Generalized Bernstein Polynomials and Symmetric Functions

Advances in Applied Mathematics 28, 17–39 (2002) doi:10.1006/aama.2002.0765, available online at http://www.idealibrary.com on

Robert P. Boyer

Department of Mathematics and Computer Science, Drexel University, Philadelphia, Pennsylvania 19104 E-mail: [email protected]

and

Linda C. Thiel

Society for Industrial and Applied Mathematics, University City Science Center, Philadelphia, Pennsylvania 19104 E-mail: [email protected]

Received February 20, 2001; accepted February 26, 2001

We begin by classifying all solutions of two natural recurrences that Bernstein polynomials satisfy. The first scheme gives a natural characterization of Stancu polynomials. In Section 2, we identify the Bernstein polynomials as coefficients in the generating function for the elementary symmetric functions, which gives a new proof of total positivity for Bernstein polynomials, by identifying the required determinants as Schur functions. In the final section, we introduce a new class of approximation polynomials based on the symplectic Schur functions. These polynomials are shown to agree with the polynomials introduced by Vaughan Jones in his work on subfactors and knots. We show that they have the same fundamental properties as the usual Bernstein polynomials: variation diminishing (whose proof uses symplectic characters), uniform convergence, and conditions for monotone convergence.  2002 Elsevier Science (USA) Key Words: Bernstein polynomials; total positivity; variation diminishing; Schur functions; symplectic group; group characters; Young tableau; unitary group; Jones polynomials.

INTRODUCTION

In this paper, we study Bernstein polynomials and other similar polynomial schemes that retain the shape of the approximating function and that

17 0196-8858/02 $35.00  2002 Elsevier Science (USA) All rights reserved. 18 boyer and thiel are computed with recurrence relations. These important properties make such approximations desirable for applications in computer-aided geometric design. We find all polynomial solutions for the Bernstein polynomial recurrence which are exhausted by the polynomials found by Stancu [17] and further investigated by many others including Goldman [8, 9, 11] who used urn models found in probability theory to develop their properties. We can visualize this recurrence with the Pascal triangle. We show that the only polynomial approximations that satisfy this Pascal recurrence and give a convergent approximation are exactly the Bernstein approximations. In particular, the Stancu approximations never converge. We give a new proof of the classical result that the Bernstein approximations are shape preserving by describing them as coefficients for the elementary symmetric functions and using the theory of the symmetric Schur functions. We study another one-dimensional approximation scheme, which comes from another family of symmetric functions: the symplectic Schur polynomials. We show that the corresponding symplectic Bernstein polynomials are also variation diminishing by showing that the polynomials form a totally positive basis. The proof that the symplectic Bernstein approximations converge uses both the ergodic theorem and the asymptotic character formula, which are described in the Appendix. This example is related to the algebra that gives the knot polynomial invariant found by Jones. This recurrence can also be visualized by a variant of the Pascal triangle. Some of our motivations for this study are the connections of polynomial approximations with special classes of symmetric functions, which are attached to the classical groups. Several other classical approximation operators such as the Baskakov and Szász–Mirakyan approximations are associated with representations of the unitary group.

1. PASCAL TRIANGLE RECURRENCE AND GENERALIZED BERNSTEIN POLYNOMIALS Recall that the Bernstein polynomials Bnt= n tk1 − tn−k, for k k 0 ≤ k ≤ n, give polynomial approximations that converge to a continuous function f on the unit interval 0 1. We express these approximations as a linear operator: n k n n k B f t= f Bnt= f tk1 − tn−k n n k k n k=0 k=0

Bernstein polynomials obey a useful recurrence relation that is important in n+1 n n computer-aided design applications: Bk t=1 − tBk t+tBk−1t. generalized bernstein polynomials 19

Stancu [17] introduced a generalized of Bernstein polynomials by n tt +α···t +k−1α1−t1−t +α···1−t +n−k−1α Snt= k k 1+α1+2α···1+n−1α

n n where α ≥ 0. For α = 0, we see that Sk t=Bk t. If we wish to empha- n size the role of the parameter, we will write Sα kt. Again, an operator can be associated to this family of polynomials: n k S f t= f Snt n n k k=0 The Stancu polynomials satisfy the recurrence: 1 − t +n − kα t +k − 1α Sn+1t= Snt+ Sn t k 1 + nα k 1 + nα k−1 We review the deﬁnitions needed to discuss the variation-diminishing property from Goldman [8]. (For a fuller discussion, see Chapter 1 of [12].)

For any ﬁnite sequence of real numbers, say T =t0t1tn, let zT denote the number of zeros in this list while vT denotes the number of sign changes in T (ignore zeros). Next, for a function g a b→R,we deﬁne

zg=sup zgt1gt2gtn

vg=sup vgt1gt2gtn where the supremum is taken over all a

n n n where a

1.1. Characterization of Polynomial Solutions Goldman used urn models to generate discrete probability distributions that give useful polynomial approximation schemes that generalized the n Bernstein polynomials. The urn polynomials Dk t, for 0 ≤ k ≤ n, have the following properties: n 0 (1.1.1) Dk t are polynomials of degree n, D0 t=1; n (1.1.2) 0 ≤ Dk t≤1, for 0 ≤ t ≤ 1; n n (1.1.3) k=0 Dk t=1; 1 (1.1.4) “normalized condition” D0 t will equal either t or 1 − t; n+1 n n n (1.1.5) “Pascal recurrence”: Dk t=ek tDk t+fk−1t n n n Dk−1t, for 0 ≤ k ≤ n + 1, where ek and fk are polynomials of degree 1 n n such that 0 ≤ ek t, fk t≤1 for 0 ≤ t ≤ 1. By convention, these polynomials are zero for k<0ork>n; n n (1.1.6) ek t+fk t=1. We will study these polynomials not through the urn model but solely through these six properties. The recurrence can be visualized by means of the Pascal triangle. The superindex n indicates the horizontal level in the triangle, while the subindex k indicates the position of the node going from left to right. The n n n linear polynomials ek t and fk t are edge weights, where ek is placed n on the edge joining n k to n + 1k, while fk is placed on the edge joining n k with n + 1k+ 1. 1 0 We ﬁnd that the polynomial D0 t=e0 t, the edge weight joining 0 0 to 1 0. If γ is a path joining 0 0 to n k, let the weight of the path wgtγ be the product of all the weights of the edges in the path γ. We observe that n Dk t= wgtγ γ is a path from 0 0 to n k generalized bernstein polynomials 21

We will study polynomial solutions whose path sum representation is path independent; that is, the value of wgtγ is the same for any path from 0 0 to n k. Examples of path-independent solutions are the Bernstein and Stancu polynomials. We give a characterization of such polynomial schemes directly from the recurrence relation. As a notational convention, we write dnt for the common value of any of the terms in the path k sum which is also equal to Dnt/ n . In particular, we observe that, for a k k m path-independent family Dj t m+1 m m m m 117 dj t=ej dj = fj−1 dj−1 where j ≤ m n Theorem 1. Let Dk t, for 0 ≤ k ≤ n n = 0 1 2, be a family of polynomials satisfying properties (1.1.1)–(1.1.6) whose path sum representation 0 is path independent, and with e0 t=t. Then n n Dk t=Sk 1 − t n 0 n Here, Sk t is the Stancu polynomial. If e0 t=1 − t,weﬁndDk t= n Sk t. Proof. For n>0, we ﬁx a value of the index k, with 0 ≤ k ≤ n − 1. By (1.1.6) and (1.1.7), with m = n and j = k + 1, we obtain

f ndn dn en = k k = 1 − en • k k+1 n k n dk+1 dk+1 0 1 We shall now assume that e0 t=t. Then it follows that e0 t evalu- 1 1 ated at t = 1 is 1. Since e0 is a linear polynomial and 0 ≤ e0 t≤1, for 0 ≤ t ≤ 1, we ﬁnd t + α e1t= 0 1 + α So, 1 − t f 1t=1 − e1t= 0 0 1 + α 1 2 2 2 By the divisibility condition, f0 f0 , we must have e0 1=1 and e0 t must be increasing. In particular, t + β e2t= for some value of β 0 1 + β So, 1 − t f 2t= 0 1 + β 22 boyer and thiel

By identity (), t + α e2t= 1 1 + β so 1 − t +β − α e2t= 2 1 − t + α

2 For e2 t to be a polynomial, β must equal 2α. This argument generalizes to give t +n − 1α ent= 0 1 +n − 1α Further, we have by induction that t +n − k − 1α ent= 0 ≤ k ≤ n − 1 k 1 +n − 1α We can now conclude that the solutions we have found are all given as n n Sk 1 − t, where Sk t is the Stancu polynomial. This ends the proof. This characterization of Stancu polynomials may be related to a result about sequences of polynomials pkx of binomial type with persistent roots, so pkx=ckx − r1x − r2···x − rk. Such sequences must have roots that form arithmetic sequences so rk =k − 1α, for some ﬁxed α [5]. n Note that the unnormalized form of S0 t, that is, pkt=tt + α···t + k − 1α, falls into this class of polynomials. Other families of polynomials of binomial type give known approximation operators but they do not satisfy the recurrence relations studied in this paper.

1.2. Convergence for Bernstein Polynomials Only We have now seen that the Pascal recurrence (1.1.5) is a very strong condition. We now consider weaker conditions. Recall the notational convention that Dnt= n dnt. In this section, we will consider a family k k k n dk t, with 0 ≤ k ≤ n, n = 0 1 2, that satisﬁes the following four conditions: n 0 (1.2.1) dk t are continuous functions on 0 1d0 t=1; n (1.2.2) 0 ≤ dk t≤1, for 0 ≤ t ≤ 1; (1.2.3) n n dnt=1; k=0 k k n n+1 n+1 (1.2.4) index raising condition: dk t=dk t+dk+1 t, where 0 ≤ k ≤ n. generalized bernstein polynomials 23

There are many more solutions to these relations than the ones in the previous section. Examples of such families are given by a continuous family of probability measures on 0 1, say t → µt. Then we define dnt as k 1 k n−k 0 x 1 − x dµt x. Further, the following theorem will explain the well- known fact that the Stancu approximations do not converge to the function for a fixed value of the parameter α. n Theorem 2. Let Dk t be a continuous family that satisfies (1.2.1)– n n (1.2.4). Then Dk t gives a convergent scheme if andonly if Dk t= n Bk t, the n kth Bernstein polynomial. Proof. We first identify solutions to the recurrence (1.2.4) as scalars, n not continuous functions. Let ak , with 0 ≤ k ≤ n, n = 0 1 2, be real numbers such that n (1) ak ≥ 0 for all k and n; 0 (2) a0 = 1; n n+1 n+1 (3) ak = ak + ak+1 , for 0 ≤ k ≤ n. n ∞ Set an = an . Then ann=0 is a completely monotone sequence; that is, k+1 −a #an ≥ 0, for all k, where #an = an+1 − an, the first forward difference operator. Further, we observe that the sequence an uniquely deter- n mines the entire sequence ak . See the work of Feller [6, Chap. VII]. The collection of all sequences satisfying (1)–(3) forms a convex set. is affinely isomorphic to the set of probability measures on the unit interval 0 1 under the correspondence a = 1 xn dνx, for a unique n 0 n 1 k n−k probability measure ν. Further, ak = 0 x 1 − x dνx. In particular, the extreme points of the convex set are given by n k n−k n ak = x 1 − x = bk x, for some fixed value of x ∈0 1 with the n corresponding measure δx. Furthermore, since the map t → Dk t is continuous, we find that the map t → νt is continuous, relative to weak convergence of measures and where νt is the unique probability measure n on 0 1 determined by Dk t Let & be the Pascal triangle path space, that is, the set of all infinite paths starting at node 0 0.Let

n Tk = ω ∈ & ω passes through n k be a cylinder set. Here, & is given as a projective limit space lim← &n, where &n is the ﬁnite set of paths from 0 0 to n k, for 0 ≤ k ≤ n, and the map from &n+1 to &n is obtained by deleting the edge from the nth to the n + 1th n n level. Since dk t0 satisﬁes the recurrence (1.2.4), Dk t determines a n n (projective limit) probability measure µt on & by µt Tk =Dk t for 24 boyer and thiel t ∈0 1. Further, the map t → µt is also continuous relative to weak convergence of measures. n k Let Sn = k=0 χ n , so we have n Tk n k S ω dµω= Dnt n n k 0 & k=0 Let f 0 1→R be continuous. Then n k f ◦ S ω dµω= f Dnt n n k 0 & k=0 which is the polynomial approximation to f given by the polynomial scheme n Dk t. We now consider

1 n n n µt Tk = Dk t= Bk x dνt x 0 So, n k n k 1 f Dnt= f Bnx dν x n k n k t k=0 k=0 0 1 n k = f Bnx dν x n k t 0 k=0 Since n f k Bnx converges uniformly to f on 0 1, we obtain k=0 n k n k 1 f Dnt → f x dν x n k 0 t k=0 0

n We conclude that the approximation Dnf , corresponding to Dk t, converges to f t, for all continuous functions f if and only if νt = δt which n n will hold if and only if Dk t=Bk t. This ends the proof. We record the following corollary of the proof.

n Corollary 3. Let Dk t be a family of continuous functions satisfying n (1.2.1)–(1.2.4). Represent Dk t as an integral relative to the Bernstein n 1 n polynomials, so Dk t= 0 Bk x dµt x, where µt is a probability measure on 0 1. Let Dn be the corresponding approximation operator n associatedwith Dk t andlet f be a continuous function on 0 1. Then 1 Dnf t converges to 0 f x dµt x. generalized bernstein polynomials 25

k For the Stancu approximations, it now follows that limn→∞ Snt = k S0 t.Fork = 2, this was veriﬁed by explicit calculation in Goldman’s paper [8]. Further, the measure m on 0 1 that corresponds to the Stancu polynomials is given by the Beta-distribution, so

1 n )1/α t−α/α 1−t−α/α k n−k Sk t= 1 − x x x 1 − x dx )t/α)1 − t/α 0

This connection of the Stancu approximation with the Beta-distribution has also been mentioned in [4].

1.3. Bernstein Polynomial andElementary Symmetric Functions We can view Bernstein polynomials as coefﬁcients of an expansion in terms of symmetric functions. See Chapter 1 of [14] for details about symmetric functions. We let enz z denote the kth elementary k 1 N symmetric function z z ···z . We start with the generating 1≤i1<···

N N j N 1 + tzk = t ej z1zN k=1 j=0 N 1 + tz N tj k = eNz z 1 + t N j 1 N k=1 j=0 1 + t N 1 t N t j 1 N−j + z = eNz z 1 + t 1 + t k 1 + t 1 + t j 1 N k=1 j=0

We now set

t 1 β = 1 − β = 1 + t 1 + t

We ﬁnd

N N j N−j N 1 − β + βzk= β 1 − β ej z1zN k=1 j=0

Note: t>0 if and only if 0 ≤ β ≤ 1. N j N−j We write bj t for t 1 − t . 26 boyer and thiel

We consider the determinant N N N b t b t ··· b t 0 0 0 1 0 N bNt bNt ··· bNt 1 0 1 1 1 N N N N bN t0 bN t1 ··· bN tN 1 − t N 1 − t N ··· 1 − t N 0 1 N t 1 − t N−1 t 1 − t N−1 ··· t 1 − t N−1 0 0 1 1 N N = N N N t0 t1 ··· tN This last determinant now equals

N N N 1 − t0 1 − t1 ···1 − tN 11··· 1 t /1 − t t /1 − t ··· t /1 − t 0 0 1 1 N N t2/1 − t 2 t2/1 − t 2 ··· t2 /1 − t 2 × 0 0 1 1 N N N N N N N N t0 /1 − t0 t1 /1 − t1 ··· tN /1 − tN

Finally, we can rewrite this determinant by a change of variables: xj = tj/1 − tj. 11··· 1 x x ··· x 0 1 N N N x2 x2 ··· x2 1 − tj 0 1 N j=0 N N N x0 x1 ··· xN

We will denote this last determinant as #. Note: 0

Let λ λ1 ≥ λ2 ≥ ··· ≥ λN ≥ 0, with λ ∈ Z. λ is called a signature.We can associate a (Young) diagram with λ which is a table with N rows with

λj boxes in row j.Asemistandard tableau over 1 2N for λ is a function T λ → 1 2N such that T is strictly increasing down any column, but is nondecreasing along any row. We can say that T has shape

λ. Finally, we can associate a monomial over z1z2zN with T by T z = zT i j i j∈λ. The Schur function Sλz1z2zN is given as a quotient of two vandermonde-type determinants:

λ +N−1 λ +N−1 λ +N−1 z 1 z 1 ··· z 1 1 2 N λ2+N−2 λ2+N−2 λ2+N−2 z z ··· zN 1 2 λ λ λ z N z N z N 1 2 N N−1 N−1 N−1 z1 z2 ··· zN N−2 N−2 N−2 z1 z2 ··· zN 11 1

It was discovered by Knuth [14] that the Schur function Sλ has the alternate representation: T Sλz1z2zN = z T is a semistandard tableau of shape λ We now can form k × k subdeterminants from #. These subdeterminants are obtained from the following matrix by a reversal of rows:   λ +k−1 λ +k−1 λ +k−1 x 1 x 1 ··· x 1  1 2 k   λ +k−2 λ +k−2 λ +k−2   x 2 x 2 ··· x 2   1 2 k        λk λk λk x1 x2 ··· xk By the Knuth form of the Schur functions, we ﬁnd that all k × k subdeterminants have the same sign. Hence, we recover the well-known result that the Bernstein approximations are variation diminishing. 28 boyer and thiel

2. SYMPLECTIC BERNSTEIN POLYNOMIALS

The above construction relating Bernstein polynomials with the symmetric Schur functions indicates the possibility of adapting these insights to other classes of symmetric functions. Here, we explore the symplectic symmetric functions which describe the characters of the symplectic group. See the book by Fulton and Harris [7] for a description of the symplectic group and an introduction to the determinantal form of the characters of the classical groups. Note that the construction given here does not require knowledge of the symplectic group. By a symplectic tableau we mean a column-strict numbering of the dia- ¯ ¯ ¯ gram λ λ1 ≥ λ2 ≥ ··· ≥ λn over the alphabet 1 < 1 < 2 < 2 < ···n

n We now claim that cj β is, in fact, a polynomial in τ = β1 − β. This will follow easily from the following observations:

1 1 2 2 (2.1) c0 β=β1 − β=τ and c1 β=β +1 − β = 1 − 2τ; n+1 n (2.2) cj+1 β=β1 − β cj β, for 0 ≤ j ≤ n; n+1 2 2 (2.3) “analogue of basic recurrence”: ck β=β +1 − β × n n ck − β1 − βck+1β, for 0 ≤ k ≤ n. To define the symplectic Bernstein polynomials requires a brief excursion into group theory in order to give the analogue of the binomial coefficients. The dimension of the irreducible representation of the symplectic group Sp2n with signature 1j, with 0 ≤ j ≤ n, is given as 2n , for j = 0 1, or j 2n − 2n , for 2 ≤ j ≤ n. We shall write this dimension as 2n j. These j j−2 dimensions (positive integers) can be displayed in a triangular form which is similar to the usual binomial coefficients in the Pascal triangle but the edges are more complicated. The usual binomial coefficients are the dimension of the irreducible representation of the unitary group UN with signature 1j. For 1 ≤ j ≤ n, we find that 2n + 1j=2n j + 1+22n j+2n j − 1 while 2n + 1n+ 1=22n n+2n n − 1 2n + 1 0=2n 0

n n We let pk τ denote the corresponding polynomial to cj β. In this n n paper, we call Pk τ=2n kpk τ a symplectic Bernstein polynomial. These polynomials have arisen already in the study of subfactors and knot theory (see Section 2.8 in [10]). n n For positivity, we require that 0 ≤ β ≤ 1. Since ck β=ck 1 − β, we need to restrict β further to the range 0 ≤ β ≤ 1/2 to ensure unique- ness. Since τ = β1 − β, we obtain the range 0 ≤ τ ≤ 1/4. In terms of t, n the restriction is 0 ≤ t ≤ 1. Note that for this range, pk τ≥0, for all 0 ≤ k ≤ n.

2.1. Symplectic Bernstein Approximation andthe Variation-Diminishing Property We now show that the symplectic Bernstein polynomials give variation- diminishing approximations. Our method is to adapt the reasoning given above relating the usual Bernstein polynomials with symmetric Schur functions. n We start with a useful algebraic identity for cj . It is easier to work with it as a function of t. 30 boyer and thiel

Lemma 4. tm−i+1 − t−m−i+1 1 cmt=tm i t − t−1 1 + t2m

m Let M = Mt1t2tm=ci t, where 0

λ∗=m − i∗ − k + 1m− i∗ − k + 1m− i∗ − k + 1

(a “rectangular” block signature). In particular, the function is given explic- itly as

∗ ∗ λi +k−i+1 −λi +k−i+1 det tj − tj k−i+1 −k−i+1 det tj − tj

By the analogue of the Knuth identity, the symplectic symmetric function can be written as a positive sum of monomials. This yields immediately that C Sλ∗ t1t2tk is positive. We conclude that the sign of M∗ will be determined by the sign of the k k −1 determinant # multiplied by j=1 tj /tj − tj , where k−i+1 −k−i+1 # = det tj − tj generalized bernstein polynomials 31

But k 2 k # = ti − tjti − tj ti − 1/t1t2 ···tk 1≤i

k k −1 We ﬁnd that # j=1 tj /tj − tj is always positive. This ends the proof. We next study how the framework for the Bernstein polynomials carries over to this situation. n Consider the convex set of all sequences ak , where 0 ≤ k ≤ n n = 0 0 1 2 of nonnegative real numbers such that a0 = 1 satisfying the consistency recurrence relation (analogue of (1.2.4)):

an = an+1 + 2an+1 + an+1 with 0 ≤ k ≤ n − 1 211 k k k+1 k+2 n n+1 n+1 an = an + 2an+1

n The extreme points of have the form pk τ, for some fixed value of τ ∈0 1/4. This is the analogue of the characterization of the normalized form of the Bernstein polynomials mentioned in the proof of Theorem 2. We now sketch a proof of this characterization. This is an old result first proven by Wassermann, Handelman, and others. For a proof in a more general situation, see [2] or the references in [10]. The proof given here is relatively self-contained. Proposition 6. The extreme points of the convex set are given by eval- n uating the symplectic Bernstein polynomials pk τ, for some value of τ ∈ 0 1/4. Proof. We first show that any extreme point must have the form of n a symplectic Bernstein polynomial. Suppose that the sequence ck is extremal. We consider an enlarged version of where we drop the 0 normalization condition that the elements must satisfy a0 = 1. The result- n ing set is now a cone. Now, if ak is nonnegative, satisfies (2.1.1), and n n n n n is dominated by ck so 0 ≤ ak ≤ ck , we find that ak = τck , for n n+1 some constant τ. In particular, if we define ak as ck+1, we find that n n ak is nonnegative, satisfies (2.1.1), and is dominated by ck . We con- n+1 n clude that ak+1 = τak , for some constant τ. Setting n = k = 0 yields 1 n+1 n n n c1 = τ. Further, since cn+1 = τcn we have cn = τ . By the relation n n+1 n+1 n+1 n cn = cn + 2cn+1 , we also obtain cn = τ 1 − 2τ. With the recur- n rence (2.1.1), we can now compute the remaining coefficients ck along the diagonals where n − k is a constant. In other words, the extreme points are given by evaluating the symplectic Bernstein polynomials at some value 32 boyer and thiel of τ. Since point evaluations are extremal, all that remains is to check n when this evaluation is nonnegative. We saw above that τ → pk τ is nonnegative for τ ∈0 1/4. By a calculation of the asymptotic character formula (see [2]), the values of τ must lie in this range to give nonnegative solutions. This ends the proof. To define the approximation operator associated with the polynomi- n als Pk τ, we need to use the framework introduced in the proof of Theorem 2. We let & denote the space of infinite paths starting at node n 0 0;weletTk denote the set of infinite paths in & that pass through node n k. For a fixed τ ∈0 1/4 we define a probability measure µτ n n on & by µτTk =Pk τ. n We associate an approximation operator to Pk τ through n k k S ω= 1 − χ n ω n T 2n 2n k k=0 Then n k k n k k S ω dµω= 1 − µ T n = 1 − Pnτ n 2n 2n k 2n 2n k & k=0 k=0 Let f 0 1/4→R be continuous. Then the approximation operator

Dnf τ is given as n k k f 1 − Pnτ 2n 2n k k=0 which can be identiﬁed with & f ◦ Snω dµω. Hence, Proposition 5 implies that the approximation operator Dnf is variation diminishing.

2.2. Convergence Properties We ﬁrst study pointwise convergence of the approximation operators. The version of the ergodic theorem given below together with the result from [2] yields the observation that for almost all paths p = kN (relative to the measure µτ) the limit kN /2N exists. In turn, this gives the following result which follows since almost everywhere pointwise convergence implies convergence in measure. Let 1>0 be given. Then we have n k k lim Pnτ 1 − − τ >1 = 0 n→∞ k 2n 2n k=0 generalized bernstein polynomials 33

We sketch the proof that Dnf τ→f τ pointwise on 0 1/4.Let M denote the maximum of f on 0 1/4. This proof follows the same pattern as the proof that the Bernstein approximations converge; however, that argument uses the law of large numbers, while here the reasoning uses a version of the ergodic theorem. A new framework is needed here since the symplectic Bernstein polynomials do not correspond to a sequence of independent random variables. Let 1>0 be given. Choose δ>0sos − t <δimplies f s−f t <1. Fix τ. Then n n Dnf τ−f τ = f τ−f xk Pk τ ≤ 31 + 32 where n n n 31 = f τ−f xk Pk τ k such that xk − τ ≤ δ and n n n 32 = f τ−f xk Pk τ k such that xk − τ >δ

Here, xn = k 1 − k . By uniform continuity and the fact that k 2n 2n n n k=0 Pk τ=1, we ﬁnd that 31 ≤ 1, while n n 221 32 ≤ 2M Pk τ k such that xk − τ >δ

We conclude that 32 → 0asn →∞. To sum up, we have now shown that if f is continuous on 0 1/4 then Dnf converges pointwise to f . We next consider uniform convergence. We ﬁx a value of τ0, with 0 < τ0 < 1/4. We will show that the convergence of Dnf is uniform on the interval 0τ0. By the inequality in (2.2.1), it is enough to show that n n Pk τ xk − τ >δ converges uniformly to 0 on 0τ0. We start with n n n n Pk τ xk − τ >δ = 2n k pk τ xk − τ >δ 2n ≤ pnτ xn − τ >δ k k k 2n = τkpn−kτ xn − τ >δ k 0 k

We now use another form for the polynomial pnτ found in Section 2.8 0 √ j 2j+1 2j+1 of [10]: p0 τ=σ+ − σ− /σ+ − σ−, where σ± =1 ± 1 − 4τ/2. 34 boyer and thiel

We now continue with the inequalities: 2n τkpn−kτ xn − τ >δ k 0 k 2n σ2n−k+1 − σ2n−k+1 k + − n = τ xk − τ >δ k σ+ − σ− 1 2n k 2n−k+1 n ≤ τ σ+ xk − τ >δ σ+ − σ− k 1 2n ≤ τkσ2n−k+1 xn − τ >δ k + k 1 − 4τ0 This last quantity can be related to the standard Bernstein polynomials. We τ note that σ+ + = 1. Then the last term of the above inequality becomes σ+ σ 2k τ k + σ2n−k xn − τ >δ k σ + k 1 − 4τ0 + σ 1 2n 2n k k 2 τ k ≤ + 1 − − τ σ2n−k 2 k 2n 2n σ + 1 − 4τ0 δ k=0 +

For convenience, we write x for τ/σ+ and so 1 − x = τ in the last expres- sion; we also use Bernstein operator notation B2nf to write it as σ 1 + B x21 − x2 − 2B x1 − xx1 − x+x21 − x2 2 2n 2n 1 − 4τ0 δ σ ≤ + M 2 n δ 1 − 4τ0 where ! ! M = !B x21 − x2−2B x1 − xx1 − x+x21 − x2! n 2n 2n ∞

We see at once that Mn → 0 so we conclude that Dnf converges uniformly to f on 0τ0. We record these results in the following theorem.

Theorem 7. Let f 0 1/4→R be continuous. Let Dnf be the polynomial approximation given by the symplectic Bernstein polynomials. Then:

(1) The approximation operators Dnf are variation diminishing.

(2) The approximations Dnf converge to f pointwise on 0 1/4 and uniformly on every subinterval 0τ0, where 0 <τ0 <τ. generalized bernstein polynomials 35

We now investigate conditions to ensure that the approximation operators converge monotonically. A consequence of this result will be uniform convergence as well. By direct calculation, we find that the symplectic Bernstein approximations do not converge monotonically for τ2 which dif- fers from the classical Bernstein case. Theorem 8. If g is an increasing, andconcave function on 0 1/4, then the sequence Dng is an increasing sequence on 0 1/4. We will write xn = k 1 − k and zn = k . k 2n 2n k 2n We will first show that Dne1τ is an increasing sequence where e1τ=τ. It is helpful to work with a simplified form of Dne1: n n n Enτ= 2n kzk pk τ k=0

Lemma 9. The sequence Enτ is increasing on 0 1/4.

Proof. We will show that En+1τ−Enτ≥0. Now, this difference is a polynomial of degree at most n + 1 so it can be expanded as a linear com- n+1 bination of the polynomials pk τ. By the variation-diminishing property of these polynomials, we observe that this difference is nonnegative on 0 1/4 if and only if all the coefficients in this linear combination are nonnegative. So, we write n+1 n+1 n+1 222 Enτ= ak pk τ k=0 To prove the lemma, it will suffice to verify that n+1 n+1 2232n + 2kzk − ak ≥ 0 We find that   2n 0zn if k = 0,  0    22n kzn +2n 1zn if k = 1,  0 1  n+1 n n ak = 2n k − 2z + 22n k − 1z  k−2 k−1   n  +2n kz if 2 ≤ k ≤ n,  k   n n 2n n − 1zn−1 + 22n 2nzn if k = n + 1. n Now, we define coefficients tk j as ratios of dimensions where 2n k − j tn = where 0 ≤ j ≤ 2 k j 2n + 2k n where we use the convention that tk j = 0ifk − j<0orifk − j>n+ 1. 36 boyer and thiel

2 n By construction, we must have j=0 tk j = 1 in all cases. To verify (2.2.3), it is equivalent to show that

n+1 n n 224 # = zk − tk j zk−j ≥ 0 j For k = 0ork = 1, we ﬁnd that # = 0. For k = n + 1, we have that # = 1 . For 2 ≤ k ≤ n, we calculate that 22n+3 kk − 1 # = n + 12n + 3nn − k + 2 In every case, # ≥ 0 so the lemma is proven. We now return to the proof of Theorem 8. Recall that a function g that is concave on an interval I must have the property that if λ0λ1λm are nonnegative and sum to 1 and x0x1xm ∈ I, then m m λjgλj≤g λjxj j=0 j=0 We observe that hx=x1 − x is increasing and concave on 0 1/4. n n n 2 n n Further, hzk =xk .Ifwk = j=0 tk j zk , then

2 n n n tk j h zk−j ≤ h wk j=0

n n+1 By (2.2.3), we have that wk ≤ zk . Since g is increasing on 0 1/4,we n n+1 also know that hwk ≤hzk . We conclude that

2 n n n+1 tk j xk ≤ xk j=0 We now observe that this last inequality is exactly the condition that the sequence Dne1 is increasing. Finally, if g is any continuous, increasing concave function on 0 1/4, we again ﬁnd that the inequalities hold:

2 2 n n n n n+1 tk j g xk ≤ g tk j xk ≤ g xk j=0 j=0

In particular, we have thus shown that the approximation operators Dng are increasing for g. This ends the proof of the theorem. generalized bernstein polynomials 37

TABLE I Normalized Symplectic Bernstein Polynomials

1 n = 0 τ 1 − 2τ n = 1 τ2 τ − 2τ2 1 − 4τ + τ2 n = 2 τ3 τ2 − 2τ3 τ − 4τ2 + τ3 1 − 6τ + 10τ2 − 4τ3 n = 3 τ4 τ3 − 2τ4 τ2 − 4τ3 + τ4 τ − 6τ2 + 10τ3 − 4τ4 1 − 8τ + 21τ2 − 20τ3 + 5τ4 n = 4

Corollary 10. Let f be a continuous function on 0 1/4. Then the symplectic Bernstein approximation operators Dng converge uniformly to f on 0 1/4. Proof. By the theorem, we know that the approximation operators for e1τ=τ and hτ=τ1 − τ are monotone increasing. Further, by Theorem 7 we also know that these approximations converge pointwise. Hence, by Dini’s theorem, the approximations converge uniformly. Finally, by the classical result of Kantovich, we know that for a sequence of positive j linear approximation operators such as Dn which converge uniformly for t , for j = 0 1 2, the approximations converge uniformly for any continuous function. n Table I gives the polynomials Pk τ, for n = 0 1 2 3 4.

FUTURE DIRECTIONS

The beginning of this study was the observation that several classical interpolation polynomial operators are given by means of generating function identities for symmetric functions. These identities may be further interpreted as traces on certain operator algebras associated with the infinite classical groups. Traces can be viewed as probability generating functions indexed by signatures (see [10]). Another application of operator algebras to problems in applied analysis can be found in [3]. The problem of convergence is attacked by means of the ergodic theorem and the asymptotic character formula, which was developed by Boyer [2], Kerov and Vershik [13], and Okounkov and Olshanski [15]. The asymptotic character formula may be interpreted as a law of large numbers for representations and so gives a natural interpretation of the first moment of the probability distribution corresponding to a trace. We saw above that the asymptotic character formula yields pointwise convergence. Interpola- tion problems require a refinement of this method because the variance or second moment governs the rate of convergence. Understanding of the 38 boyer and thiel second moment would lead to uniform convergence of the approximation operators. Because of this framework, it is very natural to study two-dimensional and higher versions of the approximation operators introduced in this paper. Finally, we note an unexpected connection of approximation theory with the work of Jones and knot polynomials. In particular, the symplectic analogue of the Bernstein polynomials described in this paper are also found in the work of Jones who used them in his development of knot invariants in the mid-1980s. It would be very interesting to discover a relationship between the total positivity of these polynomials and a geometric property of knots.

APPENDIX

We collect here two technical results which are used in this paper. Let & ) be a dynamical system, where & is the projective limit of finite sets &n and ) is a locally finite group of homeomorphisms on &. Here ) is ∞ given as a union n=0 )n, where )n is a finite group of homeomorphisms acting on &n. A probability measure µ is )-invariant if for every measurable set S we have µS=µγS, where γ ∈ ). The probability measure µ is )-ergodic if the only )-invariant elements of L∞& µ are scalars. The following version of the ergodic theorem was first introduced in [19]; see [16] for complete proofs.

Ergodic Theorem of Vershik. Let φ ∈ & andlet µ be a )-invariant, ergodic probability measure on &. Then, for almost all paths p,

1 lim φγ−1p= φ dµ N→∞ )N & γ∈)N

Asymptotic Character Formula for Schur Functions [2, 13, 15].

Given a signature λ =λ1λ2λn, let rjλ denote the length of the jth row, cjλ the length of the jth column, and λ the number of cells in λ. Then, as n →∞, the normalized symmetric Schur function

S z z 11/S 1 11 λN 1 m λN has a limit if and only if the limits of the signature statistics exist, that is, if and only if lim r λ /N = α , lim c λ /N = β , with α + N→∞ j N j N→∞ j N j j βj < ∞, and limN→∞ λN /N = γ. Set θ = γ − αj + βj. When the generalized bernstein polynomials 39 limit exists, it is given by S z z 11 m λN 1 m lim = f zk N→∞ S 1 11 λN k=1 ∞ 1 − β + β z where f z=eθz−1 j j 1 + α − α z j=1 j j This result also holds in the same form exactly for the symplectic Schur functions. This is proven in [2].

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