Ukrainian Mathematical Journal, Vol. 64, No. 5, October, 2012 (Ukrainian Original Vol. 64, No. 5, May, 2012)

SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE M. P. Prophet1 and I. A. Shevchuk2

UDC 517.5

Let P W X ! V be a projection from a real Banach space X onto a subspace V and let S  X: In this setting, one can ask if S is left invariant under P; i.e., if PS  S: If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PS  S can be characterized through a geometric description. This characterization relies heavily on the structure of S; or, more specifically, on the structure of the cone S  dual to S: In this paper, we remove the structural assumptions on S  and characterize the cases where PS  S: We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.

1. Introduction Denote the space of linear operators from a real Banach space X into a subspace V  X by L D L.X; V /: For a given subset S  X; one can look to determine those Q 2 L that leave S invariant; i.e., Q such that QS  S: There are numerous settings in which QS  S has important consequences and connections. For example, under the right conditions on S; X becomes a Banach lattice, and Q such that QS  S becomes a positive operator (see [7] for an overview). Existence of positive operators (or, more precisely, positive extensions) is employed, for example, in the classical Korovkin theorem (described in [2]) and in its many generalizations (see, e.g., [3]). A natural assumption on S is that it is a cone—a convex set closed under nonnegative scalar multiplication. And outside the Banach lattice realm, Q 2 L.X; V / such that QS  S is often called a cone-preserving map (see [8] for an extensive description). Borrowing this terminology, for a given cone S; let us denote the set of all cone-preserving operators by LS D LS .X; V /: Not surprisingly, the determination of whether or not a given Q 2 L belongs to LS can be quite difficult. Indeed, one finds in the literature that the existence of cone-preserving operators is frequently considered only in the case where X is finite-dimensional. The fact that the membership in LS is very “sensitive” to X; S; and Q certainly contributes to the difficulty. For example, there is no finite-rank operator in LS .X; V / that fixes V; where X D .C Œ0; 1; k
k1 /; S is the cone of nonnegative elements from X; and V D …2 D Œ1; x; x 2 ; the space of second-degree algebraic polynomials (spanned by f1; x; x 2 g/: However, if instead we require fixing …1 and x 2 7! .x C x 2 /=2; i.e., nearly fixing V; then such an operator does belong to LS .X; V /: Or instead, consider the fact that, while there exists no projection from X onto V D …2 preserving monotonicity, it is possible to project X1 onto V and leave the cone of monotone functions (of X1 / invariant, where X1 is the (Banach) space of C 1 functions on Œ0; 1 normed by kf kX1 WD maxfkf k1 ; kf 0 k1 g: When elements of X are to be approximated from V so that the characteristic, or shape, described by (inclusion in) S should be maintained, then we say that such a Q provides a shape-preserving approximation 1 University

of Northern Iowa, Cedar Falls, USA. Kyiv National University, Kyiv, Ukraine.

2 Shevchenko

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 674–684, May, 2012. Original article submitted December 8, 2011. 0041-5995/12/6405–0767

c 2012

Springer Science+Business Media New York

767

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whenever Q 2 LS ; and Q is referred to as a shape-preserving operator. This paper considers the problem of existence of shape-preserving operators for a given S: From the viewpoint of shape-preserving approximation, we will be primarily interested in those Q 2 L that are projections, i.e., P 2 L.X; V /

such that

PjV D idV :

Let P D P.X; V / denote the set of projections in L and let PS be the set of shape-preserving projections. The paper [5] gives a characterization of PS ¤ ¿ under so-called high-dimensional assumptions (which are explained below). As illustrated, for example, in [1, 4] and [6], there are many natural settings for which the high-dimensional assumptions are valid (and, thus, the characterization can be applied). The main goal of this paper is to consider the existence question PS ¤ ¿ without the assumptions of [5], i.e., existence under low-dimensional assumptions, and to apply our results in a specific setting. We divide this paper into four sections. Following this introductory section, we establish in Sec. 2 some basic notation involving convex cones and describe exactly our low-dimensional assumptions. In Sec. 3, we state, and subsequently prove, our main existence results. Within this section, we describe a decomposition of the subspace V that is used extensively in the consideration of shape-preserving operators. Finally, in Sec. 4, we identify a very natural setting in which the low-dimensional assumptions hold and our existence results can be applied to yield some interesting results. 2. Preliminaries and Low-Dimensional Assumptions Throughout this paper, we will denote the ball and sphere of a real Banach space X by B.X/ and S.X /; respectively. V  X will always denote a finite-dimensional subspace of X: The dual space of X is denoted, as usual, by X  : To emphasize bilinearity, we use hx; 'i to denote '.x/ for x 2 X and ' 2 X  : In a (real) topological vector space, a cone K is a convex set, closed under nonnegative scalar multiplication. K is pointed if it contains no lines. For ' 2 K; let Œ'C WD f˛' j ˛  0g: We say Œ'C is an extreme ray of K if ' D '1 C '2 implies '1 ; '2 2 Œ'C whenever '1 ; '2 2 K: We let E.K/ denote the union of all extreme rays of K: When K is a closed pointed cone of finite dimension, we always have K D co.E.K// (this need not be the case when K is infinite dimensional; indeed, we note in [6] that it is possible that E.K/ D ¿ despite K being closed and pointed). Definition 2.1. Let S  X denote a closed cone. We say that x 2 X has shape (in the sense of S / whenever x 2 S: Denote the set of projections from X onto V by P D P.X; V /: If P 2 P and PS  S; then we say that P is a shape-preserving projection; denote the set of all such projections by PS : For a given cone S; define ˇ ¶ · S  D ' 2 X  ˇ hx; 'i  0 8x 2 S : We will refer to S  as the dual cone of S: A dual is always a weak*-closed cone in X  but, in general, need not be pointed. The following lemma indicates that S  is in fact “dual” to S : Lemma 2.1. Let x 2 X: If hx; 'i  0 for all ' 2 S  ; then x 2 S: Proof. We prove the contrapositive; suppose that x 2 X is such that x 62 S: Then, since S is closed and convex, there exists a separating functional ' 2 X  and ˛ 2 R such that hx; 'i < ˛ and hs; 'i > ˛

8s 2 S:

(2.1)

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Note that we must have ˛ < 0 because 0 2 S: In fact, for every s 2 S we claim hs; 'i  0 > ˛:

(2.2)

To check this, suppose that there exists s0 2 S such that hs0 ; 'i D ˇ < 0I this would imply 

 ˛ s0 ; ' D ˛ ˇ

while ˛ s0 2 S: ˇ And this is in contradiction to (2.1). The validity of (2.2) implies that ' 2 S  ; and this completes the proof. Remark 2.1. Not surprisingly, characteristics of the cone S and the subspace V play a role in the existence of shape-preserving operators. In [5], it is assumed that both S and V have the “largest possible” dimension (the so-called high-dimensional assumptions). Specifically, it is assumed in that paper that a basis for V can be obtained from S .dim.V / D dim.V \ S // and that S  X is “so large” that the zero-functional is the only element of X  that vanishes on S (and so, roughly speaking, dim.S/ D dim.X//: This latter condition is clearly equivalent to the (geometric) condition that S  is pointed. In this paper, we look to remove the assumptions described in the note above. Specifically, throughout the remainder of this paper we make the following low-dimensional assumptions: S  is not pointed and dim.V \S/  dim.V /: By way of completeness, we note that the case S  is pointed and dim.V \ S/ < dim.V / is handled by Theorem 3.1 (below); in this case we always have PS .X; V / D ¿: Remark 2.2. We wish to distinguish between two types of (nonpointed) dual cones: those that can be made pointed and those that cannot. To this end, let S ?  X  denote the space of functionals that vanish against S and note S ?  S  : We are interested in (potentially) “sharpening” S  in the following sense: Definition 2.2. We say that S  can be sharpened if 

 S  n S ? \ S ? D ¿;

where the closure is taken with respect to the weak* topology. In this case, we define S ] WD S  n S ? : This concept of sharpening a dual cone is motivated by a simple fact: S ] is a pointed cone, with a “pre-dual” cone nearly identical to the cone S: And, as we illustrate in the next section, S ] can be employed to give a geometric characterization of when PS D ¿: 3. Main Results 3.1. General Existence Results. In this section, we give characterizations for PS ¤ ¿I the proofs of these statements are given in Sec. 3.3. To understand when PS ¤ ¿; we should consider the relationship between the shape to be preserved, S; and the range of our projection, V: Indeed, this relationship can be expressed by restricting S  to V; denoted by SjV : This consideration can often completely characterize when PS ¤ ¿:

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Definition 3.1. Let d WD dim.V /: Define V0 WD fv 2 V j hv; 'i D 0 8' 2 S  g and note V0  S: Now let k WD dim.V \ S / dim.V0 /: Fix a basis fv1 ; : : : ; vd g for V such that v1 ; : : : ; vr 62 S; V0 D ŒvrC1 ; : : : ; vd .kC2/ ; and vd .kC1/ ; : : : ; vd 2 S (where Œa1 ; : : : ; as  denotes the linear span of fa1 ; : : : ; as g/: Using this basis, we define V WD Œv1 ; : : : ; vr  and VC WD Œvd .kC1/ ; : : : ; vd  and decompose V as V D V ˚ V0 ˚ VC D Œv1 ; : : : ; vr ; vrC1 ; : : : ; vd

.kC2/ ; vd .kC1/ ; : : : ; vd :

Remark 3.1. The results presented below rely on the decomposition of V given above. Note that once the cone S  X is fixed, this decomposition is merely a convenient basis choice for V: Indeed, every Q 2 L.X; V / can be expressed in terms of this basis as

QD

d X

ui ˝ vi ;

i D1

where

Qf D

d X hf; ui ivi i D1

with ui 2 X  for each i: Using the representation, we say that the action (up to similarity) of Q on V is the matrix .hvi ; uj i/: Obviously, Q is a projection if and only if .hvi ; uj i/ D ıij : Recall that S ?  S  denotes the space of functionals that vanish against S: We say that a subspace M  X  is total over a subspace Y  X if dim.MjY / D dim.Y /: Without any assumptions on the dual cone S  ; we have the following characterization: Theorem 3.1. Let S  X be given and V D V ˚ V0 ˚ VC : Then PS .X; V / ¤ ¿ if and only if S ? is total over V and PS .X; VC / ¤ ¿: This characterization indicates that shape-preservation onto V is almost equivalent to shape-preservation onto VC : And in Sec. 3.2, we establish existence results involving VC : For the remainder of this section, we consider the case in which S  can be sharpened, i.e., the case in which S ] is defined. When a dual cone has a particular structure, the existence of shape-preserving operators can be described in terms of that structure, which we now define. Note that, in the context of our current considerations, we say that a finite (possibly) signed measure  with support E  X  is a generalized representing measure for ' 2 X  if Z hx; 'i D

hs; xidu.s/ for all x 2 X: E

A nonnegative measure  satisfying this equality is simply a representing measure. Definition 3.2. Let X be a Hausdorff space over R: We say that a pointed closed cone K  X  is simplicial if K can be recovered from its extreme rays (i.e., K D co .E.K/// and the set of extreme rays of K form an independent set (independent in the sense that any generalized representing measure for x 2 K supported on E.K/ must be a representing measure). Proposition 3.1. A pointed closed cone K  X  of finite dimension d is simplicial if and only if K has exactly d extreme rays. Theorem 3.2 ([5], Theorem 1.1). Let S   X  denote the dual cone of S  X and suppose that S  is simplicial. Then PS .X; V / ¤ ¿ if and only if the cone SjV is simplicial.

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Theorem 3.3. Let S  X be given and suppose that S ] (exists and) is simplicial. Then PS .X; V / ¤ ¿ if ] and only if S ? is total over V and SjV is simplicial. C

3.2. Preservation onto V ; V0 ; and VC :

QD

r X

! ui ˝ vi

For any Q 2 L.X; V /; we can write (using Remark 3.1)

0

d X .kC2/

˚@

i D1

1

0

d X

ui ˝ vi A ˚ @

i DrC1

1 ui ˝ vi A DW Q ˚ Q0 ˚ QC :

i Dd .kC1/

In this section, we consider these components of Q in the shape-preserving projection case. When Q is a projection, note that each component is also a projection (onto its specific range). Lemma 3.1. For a given S  X; let V D V ˚ V0 ˚ VC : Let P 2 P.X; V / be any projection. Then P0 2 PS .X; V0 /: Proof. For every f 2 S and every ' 2 S  ; we have *d hP0 f; 'i D

.kC2/ X

+ hf; ui ivi ; ' D

i DrC1

d X .kC2/

hf; ui ihvi ; 'i D 0

i DrC1

by definition of V0 : This implies, by Lemma 2.1, that P0 f 2 S; and, since P is a projection, we have P0 2 PS .X; V0 /: Lemma 3.1 is proved. Lemma 3.2. For a given S  X; let V D V ˚ V0 ˚ VC and assume that dim.V / D r ¤ 0: If

P D

d X

ui ˝ vi 2 PS .X; V /;

i D1

then u1 ; : : : ; ur 2 S ? and S ? is total over V : Proof. Let P 2 PS .X; V / and write

P D

r X

ui ˝ vi :

i D1

For every f 2 S; we know P f C P0 f C PC f 2 S: But the decomposition of V (Definition 3.1) implies

P f D

r X i D1

ui .f /vi D 0

(3.1)

M. P. P ROPHET AND I. A. S HEVCHUK

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for every f 2 S; since otherwise we would have dim.VC / > k: Now (3.1) implies that, for each i; we have ui .f / D 0 for all f 2 S and, thus, ui 2 S ? : This, together with the fact that P is a projection, i.e., ui .vj / D ıij ; implies that S ? is total over V : Lemma 3.2 is proved. Remark 3.2. When k D dim.VC / ¤ 0; note that SjV is a k-dimensional pointed cone. It is convenient to inC

terpret this cone as a subset of Rk by associating each 'jVC 2 SjV with the k-vector Œ'.vd C

We will use this association throughout the remainder of the paper. And so by construction, as a cone in the positive orthant of Rk :

T .kC1/ /; : : : ; '.vd / : we may regard SjV C

Lemma 3.3. Let S  X be given and let S  denote its dual cone. Let V D V ˚ V0 ˚ VC and assume that dim.VC / D k ¤ 0: If the .k-dimensional) cone SjV is simplicial, then PS .X; VC / ¤ ¿: C

Proof. Recall that our fixed basis of VC is given by fvd .kC1/ ; : : : ; vd g: For convenience within this proof, relabel these elements as fv1 ; : : : ; vk g: Now, by assumption, SjV has exactly k extreme rays. Label each ray as C

Œu1jVC C ; : : : ; Œuk jV C ; C

where u1jVC ; : : : ; uk jV

C

are nonzero points chosen from distinct rays. Thus, we have SjV

C

  D co Œu1jVC C ; : : : ; Œuk jV C : C

(3.2)

Define the (row) vector u WD .u1 ; : : : ; uk / 2 .S  /k ; where each ui restricts to the extreme ray Œui jVC C ; and the (column) vector v D .v1 ; : : : ; vk /T : Using this notation, note that, for any ' 2 S  ; we may write
.hv1 ; 'i; : : : ; hvk ; 'i/T D hv; 'i D hvi ; uj i c ' D M c ' ;
where M WD hvi ; uj i is a k  k matrix and c ' is the vector of nonnegative coefficients guaranteed by (3.2). Since SjV has k independent elements, the matrix M is nonsingular. Thus, we may solve for c ' and write C

c ' D M 1 hv; 'i: Let PC WD uM f 2 S and ' 2 S  ; we have

1

˝ vI obviously, P is a projection from X into VC : Moreover, for every

˝ hPC f; 'i D hf; uM

1

˛ iv; ' D hf; uiM

1

hv; 'i D hf; uic '  0

since hf; uic ' is a dot-product of two vectors with nonnegative entries. By Lemma 2.1, PC f 2 S: Lemma 3.3 is proved. Lemma 3.4. Let S  X be given and let S  denote its dual cone. Let V D V ˚ V0 ˚ VC and assume that dim.VC / D k ¤ 0: If the .k-dimensional) cone SjV is not closed, then PS .X; VC / D ¿: C

Proof. We consider the contrapositive. Let P 2 PS .X; VC / and let P  S  denote the (weak*) closure of  X  : Choose P  ' 2 P  S   P  X  and a sequence fP  'k g1  P  S  such that P  'k ! P  ': kD1 Note that, by Lemma 2.1, fP  'k g1  S  : S  is weak*-closed and, therefore, P  ' 2 S  I this implies kD1     2  P ' 2 P S since .P / D P : Thus, P  S  is closed. Note that P  S  is homeomorphic to .P  S  /jVC P S 

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and, thus, .P  S  /jVC is closed. Finally, we claim that .P  S  /jVC D SjV : To verify this, choose ' 2 S  ; C

v 2 VC and consider hv; P  'i D hP v; 'i D hv; 'i;

where the last equality follows from the fact that P is a projection. But this equation simply says that P  ' and ' agree on VC ; thus establishing the claim. From here we can conclude that SjV is closed. C

Lemma 3.4 is proved. 3.3. Proofs of Existence Results. Proof of Theorem 3.1. .)/ Let P 2 PS .X; V / and write P D P ˚ P0 ˚ PC : By Lemma 3.2, S ? is total over V : Furthermore, for every f 2 S and every ' 2 S  ; we have 0  hPf; 'i D hP f; 'i C hP0 f; 'i C hPC f; 'i D hPC ; 'i by Lemmas 3.1 and 3.2, and, therefore, PS .X; VC / ¤ ¿: .(/ Let Q D Q ˚ Q0 ˚ QC be any projection onto V and define P0 WD Q0 : Choose P1 2 PS .X; VC /I we claim that P0 ˚ P1 2 PS .X; V0 ˚ VC /:

(3.3)

The fact that this operator is shape-preserving is clear since V0  S: We need only verify that that the action of the operator on V0 ˚ VC is the identity action. Note that we need only check that P1 vanishes on V0 : But this is clear since V0  S is a linear space, P1 V0  S; and V0 \ VC D f0g: This establishes (3.3). We now focus on V : Since S ? is total over V (and assuming r WD dim.V / > 0/; there exist u1 ; : : : ; ur 2 S ? such that

P WD

r X

ui ˝ vi

i D1

is a projection onto V write

(in the case r D 0; define P

to be the zero-operator). Now with P1 chosen as above,

d X

P1 D

ui ˝ vi :

i Dd .kC1/

Again using S ? total over V ; we conclude that there exist functionals '1 ; : : : ; 'r 2 S ? such that, for each j 2 fd .k C 1/; : : : ; d g; there exist constants fc1j ; : : : ; crj g 2 R such that * vi ;

r X

+ cm;j 'm D

hvi ; uj i

for

mD1

Define ˆj WD

r X mD1

cm;j 'j

i D 1; : : : ; r:

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and note that hv; ˆj i D Let Uj WD uj C ˆj for each j D d

hv; uj i

for any v 2 V :

(3.4)

.k C 1/; : : : ; d and d X

PC WD

Ui ˝ vi :

i Dd .kC1/

We claim that P WD P ˚ P0 ˚ PC belongs to PS .X; V /: Consider first PC I note that, by construction, each ˆj  S ? vanishes on S: Thus, PC 2 PS .X; VC /; and so by (3.3), we have P0 ˚ PC 2 PS .X; V0 ˚ VC /:

(3.5)

Regarding P ; by construction this operator vanishes on S; and this, combined with (3.5), implies PS  S: To see that P has the identity action on V; we need only check that P vanishes on V0 ˚ VC and P0 ˚ PC vanishes on V : The former condition holds since the basis we use for V0 and V1 belongs to S: To establish the latter, first note that P0 vanishes on V by construction. And, by (3.4), for any v 2 V we have

PC v D

d X

hv; Ui ivi D

iDd .kC1/

d X

hv; ui C ˆi ivi D

i Dd .kC1/

d X

hv; ui

ui ivi D 0

i Dd .kC1/

by the definition of each ˆi : So PC vanishes on V : This establishes that P is a projection. Theorem 3.1 is proved. Proof of Theorem 3.3. By Theorem 3.1, the proof will be complete if we can show PS .X; VC / 6D ¿ is ] equivalent to SjV simplicial, which we now establish. Recall that S ]  S  is a pointed weak* closed cone and, C

as such, is exactly the dual cone of ˇ ¶ S1 WD x 2 X ˇ hx; i  0 8

· 2 S] :

Note that S1 contains the cone S: By Theorem 3.2, ]

S jV

C

is simplicial

” PS1 .X; VC / 6D ¿;

and, thus, we need only show that PS .X; VC / 6D ¿ ” PS1 .X; VC / 6D ¿:

(3.6)

Let P 2 PS .X; VC /I we claim that P .S1 /  S1 : From Lemma 2.1, it follows that P .S1 /  S1 if and only if P  .S ] /  S ] ; where P  denotes the adjoint of P (defined by hf; P  ui D hPf; ui for f 2 X and u 2 X  /:

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We know that P  .S ] /  S  since (via Lemma 2.1) P  S   S  and S ]  S  : Thus, we need only show that, for each 2 S ] ; nonzero P  does not vanish against S: But

P

D

k X

hvj ; iuj ;

j D1

where (via relabeling) fv1 ; : : : ; vk g  S is our fixed basis for VC : And so P  6D 0 implies hvi ; i 6D 0 for some i: Therefore, P  2 S ] ; which establishes P .S1 /  S1 : Thus, P 2 PS1 .X; VC /: To complete the proof, let P 2 PS1 .X; VC /: Arguing as above, we conclude that P  S   S  and, thus, P 2 PS .X; VC /; which establishes (3.6). Theorem 3.3 is proved. 4. Application: q-Monotone Case In this section, we consider the preservation of q-monotonicity (defined below) by a projection from X D 1:1; k
k/ onto V D …n (the subspace of algebraic polynomials of degree less than or equal to n/; where

.C q Œ

kf k WD max

j D0;:::;q

¶ .j / · kf k1 :

For s 2 N; let Ys denote the collection of s distinct points Y D fyi gsiD1 where y0 D ys < 1 D ysC1 : For q 2 N and Y 2 Ys ; define

1 < y1 < : : : <

ˇ ¶ · q SY D f 2 X ˇ . 1/j f .q/ .t /  0 whenever t 2 Œyj ; yj C1 ; j D 0; : : : ; s : q

We say that f 2 X is q-monotone (with respect to Y 2 Ys / exactly when f 2 SY : We denote by PS q the set Y of q-monotone preserving projections from X onto …n : The main point of this section is the characterization presented below. The proof of this theorem considers the (topological) consequence of restricting a dual cone to the subspace V D …n : For purposes of illustration, we include (in Sec. 4.1) two arguments that establish an existence result; Version 1 uses a “classical” approach to shape-preservation, and Version 2 utilizes the restriction of a dual cone. Theorem 4.1. Let s 2 N: Then, for Y 2 Ys ; PS q 6D ¿ ” n

s

Y

q  1:

Proof. We prove this result through induction on q: The q D 1 case is verified (for all s and n/ in the next section (see Lemma 4.1). We now proceed with the inductive step; for fixed q0 ; we assume that PS q0 6D ¿ ” n

s

Y

q0  1

(4.1)

and show that PS q0 C1 6D ¿ ” n Y

s

.q0 C 1/  1:

(4.2)

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Suppose that n s .q0 C 1/  1I then we have .n 1/ s PS q0 .X; …n 1 /: Using the notation from Sec. 3.2, we may write

q0  1; and so by (4.1) there exists P 2

Y

P D

n X1

uk ˝ vk ;

kD1

where Pf D

n X1

hf; uk ivk 2 …n

1:

kD1

Define Py WD

n X

uO k ˝ vO k ;

kD0

where uO 0 ˝ vO 0 WD ı 1 ˝ 1 and, for k > 0; uO k WD uk ı D t .D t is the differential operator), vO k WD I t ı vk .I t is the integral operator). Thus, .Py f /.t / D

n X

hf; uO k ivO k .t / D f . 1/ C

kD0

n X

hf 0 ; uk iI t .vk /

kD1

D f . 1/ C

Zt X n

Zt

0

hf ; uk ivk .x/ dx D f . 1/ C

1 kD1

Note that finally, if have Py f

.Pf 0 /.x/ dx:

1

Py W C q0 C1 Œ 1; 1 ! …n : Moreover, since P is a projection (onto …n 1 /; so is Py (onto …n /: And q C1 q q f 2 SY 0 ; then f 0 2 SY 0 ; which implies Pf 0 2 SY 0 : Hence, since .Py f /.q0 C1/ D .Pf 0 /.q0 / ; we 2 PS q0 C1 : Thus, PS q0 C1 ¤ ¿: To establish the other direction of (4.2), consider n s .q0 C 1/ > 1I Y

Y

we show that this implies PS q0 C1 D ¿: Suppose that there exists P 2 PS q0 C1 : Arguing as above, express P as Y

Y

P D

n X

uk ˝ vk ;

kD0

where vk WD x k : Define Py WD

n X1

uO k ˝ vO k ;

kD0

where uO k D uk ı I t and vO k D D t ı vk : Then Py f .t / D


n X kD0

hf; uO k ivO k .t / D D t

n X

! hI t f; uk ivk

kD1

D D t .P .I t f // :

S HAPE -P RESERVING P ROJECTIONS IN L OW-D IMENSIONAL S ETTINGS AND THE q-M ONOTONE C ASE q

q

777 q C1

Obviously, Py is a projection from C q0 onto …n 1 : If f 2 SY 0 ; then Py f 2 SY 0 since P .I t f / 2 SY 0 ; and this implies Py 2 PS q0 .X; …n 1 /: But from our supposition, we have .n 1/ s q0 > 1; which, from (4.1), Y implies PS q0 D ¿: This contradiction has resulted from assuming that P 2 PS q0 C1 ; and, therefore, we must Y

Y

have PS q0 C1 D ¿: This establishes (4.2). Y

Theorem 4.1 is proved. 4.1. Case q D 1:

In this subsection, we verify the q0 D 1 case via the following lemma:

Lemma 4.1. PS 1 .X; …n / 6D ¿ ” n

s  2:

Y

To begin, denote SY1 by SY and let S   X  denote the dual cone of SY : Recall the decomposition of V used above; relative to SY ; we write V D V ˚ V0 ˚ VC : Note that V0 is 1-dimensional and V0 D Œ1: As we will see below, dim.VC / D n sI recall from above that we may assume that SjV  Rn s : For fixed Y; put C

 D .x/ WD

s Y

.yi

x/:

i D1

Proposition 4.1. dim.VC / D maxf0; n

sg: If n Zx

vi .x/ WD

s > 0 then, for i D 1; : : : ; n

s;

t i /.t/ dt 2 SY ;

.1 1

and fv1 ; : : : ; vn s g forms a basis for VC : Let v 2 V \ SY I then, for i D 1; : : : ; s; we have v 0 .yi / D 0: Thus, if n s  0; then dim.VC / D 0: Assume that n s > 0I then, by definition of SY ; we can write v 0 .x/ D p.x/.x/ for some polynomial p: But deg./ D s; and so p 2 …n .sC1/ : Therefore, dim.VC /  n s: Finally, note that, for i D 1; : : : ; n s; Zx vi D

t i /.t/ dt 2 SY

.1 1

and are independent. Thus, VC D Œv1 ; : : : ; vn s : Note that, in this application, we have labeled the basis elements for VC as v1 ; : : : ; vn s : This departure from the labeling in the previous section is meant to simplify the notation in the current setting. Lemma 4.2. Suppose that n

s > 2: Then SjV

C

VC

Proof. Fix yj for some j 2 f1; : : : ; sg: Since n can be chosen as prescribed to include the elements

 Rn

s

s  3; it is clear from Proposition 4.1 that a basis for

Zx v1 WD

Zx .t /

1

is not closed and, thus, PSY .X; …n / D ¿:

and

v2 WD

.1 1

t 2 /.t/:

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778

Without loss, assume that .t /  0 for t 2 .yj such t; the point (or vector)

.ı 0t /jVC .t /

1 ; yj /:

And so, since SjV

is a cone, it must contain, for each

C

: Thus, by Proposition 4.1, there exists a vector .ı 0t /jVC

z D Œ1; 1; z3 ; : : : ; zn s  WD lim

.t/

t !yj

belonging to the closure of SjV : Now, by way of contradiction, let us suppose that there exists ' 2 S  such that 'jVC D z: Note that

C

0

Zx

1 D '.v1 / D ' @

1

0

Zx

.t/A D '.v2 / D ' @ 1

1 .1

t 2 /.t/A ;

(4.3)

1

which implies 0

Zx

'@

1 t 2 .t/A D 0:

1

Moreover, for every even integer   2; we have Zx

Zx

t  .t / 2 S

and

1

since t 2

.t 2

t  /.t/ 2 S

1

t   0 on Œ 1; 1: And thus, for every ; 0

Zx

'@

1 t  .t/A D 0:

(4.4)

1

y y For convenience, assume that yj D 0: Define .x/ by .x/ D x .x/: Let TO .x/ be an odd Tchebyshev polynomial of (arbitrary odd) degree d: Consider the polynomial Zx p.x/ WD

y 2 XI TO 

1

the norm kpk is clearly bounded independent of d: But by (4.3) and (4.4) we find ˇ 0 0 1 1ˇ ˇ 0 ˇ ˇ ˇ Zx X d d Zx ˇ ˇ ˇ ˇ B B Cˇ ˇ B X iC y j'.p/j D ˇ' @ @ ci t A .t/Aˇ D ˇ' @ ci t i ˇ ˇ ˇ i D1 i D1 1 ˇ ˇ ˇ 1 i odd

i odd

1ˇ ˇ ˇ 1 Cˇ Aˇ D d ˇ ˇ

S HAPE -P RESERVING P ROJECTIONS IN L OW-D IMENSIONAL S ETTINGS AND THE q-M ONOTONE C ASE

779

since jc1 j D d: This implies that ' is unbounded and thus cannot be an element of S  : Therefore, SjV

is not

C

closed. Consequently, by Lemma 4.2 and Corollary 3.4, we have PSY .X; VC / D ¿ and, thus, PSY .X; V / D ¿ by Theorem 3.1. Lemma 4.2 is proved. Lemma 4.3. Suppose that n

s  2: Then PSY .X; V / ¤ ¿:

Proof (Version 1). Set ysC2 WD y0 D 1: Fix n 2 N; n s  2: For each g 2 C Œ 1; 1; denote by Ln 1 .x; g/ WD L.x; gI y1 ; : : : ; yn / the Lagrange polynomial of degree < n that interpolates g at yj ; j D 1; : : : ; n: First, we note that the operator P 2 L.C 1 Œ 1; 1; …n / defined by Zx .P g/.x/ WD g.0/ C

Ln

1 .t; g

0

/dt;

0

is a projection, i.e., P 2 P.C 1 Œ 1; 1; …n /: This readily follows from the fact that, for each pn have Ln

 pn

1 .x; pn 1 /

1

2 …n

1;

we

1 .x/:

So, to end the proof we have to check that if f 2 SY ; then .Pf / 2 SY as well, or, which is the same, Ln

1 .x; f

0

/.x/  0;

x 2 Œ 1; 1;

(4.5)

where .x/ WD

s Y

.yj

x/:

j D1

Indeed, if n  s; then Ln 1 .x; f 0 /  0; which yields (4.5). If n D s C 1; then Ln 1 .x; f 0 / D A.x/; where A  0; which yields (4.5). Finally, if n D s C 2; then Ln 1 .x; f 0 / D .ax C b/.x/: Let us show that ax C b  0; If x D

x 2 Œ 1; 1:

(4.6)

1; then aCb D

Ln

1; f 0 / f 0 . 1/ D  0: . 1/ . 1/ 1.

Similarly, a C b  0: Thus, (4.6) holds, which yields (4.5). Proof (Version 2). We claim that (regardless of the value n s/ S ? is total over V : Indeed, note that, in our setting, we have r WD dim.V / D minfs; ng and V D Œx; x 2 ; : : : ; x r : And since fıy0 i gsiD1  S ? ; we conclude that S ? is total over V : Now in the case n s  0 we have dim.VC / D 0; and so trivially PS .X; VC / ¤ ¿ since the zero-operator belongs to this set. Suppose that n s > 0I by Proposition 4.1, n s is exactly the dimension of SjV : We claim that, in the cases n s D 1; 2; the cone SjV is simplicial. This is clear in the case n

C

C

s D 1 since every 1-dimensional pointed cone is (trivially) simplicial. For n

s D 2; note

M. P. P ROPHET AND I. A. S HEVCHUK

780

that a 2-dimensional pointed cone is simplicial if and only if it is closed. We now show that SjV Recall that SjV

C

C

 R2 is closed.

belongs to the positive quadrant of R2 : And it will suffice to show that, for some basis for VC ;

there exist functionals '1 ; '2 2 S  such that .'i /jVC belongs to the ray determined by e i (the standard basis element) for i D 1; 2: To this end, note that Zx v1 WD

Zx .t

1/.t/

and

v2 WD

1

.t C 1/.t/ 1

are elements of S and form a basis for VC : Moreover .ı 0 1 /jVC D Œa; 0 and .ı10 /jVC D Œ0; b for some a; b > 0: Therefore, SjV is exactly the positive quadrant of R2 : Thus, in the cases n s D 1; 2; we have SjV simplicial, C

C

which implies that PS .X; VC / ¤ ¿ by Theorem 3.3. By Theorem 3.1, we conclude that PS .X; V / ¤ ¿: Lemma 4.3 is proved. REFERENCES 1. B. Chalmers, D. Mupasiri, and M. P. Prophet, “A characterization and equations for minimal shape-preserving projections,” J. Approxim. Theory, 138, 184–196 (2006). 2. E. W. Cheney, Introduction to Approximation Theory, Chelsea, New York (1982). 3. K. Donner, Extension of Positive Operators and Korovkin Theorems, Springer, Berlin (1982). 4. G. Lewicki and M. P. Prophet, “Minimal multi-convex projections,” Stud. Math., 178, No. 2, 99–124 (2007). 5. D. Mupasiri and M. P. Prophet, “A note on the existence of shape-preserving projections,” Rocky Mountain J. Math., 37, No. 2, 573–585 (2007). 6. D. Mupasiri and M. P. Prophet, “On the difficulty of preserving monotonicity via projections and related results,” Jaen J. Approxim., 2, No. 1, 1–12 (2010). 7. H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York (1974). 8. H. Schneider and B. Tam, “On the core of a cone-preserving map,” Trans. Amer. Math. Soc., 343, No. 2, 479–524 (1994).