 2)}{4(2k − 3)}x^{k−2} − · · · .  (4.4)

Proof: Since p_{0}^{(n)}(x) = 1 = p_{0}^{∗}(x), p_{1}^{(n)}(x) = x − 1/2 = p_{1}^{∗}(x), it is sufficient to prove the result for k ≥ 2. To show this we prove by strong induction on j and k ≥ 2 that lim_{n→∞} c(k − j, k, n) exists and is given by (4.3). Since c(k, k, n) = 1, this result holds for j = 0 and all values of k (where as usual the empty product in (4.3) is interpreted as 1). Suppose it is true for lim_{n→∞} c(k − i, k, n), i = 0, …, j − 1, where 0 < j ≤ k. Since

(n − k + 1)_{j} − n^{j} = \frac{1}{2}j(j − 2k + 1) n^{j−1} + lower order powers of n, j > 0,

taking the limit as n → ∞ of both sides of

c(k − j, k, n) = ∑_{i=0}^{j−1} \frac{n^{i} S(k − i, k − j) c(k − i, k, n)}{(n − k + 1)_{j} − n^{j}},

and using the induction hypothesis gives

lim_{n→∞} c(k − j, k, n) = \frac{2S(k − j + 1, k − j)}{j(j − 2k + 1)}c^{∗}(k − j + 1, k)

= \frac{(k + 1 − j)(k − j)}{j(j − 2k + 1)} ∏_{i=1}^{j−1} \frac{(k + 1 − i)(k − i)}{i(i − 2k + 1)}

= ∏_{i=1}^{j} \frac{(k + 1 − i)(k − i)}{i(i − 2k + 1)},

which completes the induction.

We now show the p_{k}^{∗} are closely related to the Jacobi polynomials P_{k}^{(α,β)}. These are by definition the orthogonal polynomials with respect to the weight (1 − t)^{α}(1 + t)^{β} on the interval t ∈ [−1, 1], see, e.g., [E53:vol.2,p.168–173].

Theorem 4.5 (Identification of the p_{k}^{∗}). It is immediate that

p_{0}^{∗}(x) = 1,  p_{1}^{∗}(x) = x − 1/2.  (4.6)

Moreover,

p_{k}^{∗}(x) = \frac{k!(k − 2)!}{(2k − 2)!}x(x − 1)P_{k−2}^{(1,1)}(2x − 1),  k ≥ 2.  (4.7)

Proof: Suppose that k ≥ 2. Then

c^{∗}(j, k) = ∏_{i=1}^{k−j} \frac{(k + 1 − i)(k − i)}{i(i − 2k + 1)} = \frac{(−k)_{k−j}(1 − k)_{k−j}}{(k − j)!(2 − 2k)_{k−j}} ,

and so, since c^{∗}(0, k) = 0,

\begin{aligned}p_{k}^{∗}(x) & = ∑_{j=0}^{k} c^{∗}(j, k)x^{j} = ∑_{j=1}^{k} \frac{(−k)_{k−j}(1 − k)_{k−j}}{(k − j)!(2 − 2k)_{k−j}} x^{j} \\ & = x ∑_{j=0}^{k−1} \frac{(−k)_{k−1−j}(1 − k)_{k−1−j}}{(k − 1 − j)!(2 − 2k)_{k−1−j}} x^{j}. \end{aligned}

Next use

(a)_{n−j} = \frac{(a)_{n}(−1)^{j}}{(1 − a − n)_{j}}

with n = k − 1 to obtain

\begin{aligned}p_{k}^{∗}(x) & = x\frac{(−k)_{k−1}(1 − k)_{k−1}}{(k − 1)!(2 − 2k)_{k−1}} ∑_{j=0}^{k−1} \frac{(1 − k)_{j}(k)_{j}}{(2)_{j}j!} x^{j} \\ & = x(−1)^{k−1} \frac{k!(k − 1)!}{(2k − 2)!}_{2}F_{1}(1 − k, k; 2; x). \end{aligned}

Apply Euler’s transformation [E53:vol.1,p.64(23)], to get

\begin{aligned}p_{k}^{∗}(x) & = x(x − 1)(−1)^{k} \frac{k!(k − 1)!}{(2k − 2)!}_{2}F_{1}(2 − k, k + 1; 2; x) \\ & = x(x − 1)\frac{k!(k − 2)!}{(2k − 2)!}P_{k−2}^{(1,1)}(2x − 1). \end{aligned}  (4.8)

This proves the result.

It is interesting to compare this result with the spectral properties of the Bernstein– Durrmeyer operator, which is a self adjoint operator on L_{2}[0, 1], defined by

M_{n}f(x) := ∑_{k=0}^{n} p_{k,n}(x) (n + 1) ∫_{0}^{1} f(t)p_{k,n}(t) dt, 0 ≤ x ≤ 1,

where

p_{k,n}(x) := \binom{n}{k}x^{k}(1 − x)^{n−k}, 0 ≤ k ≤ n.

Derriennic [D81] showed that the eigenvalues of M_{n} are

λ_{k,n} = \frac{n!}{(n − k)!} \frac{(n + 1)!}{(n + k + 1)!}, k = 0, 1, …, n,

and the corresponding eigenfunctions are the Legendre polynomials

P_{k}(2x − 1) := P_{k}^{(0,0)}(2x − 1), k = 0, 1, …, n.

Notice that these are independent of n. Similarly, in [BX91] it was shown for Bernstein– Durrmeyer operators with Jacobi weights w^{(α,β)}(x) := x^{α}(1 − x)^{β}, α, β > −1, the eigenfunctions are the Jacobi polynomials P_{k}^{(α,β)}(2x − 1).

The limiting dual functionals

By (1.3) and Theorem 4.5, it follows that

λ_{k}^{(n)} → 1, p_{k}^{(n)} → p_{k}^{∗}, n → ∞,

whilst the Weierstrass density theorem implies for all f ∈ C[0, 1] that

B_{n}f = ∑_{k=0}^{n} λ_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f) → f, n → ∞.

We now use these facts to investigate the limiting behaviour of μ_{k}^{(n)}(f) as n → ∞. Let L denote the operator of linear interpolation at 0 and 1, i.e.,

Lf(x) := (1 − x)f(0) + xf(1).  (4.9)

Lemma 4.10 (Orthogonal expansion). Each f ∈ C[0, 1] satisfying

∫_{0}^{1}(f(x) − Lf(x))^{2} \frac{dx}{x(1 − x)} < ∞,  (4.11)

or equivalently

∫_{0}^{1} \frac{(f(x) − f(0))^{2}}{x} dx < ∞, ∫_{0}^{1} \frac{(f(x) − f(1))^{2}}{1 − x} dx < ∞,  (4.12)

can be uniquely represented by a series of the form

f = ∑_{k=0}^{∞} p_{k}^{∗}μ_{k}^{∗}(f),  (4.13)

where the convergence of ∑_{k=2}^{∞} p_{k}^{∗}μ_{k}^{∗}(f) above is in the L_{2}(dx/x(1 − x))–norm, and the linear functionals μ_{k}^{∗} are defined by

μ_{0}^{∗}(f) := (f(0) + f(1))/2,  μ_{1}^{∗}(f) := f(1) − f(0),  (4.14)

μ_{k}^{∗}(f) := \frac{1}{2}\binom{2k}{k} \{(−1)^{k}f(0) + f(1) − k ∫_{0}^{1} f(x)P_{k−2}^{(1,1)}(2x − 1)dx\} ,  k ≥ 2. (4.15)

If f is differentiable on [0, 1], then

μ_{k}^{∗}(f) = \frac{1}{2}\binom{2k}{k} ∫_{0}^{1} f′(x)P_{k−1}(2x − 1)dx,  k ≥ 2,  (4.16)

where {P_{k}(x)}_{k≥0} are the Legendre polynomials.

Proof: Define inner products by

⟨g, h⟩ := ∫_{0}^{1} g(x)h(x) dx, ⟨⟨g, h⟩⟩ := ∫_{0}^{1} g(x)h(x) x(1 − x)dx.

Suppose that f satisfies the hypotheses of the lemma, and let

g(x) := \frac{f(x) − Lf(x)}{x(1 − x)}.

Then g ∈ L_{2}(x(1 − x)dx, [0, 1]), and so has a unique representation

g = ∑_{j=0}^{∞}⟨⟨g, g_{j}⟩⟩g_{j},

where

g_{j}(x) := \sqrt{\frac{(j + 2)(2j + 3)}{j + 1}}P_{j}^{(1,1)}(2x − 1), j = 0, 1, 2, …

are orthonormal Jacobi polynomials with respect to the weight function x(1 − x) on [0, 1]. This can be rewritten as

\frac{f(x) − Lf(x)}{x(1 − x)} = ∑_{j=0}^{∞}⟨⟨g, g_{j}⟩⟩g_{j}(x) = ∑_{j=0}^{∞}⟨f − Lf, g_{j}⟩g_{j}(x)

= ∑_{j=0}^{∞} \frac{(j + 2)(2j + 3)}{j + 1}⟨f − Lf, P_{j}^{(1,1)}(2 · −1)⟩P_{j}^{(1,1)}(2x − 1)

= \frac{1}{x(x − 1)} ∑_{j=0}^{∞} \frac{j + 2}{2}\binom{2j + 4}{j + 2}⟨f − Lf, P_{j}^{(1,1)}(2 · −1)⟩p_{j+2}^{∗}(x),

which gives

f(x) = Lf(x) − ∑_{k=2}^{∞} \frac{k}{2} \binom{2k}{k}⟨f − Lf, P_{k−2}^{(1,1)}(2 · −1)⟩p_{k}^{∗}(x).  (4.17)

Since

Lf(x) = \frac{f(0) + f(1)}{2} + (f(1) − f(0))(x − 1/2) = μ_{0}^{∗}(f) p_{0}^{∗}(x) + μ_{1}^{∗}(f) p_{1}^{∗}(x),

and

\begin{aligned}⟨Lf, P_{k−2}^{(1,1)}(2 · −1)⟩ & = f(0) ∫_{0}^{1}(1 − x)P_{(k−2)}^{(1,1)}(2x − 1)dx + f(1) ∫_{0}^{1} xP_{k−2}^{(1,1)}(2x − 1)dx \\ & = f(0)\frac{(−1)^{k}}{k} + \frac{f(1)}{k}, \end{aligned}

we obtain (4.13) from (4.17), with the convergence as asserted. Equation (4.16) follows from (4.15) using

P_{k−2}^{(1,1)}(2x − 1) = \frac{2}{k} P′_{k−1}(2x − 1), k ≥ 2,

and integrating by parts.

Clearly, condition (4.11) is satisfied when f is differentiable at 0 and 1, and so we have

f = ∑_{k=0}^{s} p_{k}^{∗}μ_{k}^{∗}(f),  ∀f ∈ Π_{s},  (4.18)

and

μ_{k}^{∗}(Π_{k−1}) = 0,  k = 1, 2, ….  (4.19)

These facts are now used to investigate the limiting behaviour of the dual functionals.

Theorem 4.20 (Limits of the dual functionals). For every f ∈ Π,

lim_{n→∞} μ_{k}^{(n)}(f) = μ_{k}^{∗}(f).  (4.21)

Proof: We prove (4.21) holds for f ∈ Π_{k+r}, r = 0, 1, 2, …, by strong induction on r (with the result holding for all k). Recall that

λ_{k}^{(n)} → 1,  n → ∞.  (4.22)

First suppose f ∈ Π_{k} (r = 0). Because B_{n} is degree reducing, we have

B_{n}f = ∑_{j=0}^{k} λ_{j}^{(n)}p_{j}^{(n)}μ_{j}^{(n)}(f) → f = ∑_{j=0}^{k} p_{j}^{∗}μ_{j}^{∗}(f),  n → ∞.  (4.23)

Since the convergence in (4.23) takes place in the finite–dimensional space Π_{k}, we may equate coefficients of x^{k} to obtain

λ_{k}^{(n)}μ_{k}^{(n)}(f) → μ_{k}^{∗}(f),  n → ∞,  (4.24)

which by (4.22) gives (4.21).

Now suppose f ∈ Π_{k+r}. Since B_{n} is degree reducing, we have

B_{n}f = ∑_{j=0}^{k+r} λ_{j}^{(n)}p_{j}^{(n)}μ_{j}^{(n)}(f) → f = ∑_{j=0}^{k+r} p_{j}^{∗}μ_{j}^{∗}(f), n → ∞.

Since the convergence above is in Π_{k+r}, equating coefficients of x^{k} gives

λ_{k}^{(n)}μ_{k}^{(n)}(f) + ∑_{j=1}^{r} λ_{k+j}^{(n)}c(k, k + j, n)μ_{k+j}^{(n)}(f) → μ_{k}^{∗}(f) + ∑_{j=1}^{r} c^{∗}(k, k + j)μ_{k+j}^{∗}(f), n → ∞.

(4.25)

By the inductive hypothesis together with (4.2) and (4.22), we have

∑_{j=1}^{r} λ_{k+j}^{(n)}c(k, k + j, n)μ_{k+j}^{(n)}(f) → ∑_{j=1}^{r} c^{∗}(k, k + j)μ_{k+j}^{∗}(f),

and so (4.25) gives (4.24) as before. This completes the induction.

The first few μ_{k}^{∗} are listed in the appendix.

Remark. We conjecture that (4.21) holds for all f ∈ C[0, 1], which is equivalent to the sequence (∥μ_{k}^{(n)}∥)_{n=0}^{∞} being bounded.

5. Application to iterates of the Bernstein operator

There have been a number of papers dealing with iterates of the Bernstein operator: Kelisky and Rivlin [KR67], Karlin and Ziegler [KZ70], Micchelli [M73], da Silva [Si85], Gonska and Zhou [GZ94], Sevy [Se95] and Wenz [W97]. We now investigate those results relevant to this work in terms of the diagonal form of B_{n}. By Theorem 2.3,

B_{n}^{j}f = ∑_{k=0}^{n}(λ_{k}^{(n)})^{j} p_{k}^{(n)} μ_{k}^{(n)}(f),  ∀f ∈ C[0, 1], j = 0, 1, 2, ….  (5.1)

Theorem 1 of [KR67] is that

lim_{j→∞} B_{n}^{j}f = Lf, ∀f ∈ C[0, 1],

where Lf is defined by (4.9). This follows immediately from (5.1) since λ_{k}^{(n)} < 1, k ≥ 2. More generally we have:

Corollary 5.2 (Limits for n fixed). Suppose (g_{j}) is a sequence of polynomials, with

lim_{j→∞} g_{j}(λ_{k}^{(n)}) = G(k, n), k = 0, 1, …, n,

then

lim_{j→∞} g_{j}(B_{n}) f = ∑_{k=0}^{n} G(k, n)p_{k}^{(n)}μ_{k}^{(n)}(f),  ∀f ∈ C[0, 1],  (5.3)

with the convergence above in the uniform norm.

Proof: The appropriate linear combination of (5.1) gives

g_{j}(B_{n})f = ∑_{k=0}^{n} g_{j}(λ_{k}^{(n)}) p_{k}^{(n)} μ_{k}^{(n)}(f),  ∀f ∈ C[0, 1].  (5.4)

Taking the limit j → ∞ then gives (5.3).

Let L_{n} denote the operator of Lagrange interpolation at the equally spaced points {0, 1/n, 2/n, …, 1},

L_{n}f(x) := ∑_{k=0}^{n}\{∏_{{j=0 \atop j≠k}}^{n} \frac{x − \frac{j}{n}}{\frac{k}{n} − \frac{j}{n}} \}f(\frac{k}{n}).

The biorthogonality condition (2.8) implies that f ↦ ∑_{k=0}^{n} p_{k}^{(n)}μ_{k}^{(n)}(f) reproduces Π_{n}, and so

L_{n}f = ∑_{k=0}^{n} p_{k}^{(n)}μ_{k}^{(n)}(L_{n}f) = ∑_{k=0}^{n} p_{k}^{(n)}μ_{k}^{(n)}(f).  (5.5)

In Sevy [Se95] the particular sequence of polynomials

g_{j}(x) := 1 − (1 − x)^{j}, j = 1, 2, …

was considered. For these

lim_{j→∞} g_{j}(λ_{k}^{(n)}) = 1,  (5.6)

(since 0 ≤ 1 − λ_{k}^{(n)} < 1) so that Corollary 5.2, together with (5.5), gives

lim_{j→∞} (1 − (1 − B_{n})^{j}) f = ∑_{k=0}^{n} p_{k}^{(n)}μ_{k}^{(n)}(f) = L_{n}f,  ∀f ∈ C[0, 1],  (5.7)

which is [Se95:Th.1] (also see Wenz [W97:Th.3] for a generalisation to the Bernstein– Schoenberg operator). The above operator has been studied by many, sometimes viewed as an iterated Boolean sum of B_{n}

⊕^{j}B_{n} = g_{j}(B_{n}) = 1 − (1 − B_{n})^{j}

(see Gonska and Zhou [GZ94] and the references therein). Since

1 − λ_{k}^{(n)} = O(\frac{1}{n}), n → ∞,

using (5.4) and (5.5) one obtains that ∀f ∈ C[0, 1]

∥L_{n}f − ⊕^{j}B_{n}f∥_{∞} = ∥ ∑_{k=0}^{n}(1 − λ_{k}^{(n)})^{j}p_{k}^{(n)}μ_{k}^{(n)}(f)∥_{∞} = ∥L_{n}f∥_{∞}O(\frac{1}{n^{j}} ),  n → ∞. (5.8)

From (5.8), and its pointwise analog, it is then possible to obtain approximation order results for ⊕^{j}B_{n} (large n) by appropriately modifying those for L_{n} (cf. [GZ94:Th.1]).

Lemma 5.9 (Limits of powers of the eigenvalues). Suppose that j_{n} is a sequence of positive integers with

lim_{n→∞} \frac{j_{n}}{n} = t,  (5.10)

then

lim_{n→∞}(λ_{k}^{(n)})^{j_{n}} = e^{− \frac{1}{2} k(k−1)t},  ∀k, 0 ≤ t < ∞,  (5.11)

and

lim_{n→∞}(λ_{k}^{(n)})^{j_{n}} = 0,  ∀k ≥ 2, t = ∞.  (5.12)

Proof: Let

y = (λ_{k}^{(n)})^{j_{n}−nt} = ((1 − \frac{1}{n})(1 − \frac{2}{n}) ⋯ (1 − \frac{k − 1}{n}))^{j_{n}−nt}.

Then

log y = (j_{n} − nt)(log(1 − \frac{1}{n}) + log(1 − \frac{2}{n}) + ⋯ log(1 − \frac{k − 1}{n}))

= (\frac{j_{n}}{n} − t)(−\frac{k(k − 1)}{2} + O(\frac{1}{n}))

→ 0, (n → ∞).

Therefore

lim_{n→∞}(λ_{k}^{(n)})^{j_{n}−nt} = lim_{n→∞} y = 1.  (5.13)

But

lim_{n→∞}(λ_{k}^{(n)})^{nt} = lim_{n→∞}(1 − \frac{1}{n})^{nt}(1 − \frac{2}{n})^{nt} ⋯ (1 − \frac{k − 1}{n})^{nt} = e^{− \frac{1}{2} k(k−1)t}.  (5.14)

Combining (5.13) and (5.14) gives (5.11). Let k → ∞ to obtain (5.12).

Theorem 2 of [KR67] shows that if (5.10) holds, then B_{n}^{j_{n}}(e_{s}) converges to a polynomial of degree s which is given explicity. We offer the following extension of this result.

Corollary 5.15 (Limits for 0 ≤ t ≤ ∞). Suppose that

lim_{n→∞} \frac{j_{n}}{n} = t,

then for 0 ≤ t < ∞

\begin{aligned}lim_{n→∞} B_{n}^{j_{n}}f & = ∑_{k=0}^{s} e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}μ_{k}^{∗}(f) \\ & = ∑_{k=0}^{∞} e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}μ_{k}^{∗}(f),  ∀f ∈ Π_{s}, \end{aligned}  (5.16)

and for t = ∞

lim_{n→∞} B_{n}^{j_{n}}f = Lf = ∑_{k=0}^{1} p_{k}^{∗}μ_{k}^{∗}(f)  ∀f ∈ Π,  (5.17)

with the convergence in (5.16) and (5.17) being in the uniform norm.

Proof: Suppose that f ∈ Π_{s}. Since B_{n} is degree reducing (5.1) gives

B_{n}^{j_{n}}f = ∑_{k=0}^{s}(λ_{k}^{(n)})^{j_{n}}p_{k}^{(n)}μ_{k}^{(n)}(f), n ≥ s.

Take the limit as n → ∞ in the above, and use Lemma 5.9 , Theorem 4.1 and Theorem 4.20 to obtain (5.17) and the first equality in (5.16). The second equality in (5.16) follows from (4.19).

Since the operators B_{n}^{j_{n}} : C[0, 1] → C[0, 1] have norm 1 and the polynomials are dense in C[0, 1], the limits in (5.16) and (5.17) exist for all f ∈ C[0, 1]. It has been shown that there exists a semigroup {B_{t} : t ≥ 0} of class (C_{0}) on C[0, 1] such that

lim_{n→∞} B_{n}^{j_{n}}f = B_{t}f,  whenever lim_{n→∞} \frac{j_{n}}{n} = t,  (5.18)

for all t ≥ 0 and f ∈ C[0, 1], and {B_{t} : t ≥ 0} is a positive contraction semigroup (see Karlin and Ziegler, [KZ70] and Micchelli [M73:Th.3.1]). Corollary 5.15 gives an explicit representation of this semigroup on the polynomials. This representation can be extended trivially to C[0, 1] in the case (5.17) and in the case (5.16) as follows.

Corollary 5.19 (Representation of {B_{t}}). The semigroup {B_{t} : t ≥ 0} defined by (5.18) has the representation

B_{t}f(x) = Lf(x) + x(1 − x) ∫_{0}^{1} G_{t}(x, y)(f − Lf)(y) dy,  ∀f ∈ C[0, 1],  (5.20)

where the kernel G_{t} is given by

G_{t}(x, y) := ∑_{k=2}^{∞} \frac{k(2k − 1)}{k − 1}e^{− \frac{1}{2} k(k−1)t}P_{k−2}^{(1,1)}(2x − 1)P_{k−2}^{(1,1)}(2y − 1).  (5.21)

Proof: Suppose first that f ∈ Π. From (4.14), μ_{0}^{∗}(f − Lf) = μ_{1}^{∗}(f − Lf) = 0, while by (4.19), μ_{k}^{∗}(Lf) = 0, k ≥ 2. Therefore (5.16) gives

B_{t}f = B_{t}(Lf) + B_{t}(f − Lf)

= ∑_{k=0}^{∞} e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}μ_{k}^{∗}(Lf) + ∑_{k=0}^{∞} e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}μ_{k}^{∗}(f − Lf)

= ∑_{k=0}^{1} p_{k}^{∗}μ_{k}^{∗}(Lf) + ∑_{k=2}^{∞} e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}μ_{k}^{∗}(f − Lf).

By (4.14) and (4.15), this becomes

B_{t}f = Lf + ∑_{k=2}^{∞} \frac{−k}{2} \binom{2k}{k}e^{− \frac{1}{2} k(k−1)t}p_{k}^{∗}(x) ∫_{0}^{1} P_{k−2}^{(1,1)}(2y − 1)(f − Lf)(y)dy.

Using the dominated convergence theorem and Theorem 4.5 , this gives

\begin{aligned}B_{t}f(x) & = Lf(x) \\ & + x(1 − x) ∫_{0}^{1} ∑_{k=2}^{∞} \frac{k(2k − 1)}{k − 1}e^{− \frac{1}{2} k(k−1)t}P_{k−2}^{(1,1)}(2x − 1)P_{k−2}^{(1,1)}(2y − 1)(f − Lf)(y)dy. \end{aligned}

This proves the result for f ∈ Π. The extension of (5.20) to all f ∈ C[0, 1] now follows from the density of Π in C[0, 1].

This representation of the semigroup {B_{t}} was given in [KZ70:(4.7)] and da Silva [Si85] (where total positivity properties of the kernel G_{t} are investigated).

The infinitesimal generator of the semigroup {B_{t}} is

Af := lim_{t→0^{+}} \frac{B_{t}f − f}{t} ,

whenever this limit exists. On the polynomials (4.13) and (5.16) give

Af = lim_{t→0^{+}} ∑_{k=2}^{∞} \frac{e^{− \frac{1}{2} k(k−1)t} − 1}{t}p_{k}^{∗}μ_{k}^{∗}(f) = ∑_{k=2}^{∞} −\frac{1}{2}k(k − 1)p_{k}^{∗}μ_{k}^{∗}(f),  ∀f ∈ Π. (5.22)

In [KZ70] it is shown that

Af(x) = \frac{1}{2}x(1 − x)D^{2}f(x),  ∀f ∈ C^{2}[0, 1].  (5.23)

Since

x(x − 1)D^{2}p_{k}^{∗}(x) = k(k − 1)p_{k}^{∗}(x), ∀k ≥ 2,

it is clear (5.22) and (5.23) are consistent. It is possible to extend (5.22) to C[0, 1] in the following way.

Corollary 5.24 (Infinitesimal generator of {B_{t}}). The infinitesimal generator A of the semigroup {B_{t} : t ≥ 0} defined by (5.18) has the representation

Af(x) = \frac{1}{2}x(x − 1) ∫_{0}^{1} G(x, y)(f − Lf)(y) dy,  ∀f ∈ C[0, 1],  (5.25)

where the kernel G is given by

G(x, y) := ∑_{k=2}^{∞} k^{2}(2k − 1)P_{k−2}^{(1,1)}(2x − 1)P_{k−2}^{(1,1)}(2y − 1).  (5.26)

It can also be represented by (5.22) and (5.23).

Proof: Suppose first that f ∈ Π. It follows from (5.22) that

Af = A(Lf) + A(f − Lf) = −\frac{1}{2} ∑_{k=2}^{∞} k(k − 1)p_{k}^{∗}μ_{k}^{∗}(f − Lf),

and so Theorem 4.5 and (4.15) give

Af(x)

= −\frac{1}{2} ∑_{k=2}^{∞} k(k − 1)p_{k}^{∗}(x)\frac{−k}{2} \binom{2k}{k} ∫_{0}^{1} P_{k−2}^{(1,1)}(2y − 1)(f − Lf)(y) dy

= \frac{1}{2}x(x − 1) ∫_{0}^{1} ∑_{k=2}^{∞} k^{2}(2k − 1)P_{k−2}^{(1,1)}(2x − 1)P_{k−2}^{(1,1)}(2y − 1)(f − Lf)(y) dy.

This gives (5.25) for f ∈ Π. The extension of (5.25) to all f ∈ C[0, 1] now follows from the density of Π in C[0, 1].

Iterates of all orders

As in [KR67:§4], it is possible to use (5.1) to define the iterates of B_{n} of all orders −∞ < t < ∞, in a manner consistent with the case when t is a nonnegative integer, namely

B_{n}^{t}f := ∑_{k=0}^{n}(λ_{k}^{(n)})^{t}p_{k}^{(n)}μ_{k}^{(n)}(f), ∀f ∈ C[0, 1].

By (5.5), the iterate B_{n}^{0} eqauls L_{n} (Lagrange interpolation at equally spaced points, which is the identity on Π_{n}). The inverse B_{n}^{−1} restricted to Π_{n} has been studied by Sablonnière [S92] (see next section).

6. Application to Bernstein quasi–interpolants

By (5.5), the operator of Lagrange interpolation at equally spaced points can be written as

L_{n}f = ∑_{k=0}^{n} p_{k}^{(n)}μ_{k}^{(n)}(f), ∀f ∈ C[0, 1],

while the Bernstein operator is

B_{n}f = ∑_{k=0}^{n} λ_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f), ∀f ∈ C[0, 1].

In this way the Bernstein operator can be thought of as being obtained from the Lagrange interpolant by ‘damping out’ the p_{k}^{(n)} coefficient (frequency) by the amount 0 < λ_{k}^{(n)} < 1, k ≥ 2. The failure of Lagrange interpolation at n equally spaced points to converge for all continuous functions (whilst the Bernstein approximants do) might then be explained by its failure to sufficiently damp out the highly oscillatory polynomials p_{k}^{(n)} (cf (3.6)). It is then natural to consider operators of the form

A_{n,α}f = ∑_{k=0}^{n} α_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f),  ∀f ∈ C[0, 1],  (6.1)

for other amounts of damping α_{k}^{(n)} ∈ IR. Remember that A_{n,α}f depends only on the values f(0), f(1/n), …, f(1). These operators are automatically degree reducing, and reproduce the linear polynomials if and only if α_{0}^{(n)} = α_{1}^{(n)} = 1. Indeed (6.1) is the diagonalised form of these operators. The linear combinations of iterates of the Bernstein operator considered in Section 6 (including iterated Boolean sums) are operators of this type. Presumably, by choosing the quantities α_{k}^{(n)} appropriately it should be possible to construct approximation processes inheriting some of the desirable properties of L_{n} (such as interpolation) and of B_{n} (like convergence as n → ∞ for all f ∈ C[0, 1]). We now suggest a few possibilities.

1. Continuous families. The eigenvalues of L_{n} could be continuously transformed via some parameter 0 ≤ t ≤ 1 into those for B_{n}, giving a family of operators A_{n,t}, which depends continuously on t, with endpoints A_{n,0} = L_{n} and A_{n,1} = B_{n}. Depending on the properties of the operator desired (or the smoothness of f) an appropriate value of t could then be chosen. A couple of such schemes for changing the eigenvalues (and corresponding operator) are

A_{n,t}f := ∑_{k=0}^{n}((1 − t) + tλ_{k}^{(n)})p_{k}^{(n)}μ_{k}^{(n)}(f) = (1 − t)L_{n}f + tB_{n}f, ∀f ∈ C[0, 1],

and

A_{n,t}f := ∑_{k=0}^{n}(λ_{k}^{(n)})^{t}p_{k}^{(n)}μ_{k}^{(n)}(f), ∀f ∈ C[0, 1].

2. Polynomial reproduction. The first j + 1 (0 ≤ j ≤ n) eigenvalues could be set to 1. This then gives an operator which reproduces Π_{j}, e.g.,

A_{n,j}f := ∑_{k=0}^{j} p_{k}^{(n)}μ_{k}^{(n)}(f) + ∑_{k=j+1}^{n} λ_{k}^{(n)}p_{k}^{(n)}μ_{k}^{(n)}(f), ∀f ∈ C[0, 1].

For this choice A_{n,0} = A_{n,1} = B_{n} and A_{n,n} = L_{n}. This property, together with reproduction of Π_{j} is shared with the left Bernstein quasi–interpolant of order j of Sablonnière [S92],

B_{n}^{(j)} := A_{n}^{(j)} ∘ B_{n},

where A_{n}^{(j)} is a truncated version of B_{n}^{−1} thought of as a differential operator on Π_{n}. It is not clear to the authors at this point how these two similar operators are related.

Operators which reproduce Π_{j} can also be obtained by taking affine combinations of Bernstein operators B_{n} of various degrees n, see, e.g., [B53], [DT87] and [Z95].

3. Adaptive methods. Since p_{k}^{(n)} is an eigenfunction of (6.1) with eigenvalue α_{k}^{(n)} the limiting properties of p_{k}^{(n)} and μ_{k}^{(n)} imply that for A_{n,α}f converge as n → ∞ for all continuous f (p_{k}^{∗} in particular) it is necessary that α_{k}^{(n)} → 1. If the rate of convergence of α_{k}^{(n)} → 1 is too fast (as in the case of Lagrange interpolation L_{n}, when α_{k}^{(n)} = 1) then A_{n,α}f fails to converge for some f. Hence it seems the approximation properties of A_{n,α} (large n) are controlled by the rates at which α_{k}^{(n)} → 1, as n → ∞.

A more detailed analysis of these questions seems worthy of further study.

7. The spectrum of the Kantorovich operator

Recall the Kantorovich operator K_{n} : L_{1}[0, 1] → C[0, 1], n = 1, 2, …, which is defined by

K_{n}f(x) := ∑_{k=0}^{n} \binom{n}{k}x^{k}(1 − x)^{n−k}(n + 1) ∫_{k/(n+1)}^{(k+1)/(n+1)} f(t) dt.

It satisfies

K_{n}(Df) = D(B_{n+1}f), ∀f ∈ C^{1}[0, 1],

and so in particular

K_{n}(Dp_{k+1}^{(n+1)}) = D(B_{n+1}p_{k+1}^{(n+1)}) = λ_{k+1}^{(n+1)}(Dp_{k+1}^{(n+1)}), k = 0, 1, …, n,

i.e., λ_{k+1}^{(n+1)} is an eigenvalue of K_{n} with corresponding eigenfunction Dp_{k+1}^{(n+1)}.

Corollary 7.1 (Eigenstructure of K_{n}). The eigenvalues of the Kantorovich operator K_{n} are

ν_{k}^{(n)} := λ_{k+1}^{(n+1)} = \frac{n!}{(n − k)!} \frac{1}{(n + 1)^{k}} , k = 0, 1, …, n,

and the corresponding eigenfunctions are polynomials of exact degree k given by

q_{k}^{(n)} := Dp_{k+1}^{(n+1)} (these have leading coefficient k + 1).

The eigenvalues of K_{n} are distinct

ν_{0}^{(n)} = 1 > ν_{1}^{(n)} = \frac{n}{(n + 1)} > ν_{2}^{(n)} = \frac{n(n − 1)}{(n + 1)^{2}} > ⋯ > ν_{n}^{(n)} = \frac{n!}{(n + 1)^{n}} > 0.

Many of the previous results for the Bernstein operator can now be adapted to the Kantorovich operator. For example, its diagonal form is

K_{n}f = ∑_{k=0}^{n} ν_{k}^{(n)} q_{k}^{(n)}μ_{k+1}^{(n+1)}(D^{−1}f), ∀f ∈ L_{1}[0, 1],

where

D^{−1}f(x) := ∫_{0}^{x} f,

and the analogue of (5.7) is

lim_{j→∞} (1 − (1 − K_{n})^{j}) f = D (L_{n+1}(D^{−1}f)) ,  ∀f ∈ L_{1}[0, 1].  (7.2)

The limit in (7.2) is the area matching map which interpolates from Π_{n} to the data

∫_{0}^{1/(n+1)} f, ∫_{1/(n+1)}^{2/(n+1)} f, ∫_{2/(n+1)}^{3/(n+1)} f, ⋯, ∫_{(n−1)/(n+1)}^{n/(n+1)} f, ∫_{n/(n+1)}^{1} f.

8. Multivariate Bernstein operators

The eigenstructure of the tensor product Bernstein operators (see [L53:p.51]) can be deduced from that of the univariate operator in the usual way. For simplicity, we illustrate this for the bivariate tensor product Bernstein operator B_{n} ⊗ B_{m} : C([0, 1]^{2}) → C([0, 1]^{2}), which is defined by

(B_{n} ⊗ B_{m})f(x) := ∑_{i=0}^{n} ∑_{j=0}^{m} \binom{n}{i}\binom{m}{j}x^{i}(1 − x)^{n−i}y^{j}(1 − y)^{m−j}.

Corollary 8.1 (Eigenstructure of B_{n} ⊗ B_{m}). The eigenvalues of B_{n} ⊗ B_{m} are

λ_{i,j}^{(n,m)} := λ_{i}^{(n)}λ_{j}^{(m)} = \frac{n!}{(n − i)!} \frac{m!}{(m − j)!} \frac{1}{n^{i}m^{j}} , i = 0, 1, …, n, j = 0, 1, …, m,

and the corresponding eigenfunctions p_{i,j}^{(n,m)} are given by

p_{i,j}^{(n,m)}(x, y) := p_{i}^{(n)}(x)p_{j}^{(m)}(y).

It follows from the analogue of (1.2), that the multivariate Bernstein operator B_{n} on a simplex S ⊂ IR^{d} (see [L53:p.51]) has the same spectra as the univariate operator, and the λ_{k}^{(n)}–eigenspaces has dimension \binom{k+d−1}{d−1}. Thus it is diagonalisable. More detailed results, including computational formulæ and symmetry properties of the eigenspaces can be found in [CW99].

9. Appendix

The eigenfunctions

The first 4 eigenfunctions are independent of n p_{0}^{(n)}(x) = 1, p_{1}^{(n)}(x) = x−1/2, p_{2}^{(n)}(x) = x(x−1), p_{3}^{(n)}(x) = x(x−1)(x−1/2), and the others depend on n. Here are the next 3 eigenfunctions in the factored form (2.11)

\begin{aligned}p_{4}^{(n)}(x) & = x(x − 1) ((x − 1/2)^{2} + \frac{2 − n}{4(5n − 6)}) , \\ p_{5}^{(n)}(x) & = x(x − 1)(x − 1/2) ((x − 1/2)^{2} + \frac{8 − 3n}{4(7n − 12)}) , \\ p_{6}^{(n)}(x) & = x(x − 1)((x − 1/2)^{4} + \frac{10 − 3n}{2(9n − 20)}(x − 1/2)^{2} \\ & + \frac{(n − 2)(n − 4)(6n^{2} − 23n + 40)}{16(9n − 20)(14n^{3} − 71n^{2} + 154n − 120)}). \end{aligned}  (9.1)

Using (2.7) the first few coefficients in (2.6) are

\begin{matrix}c(k, k, n) = 1,  0 ≤ k ≤ n, \\ c(k − 1, k, n) = −\frac{k}{2} ,  1 ≤ k ≤ n, \\ c(k − 2, k, n) = \frac{1}{24} \frac{k(k − 1)(k − 2)(6n + 5 − 3k)}{−k^{2} + 3k + 2nk − 2 − 3n},  2 ≤ k ≤ n, \\ c(k − 3, k, n) = −\frac{1}{48} \frac{k(k − 1)(k − 2)(k − 3)(2n + 2 − k)}{−k^{2} + 3k + 2nk − 2 − 3n},  3 ≤ k ≤ n. \end{matrix}  (9.2)

The dual functionals

The matrix P of Theorem 2.3 can be inverted as follows. Lemma 9.3 (Finding V). The j–th row of the matrix V = P^{−1} of Theorem 2.3, whose k–th column gives the coefficients of the dual functional μ_{k}^{(n)}, i.e.,

μ_{k}^{(n)}(f) = ∑_{j=0}^{n} v(j, k, n)f(\frac{j}{n}), k = 0, …, n,

can be calculated using the recurrence

\begin{aligned}v(j, n, n) & = (−1)^{n−j} \frac{n^{n}}{j!(n − j)!}, \\ v(j, n − k, n) & = \frac{(−1)^{n−j−k}n^{n−k}}{(n − j − k)!j!} \\ & − ∑_{s=0}^{k−1} \frac{k!}{s!}n^{s−k} v(j, n − s, n) c(n − k, n − s, n), k = 1, …, n, \end{aligned} (9.4)

where

\frac{1}{(n − k − j)!} := 0, j > n − k.

Proof: Let ℓ_{j}^{(n)} be the (Lagrange) polynomial of degree n satisfying ℓ_{j}^{(n)}(i/n) = δ_{ij}, i.e.,

ℓ_{j}^{(n)}(x) := ∏_{{i=0 \atop i≠j}}^{n} \frac{(x − i/n)}{(j/n − i/n)}.

Apply B_{n} in the form (1.1) and in the diagonal form (2.4) to ℓ_{j}^{(n)}, and equate the results to obtain

( \begin{matrix}n \\ j \end{matrix} ) x^{j}(1 − x)^{n−j} = ∑_{k=0}^{n} λ_{k}^{(n)}p_{k}^{(n)}(x)v(j, k, n),  j = 0, …, n.  (9.5)

Equating the coefficients of x^{n−k}, k = 0, …, n in (9.5) gives

\begin{matrix}(−1)^{n−j−k} ( \begin{matrix}n \\ j \end{matrix} ) ( \begin{matrix}n − j \\ n − j − k \end{matrix} ) \\ = ∑_{s=0}^{k−1} λ_{n−s}^{(n)} v(j, n − s, n) c(n − k, n − s, n) + λ_{n−k}^{(n)} v(j, n − k, n). \end{matrix}

For k = 0 this gives the first equation in (9.4), and for k = 1, …, n this can be solved for v(j, n − k, n) as follows

v(j, n − k, n)

= \frac{1}{λ_{n−k}^{(n)}} \{(−1)^{n−j−k} ( \begin{matrix}n \\ j \end{matrix} ) ( \begin{matrix}n − j \\ n − j − k \end{matrix} ) − ∑_{s=0}^{k−1} λ_{n−s}^{(n)} v(j, n − s, n) c(n − k, n − s, n)\}

= \frac{(−1)^{n−j−k}n^{n−k}}{j!(n − k − j)!} − ∑_{s=0}^{k−1} \frac{k!}{s!}n^{s−k} v(j, n − s, n) c(n − k, n − s, n).

giving (9.4). The columns of V correspond to the dual functionals μ_{k}, and the rows of P to the eigenfunctions p_{k}.

The first few matrices V are

( \begin{matrix}1/2 & −1 \\ 1/2 & 1 \end{matrix} ) , ( \begin{matrix}1/2 & −1 & 2 \\ 0 & 0 & −4 \\ 1/2 & 1 & 2 \end{matrix} ) , ( \begin{matrix}1/2 & −1 & 9/4 & −9/2 \\ 0 & 0 & −9/4 & 27/2 \\ 0 & 0 & −9/4 & −27/2 \\ 1/2 & 1 & 9/4 & 9/2 \end{matrix} )

( \begin{matrix}1/2 & −1 & 50/21 & −16/3 & 32/3 \\ 0 & 0 & −32/21 & 32/3 & −128/3 \\ 0 & 0 & −12/7 & 0 & 64 \\ 0 & 0 & −32/21 & −32/3 & −128/3 \\ 1/2 & 1 & 50/21 & 16/3 & 32/3 \end{matrix} )

( \begin{matrix}1/2 & −1 & 375/152 & −1625/276 & 625/48 & −625/24 \\ 0 & 0 & −175/152 & 2375/276 & −625/16 & 3125/24 \\ 0 & 0 & −25/19 & 250/69 & 625/24 & −3125/12 \\ 0 & 0 & −25/19 & −250/69 & 625/24 & 3125/12 \\ 0 & 0 & −175/152 & −2375/276 & −625/16 & −3125/24 \\ 1/2 & 1 & 375/152 & 1625/276 & 625/48 & 625/24 \end{matrix} )

( \begin{matrix}1/2 & −1 & 2681/1060 & −63/10 & 252/17 & −162/5 & 324/5 \\ 0 & 0 & −981/1060 & 36/5 & −594/17 & 648/5 & −1944/5 \\ 0 & 0 & −225/212 & 9/2 & 108/17 & −162 & 972 \\ 0 & 0 & −115/106 & 0 & 468/17 & 0 & −1296 \\ 0 & 0 & −225/212 & −9/2 & 108/17 & 162 & 972 \\ 0 & 0 & −981/1060 & −36/5 & −594/17 & −648/5 & −1944/5 \\ 1/2 & 1 & 2681/1060 & 63/10 & 252/17 & 162/5 & 324/5 \end{matrix} )

For example,

μ_{3}^{(4)}(f) = −16/3f(0) + 32/3f(1/4) + 0f(1/2) − 32/3f(3/4) + 16/3f(1).

The dual functionals can also be computed using the symmetry properties of μ_{k}^{(n)} and p_{k}^{(n)}. For example, the dual functional μ_{n−1}^{(n)}, n ≥ 1 must come from the 2–dimensional subspace (of span{f ↦ f(j/n) : j = 0, 1, …, n}) consisting of those functionals which annihilate Π_{n−2}. A basis for this space is

{[0, 1/n, …, (n − 2)/n, (n − 1)/n], [1/n, 2/n, …, (n − 1)/n, 1]}.

Here [x_{0}, x_{1}, …, x_{k}] denotes the divided difference at the points x_{0}, x_{1}, …, x_{k}, which if equally spaced equals

[x, x + h, x + 2h, …, x + kh]f = \frac{1}{k!h^{k}} Δ_{h}^{k}f(x) = \frac{1}{k!h^{k}} ∑_{i=0}^{k}(−1)^{k−i} ( \begin{matrix}k \\ i \end{matrix} ) f(x + ih). (9.6)

By the symmetry conditions (2.10) and (9.6), μ_{n−1}^{(n)} must be a scalar multiple of

[0, 1/n, …, (n − 2)/n, (n − 1)/n] + [1/n, 2/n, …, (n − 1)/n, 1],  (9.7)

which (by the symmetry conditions) annihilates p_{n}^{(n)}. For (9.7) to satisfy the condition μ_{n−1}^{(n)}(p_{n−1}^{(n)}) = 1, its scalar multiplier must be 1/2. Continuing to argue along these lines leads to the following.

Theorem 9.8. The dual functionals in (2.4) can be expressed as follows

μ_{0}^{(n)}(f) = \frac{1}{2} (f(0) + f(1)) , n ≥ 0,

μ_{1}^{(n)}(f) = f(1) − f(0), n ≥ 1,

μ_{n−3}^{(n)}(f) = \frac{1}{8} \frac{n(n + 1)}{n^{2} − 6}[0, \frac{1}{n}, …, \frac{n − 3}{n}]f

+ \frac{1}{8} \frac{(3n + 8)(n − 3)}{n^{2} − 6}[\frac{1}{n}, \frac{2}{n}, …, \frac{n − 2}{n}]f

+ \frac{1}{8} \frac{(3n + 8)(n − 3)}{n^{2} − 6}[\frac{2}{n}, \frac{3}{n}, …, \frac{n − 1}{n}]f,

+ \frac{1}{8} \frac{n(n + 1)}{n^{2} − 6}[\frac{3}{n}, \frac{4}{n}, …, 1]f, n ≥ 3,

μ_{n−2}^{(n)}(f) = \frac{1}{12} \frac{(n + 1)(3n − 2)}{n^{2} − 2}[0, \frac{1}{n}, …, \frac{n − 2}{n}]f

+ \frac{1}{6} \frac{(3n + 5)(n − 2)}{n^{2} − 2}[\frac{1}{n}, \frac{2}{n}, …, \frac{n − 1}{n}]f

+ \frac{1}{12} \frac{(n + 1)(3n − 2)}{n^{2} − 2}[\frac{2}{n}, \frac{3}{n}, …, 1]f, n ≥ 2,

μ_{n−1}^{(n)}(f) = \frac{1}{2} ([0, \frac{1}{n}, …, \frac{n − 1}{n}]f + [\frac{1}{n}, \frac{2}{n}, …, 1]f) , n ≥ 1,

μ_{n}^{(n)}(f) = [0, \frac{1}{n}, \frac{2}{n}, …, 1]f, n ≥ 0.

Formulæ for μ_{n−4}^{(n)}, μ_{n−5}^{(n)}, … can, in principle, also be obtained by this method, but are more complicated. There appears to be no simple closed form.

The limiting dual functionals

Using (4.14) and (4.15) to compute the first few μ_{k}^{∗} gives

μ_{0}^{∗}(f) = \frac{1}{2}f(0) + \frac{1}{2}f(1),

μ_{1}^{∗}(f) = −f(0) + f(1), μ_{2}^{∗}(f) = 3f(0) + 3f(1) − 6 ∫_{0}^{1} f(x) dx, μ_{3}^{∗}(f) = −10f(0) + 10f(1) − 120 ∫_{0}^{1} f(x)(x − 1/2) dx, μ_{4}^{∗}(f) = 35f(0) + 35f(1) − 2100 ∫_{0}^{1} f(x)(x^{2} − x + 1/5) dx, μ_{5}^{∗}(f) = −126f(0) + 126f(1) − 35280 ∫_{0}^{1} f(x)(x^{3} − 3/2x^{2} + 9/14x − 1/14) dx, μ_{6}^{∗}(f) = 462f(0) + 462f(1) − 582120 ∫_{0}^{1} f(x)(x^{4} − 2x^{3} + 4/3x^{2} − 1/3x + 1/42) dx.

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