Shape-preserving projections in low-dimensional settings and the q -monotone case
Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence o...
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| author | Prophet, M. P. Shevchuk, I. A. Профет, М. П. Шевчук, І. О. |
| author_facet | Prophet, M. P. Shevchuk, I. A. Профет, М. П. Шевчук, І. О. |
| author_sort | Prophet, M. P. |
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| datestamp_date | 2020-03-18T19:30:33Z |
| description | Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$.
In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$.
If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ 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, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$.
We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization. |
| first_indexed | 2026-03-24T02:26:45Z |
| format | Article |
| fulltext |
UDC 517.5
M. P. Prophet (Univ. Northern Iowa, USA),
I. A. Shevchuk (Kyiv Nat. Taras Shevchenko Univ., Ukraine)
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS
AND THE q-MONOTONE CASE
ФОРМОЗБЕРIГАЮЧI ПРОЕКЦIЇ У МАЛОВИМIРНIЙ ПОСТАНОВЦI
ТА q-МОНОТОННИЙ ВИПАДОК
Let P : 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, шe 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 which (lacks structure and thus) serves as an application for our characterization.
Нехай P : X → V — проекцiя дiйсного банахового простору X на пiдпростiр V i, крiм того, S ⊂ X. У цiй постановцi
виникає питання: чи є S лiвоiнварiантним пiд дiєю P , тобто чи має мiсце вкладення PS ⊂ S? Якщо пiдпростiр
V є скiнченновимiрним, а S є конусом iз певною структурою, то вкладення PS ⊂ S може бути охарактеризовано
шляхом геометричного опису. Ця характеризацiя iстотно залежить вiд структури S, або, точнiше, вiд структури
конуса S∗, спряженого до S. У цiй роботi усунено структурнi припущення щодо S∗ i охарактеризовано випадки, у
яких PS ⊂ S. Вiдзначено, що (так звана) q-монотонна форма утворює конус, який (не має структури i тому) може
бути використаний для застосування нашої характеризацiї.
1. Introduction. Denote the space of linear operators from real Banach space X into subspace
V ⊂ X by L = L(X,V ). For a given subset S ⊂ X, one can look to determine those Q ∈ L
which leave S invariant; i.e., those 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 Korovkin’s classical theorem (described in [2]) and in its many generalizations (see,
for example, [3]).
A natural assumption on S is that it is a cone — a convex set, closed under nonnegative scalar
multiplication. And outside of the Banach lattice realm, Q ∈ L(X,V ) such that QS ⊂ S is often
called a cone-preserving map (see [8] for an extensive description). Borrowing this terminology,
for given cone S let us denote the set of all cone-preserving operators by LS = LS(X,V ). Not
surprisingly, the determination of whether or not a given Q ∈ L belongs to LS can be quite difficult.
Indeed, one finds in the literature that existence of cone-preserving operators is frequently considered
only in the case in which X is finite-dimensional. The fact that 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 ) which fixes V, where X = (C[0, 1], ‖·‖∞), S is the cone of nonnegative elements from X
and V = Π2 = [1, x, x2], the space of second-degree algebraic polynomials (spanned by {1, x, x2}).
However, if instead we require fixing Π1 and x2 7→ (x + x2)/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 = Π2 preserving monotonicity, it is possible to project X1 onto V and leave the cone
c© M. P. PROPHET, I. A. SHEVCHUK, 2012
674 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE 675
of monotone functions (of X1) invariant, where X1 is the (Banach) space of C1 functions on [0, 1]
normed by ‖f‖X1 := max{‖f‖∞, ‖f ′‖∞}.
When elements of X are to approximated from V such that the characteristic, or shape, de-
scribed by (inclusion in) S should be maintained, then we say such a Q provides a shape-preserving
approximation whenever Q ∈ 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 ∈ L that projections,
i.e.,
P ∈ L(X,V ) such that P|V = idV .
Let P = 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 6= ∅ 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 6= ∅ without the assumptions
of [5], that is, 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
Section 2 some basic notation involving convex cones and describe exactly our low-dimensional
assumptions. In Section 3 we state, and subsequently prove, our main existence results. Within this
section we describe a decomposition of subspace V which is used extensively in the consideration
of shape-preserving operators. Finally in Section 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 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
bi-linearity, use 〈x, ϕ〉 to denote ϕ(x) for x ∈ X and ϕ ∈ 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 ϕ ∈ K, let [ϕ]+ := {αϕ | α ≥ 0}. We say [ϕ]+ is an extreme ray of K if ϕ = ϕ1 +ϕ2
implies ϕ1, ϕ2 ∈ [ϕ]+ whenever ϕ1, ϕ2 ∈ 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 = 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) = ∅ despite K being closed and pointed).
Definition 2.1. Let S ⊂ X denote a closed cone. We say that x ∈ X has shape (in the sense
of S) whenever x ∈ S. Denote the set of projections from X onto V by P = P(X,V ). If P ∈ P
and PS ⊂ S then we say P is a shape-preserving projection; denote the set of all such projections
by PS . For a given cone S, define
S∗ =
{
ϕ ∈ X∗
∣∣ 〈x, ϕ〉 ≥ 0 ∀x ∈ 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.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
676 M. P. PROPHET, I. A. SHEVCHUK
Lemma 2.1. Let x ∈ X. If 〈x, ϕ〉 ≥ 0 for all ϕ ∈ S∗ then x ∈ S.
Proof. We prove the contrapositive; suppose x ∈ X such that x 6∈ S. Then, since S is closed and
convex, there exists a separating functional ϕ ∈ X∗ and α ∈ R such that 〈x, ϕ〉 < α and
〈s, ϕ〉 > α ∀s ∈ S. (2.1)
Note that we must have α < 0 because 0 ∈ S. In fact, for every s ∈ S we claim
〈s, ϕ〉 ≥ 0 > α. (2.2)
To check this, suppose there exists s0 ∈ S such that 〈s0, ϕ〉 = β < 0; this would imply〈
α
β
s0, ϕ
〉
= α
while
α
β
s0 ∈ S. And this is in contradiction to (2.1). The validity of (2.2) implies that ϕ ∈ 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 ‘largest
possible’ dimension (the so-called high-dimensional assumptions). Specifically, in that paper it is
assumed that a basis for V can be obtained from S (dim (V ) = 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) = 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 ) = ∅.
Remark 2.2. We wish to distinguish between two types of (non-pointed) dual cones: those
which can be made pointed and those which 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∗ \ S⊥
)
∩ S⊥ = ∅
where the closure is taken with respect to the weak* topology. In this case, we define S] := S∗ \ 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 cone S. And, as we illustrate in the next section, S] can be
employed to give a geometric characterization of when PS = ∅.
3. Main results. 3.1. General existence results. In this section we give characterizations for
PS 6= ∅; the proofs of these statements are given in Subsection 3.3. To understand when PS 6= ∅,
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 S∗|V . This
consideration can often completely characterize when PS 6= ∅.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE 677
Definition 3.1. Let d := dim (V ). Define V0 := {v ∈ V | 〈v, ϕ〉 = 0 ∀ϕ ∈ S∗} and note
V0 ⊂ S. Now let k := dim (V ∩S)−dim (V0). Fix a basis {v1, . . . , vd} for V such that v1, . . . , vr 6∈
6∈ S, V0 = [vr+1, . . . , vd−(k+2)], and vd−(k+1), . . . , vd ∈ S (where [a1, . . . , as] denotes the linear
span of {a1, . . . , as}). Using this basis, we define V− := [v1, . . . , vr] and V+ := [vd−(k+1), . . . , vd]
and decompose V as
V = V− ⊕ V0 ⊕ V+ = [v1, . . . , vr, vr+1, . . . , vd−(k+2), vd−(k+1), . . . , vd].
Remark 3.1. The following results 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 ∈ L(X,V ) can be expressed in terms of this basis as
Q =
d∑
i=1
ui ⊗ vi, where Qf =
d∑
i=1
〈f, ui〉vi
with ui ∈ X∗ for each i. Using the representation, we say that the action (up to similarity) of Q on
V is the matrix (〈vi, uj〉). Evidently Q is a projection if and only if (〈vi, uj〉) = δij .
Recall that S⊥ ⊂ S∗ denotes the space of functionals that vanish against S. We say subspace
M ⊂ X∗ is total over subspace Y ⊂ X if dim (M|Y ) = 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 = V− ⊕ V0 ⊕ V+. Then PS(X,V ) 6= ∅ if and only
if S⊥ is total over V− and PS(X,V+) 6= ∅.
This characterization indicates that shape-preservation onto V is almost equivalent to shape-
preservation onto V+. And in Subsection 3.2, we establish existence results involving V+. 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, 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 a finite (possibly) signed measure µ with support E ⊂ X∗ is a generalized
representing measure for ϕ ∈ X∗ if 〈x, ϕ〉 =
∫
E
〈s, x〉 du(s) for all x ∈ X. A nonnegative measure
µ 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 = 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 ∈ 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 S∗
is simplicial. Then PS(X,V ) 6= ∅ if and only if the cone S∗|V is simplicial.
Theorem 3.3. Let S ⊂ X be given and suppose S] (exists and) is simplicial. Then PS(X,V ) 6=
6= ∅ if and only if S⊥ is total over V and S]|V+
is simplicial.
3.2. Preservation onto V−, V0, V+. For any Q ∈ L(X,V ) we can write (using Remark 3.1)
Q =
(
r∑
i=1
ui ⊗ vi
)
⊕
d−(k+2)∑
i=r+1
ui ⊗ vi
⊕
d∑
i=d−(k+1)
ui ⊗ vi
=: Q− ⊕Q0 ⊕Q+.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
678 M. P. PROPHET, I. A. SHEVCHUK
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 = V−⊕V0⊕V+. Let P ∈ P(X,V ) be any projection.
Then P0 ∈ PS(X,V0).
Proof. For every f ∈ S and every ϕ ∈ S∗ we have
〈P0f, ϕ〉 =
〈
d−(k+2)∑
i=r+1
〈f, ui〉vi, ϕ
〉
=
d−(k+2)∑
i=r+1
〈f, ui〉〈vi, ϕ〉 = 0
by definition of V0. This implies, by Lemma 2.1, that P0f ∈ S and, since P is a projection, we have
P0 ∈ PS(X,V0).
Lemma 3.1 is proved.
Lemma 3.2. For a given S ⊂ X, let V = V− ⊕ V0 ⊕ V+ and assume dim (V−) = r 6= 0. If
P =
∑d
i=1
ui ⊗ vi ∈ PS(X,V ) then u1, . . . , ur ∈ S⊥ and S⊥ is total over V−.
Proof. Let P ∈ PS(X,V ) and write P− =
∑r
i=1
ui ⊗ vi. For every f ∈ S we know
P−f + P0f + P+f ∈ S.
But the decomposition of V (Definition 3.1) implies
P−f =
r∑
i=1
ui(f)vi = 0, (3.1)
for every f ∈ S, since otherwise we would have dim (V+) > k. Now (3.1) implies that for each i,
ui(f) = 0 for all f ∈ S and thus ui ∈ S⊥. This, together with the fact that P is a projection, i.e.,
ui(vj) = δij , implies that S⊥ is total over V−.
Lemma 3.2 is proved.
Remark 3.2. When k = dim (V+) 6= 0, note that S∗|V+
is a k-dimensional pointed cone. It is
convenient to interpret this cone as a subset of Rk by associating each ϕ|V+ ∈ S
∗
|V+
with the k-vector
[ϕ(vd−(k+1)), . . . , ϕ(vd)]
T . We will use this association throughout the remainder of the paper. And
so by construction, we may regard S∗|V+
as a cone in the positive orthant of Rk.
Lemma 3.3. Let S ⊂ X be given and let S∗ denote its dual cone. Let V = V−⊕ V0⊕ V+ and
assume dim (V+) = k 6= 0. If the (k-dimensional ) cone S∗|V+
is simplicial then PS(X,V+) 6= ∅.
Proof. Recall that our fixed basis of V+ is given by {vd−(k+1), . . . , vd}. For convenience within
this proof, relabel these elements as {v1, . . . , vk}. Now, by assumption, S∗|V+
has exactly k extreme
rays. Label each ray as
[u1|V+ ]+, . . . , [uk |V+ ]+,
where u1|V+ , . . . , uk |V+ are non-zero points chosen from distinct rays. Thus we have
S∗|V+
= co
(
[u1|V+
]+, . . . , [uk|V+
]+
)
. (3.2)
Define the (row) vector u := (u1, . . . , uk) ∈ (S∗)k, where each ui restricts to extreme ray [ui|V+ ]+,
and the (column) vector v = (v1, . . . , vk)
T . Using this notation, note that for any ϕ ∈ S∗ we may
write
(〈v1, ϕ〉, . . . , 〈vk, ϕ〉)T = 〈v, ϕ〉 = (〈vi, uj〉) cϕ = Mcϕ,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE 679
where M := (〈vi, uj〉) is a k × k matrix and cϕ is the vector of nonnegative coefficients guaranteed
by (3.2). Since S∗|V+
has k independent elements, matrix M is non-singular. Thus we may solve for
cϕ and write cϕ = M−1〈v, ϕ〉. Let P+ := uM−1 ⊗ v; obviously P is a projection from X into V+.
Moreover, for every f ∈ S and ϕ ∈ S∗ we have
〈P+f, ϕ〉 =
〈
〈f,uM−1〉v, ϕ
〉
= 〈f,u〉M−1〈v, ϕ〉 = 〈f,u〉cϕ ≥ 0
since 〈f,u〉cϕ is a dot-product of two vectors with nonnegative entries. By Lemma 2.1, P+f ∈ S.
Lemma 3.3 is proved.
Lemma 3.4. Let S ⊂ X be given and let S∗ denote its dual cone. Let V = V−⊕ V0⊕ V+ and
assume dim (V+) = k 6= 0. If the (k-dimensional ) cone S∗|V+
is not closed then PS(X,V+) = ∅.
Proof. We consider the contrapositive. Let P ∈ PS(X,V+) and let P ∗S∗ denote the (weak*)
closure of P ∗S∗ ⊂ X∗. Choose P ∗ϕ ∈ P ∗S∗ ⊂ P ∗X∗ and a sequence {P ∗ϕk}∞k=1 ⊂ P ∗S∗ such
that P ∗ϕk → P ∗ϕ. Notice, by Lemma 2.1, {P ∗ϕk}∞k=1 ⊂ S∗. S∗ is weak*-closed and therefore
P ∗ϕ ∈ S∗; this implies P ∗ϕ ∈ P ∗S∗ since (P ∗)2 = P ∗. Thus P ∗S∗ is closed. Note that P ∗S∗ is
homeomorphic to (P ∗S∗)|V+ and thus (P ∗S∗)|V+ is closed. Finally, we claim (P ∗S∗)|V+ = S∗|V+
. To
verify this, choose ϕ ∈ S∗, v ∈ V+ and consider
〈v, P ∗ϕ〉 = 〈Pv, ϕ〉 = 〈v, ϕ〉,
where the last equality follows from the fact that P is a projection. But this equation simply says that
P ∗ϕ and ϕ agree on V+, thus establishing the claim. From here we can conclude that S∗|V+
is closed.
Lemma 3.4 is proved.
3.3. Proofs of existence results. Proof of Theorem 3.1. (⇒) Let P ∈ PS(X,V ) and write
P = P− ⊕ P0 ⊕ P+. By Lemma 3.2, S⊥ is total over V−. Furthermore, for every f ∈ S and every
ϕ ∈ S∗ we have
0 ≤ 〈Pf, ϕ〉 = 〈P−f, ϕ〉+ 〈P0f, ϕ〉+ 〈P+f, ϕ〉 = 〈P+, ϕ〉
by Lemmas 3.1 and 3.2 and therefore PS(X,V+) 6= ∅.
(⇐) Let Q = Q− ⊕ Q0 ⊕ Q+ be any projection onto V and define P0 := Q0. Choose P1 ∈
PS(X,V+); we claim
P0 ⊕ P1 ∈ PS(X,V0 ⊕ V+). (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⊕V+ 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, P1V0 ⊂ S and V0∩V+ = {0}. This establishes
(3.3). We now focus on V−. Since S⊥ is total over V− (and assuming r := dim (V−) > 0), there exist
u1, . . . , ur ∈ S⊥ such that P− :=
∑r
i=1
ui⊗ vi is a projection onto V− (in the case r = 0 define P−
to be the zero-operator). Now with P1 chosen as above, write P1 =
∑d
i=d−(k+1)
ui⊗vi. Again using
S⊥ total over V−, there exist functionals ϕ1, . . . , ϕr ∈ S⊥ such that for each j ∈ {d−(k+1), . . . , d},
there exist constants {c1j , . . . , crj} ∈ R such that〈
vi,
r∑
m=1
cm,jϕm
〉
= −〈vi, uj〉 for i = 1, . . . , r.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
680 M. P. PROPHET, I. A. SHEVCHUK
Define Φj :=
∑r
m=1
cm,jϕj and note that
〈v,Φj〉 = −〈v, uj〉 for any v ∈ V−. (3.4)
Let Uj := uj + Φj for each j = d− (k + 1), . . . , d and P+ :=
∑d
i=d−(k+1)
Ui ⊗ vi. We claim
P := P− ⊕ P0 ⊕ P+
belongs to PS(X,V ). Consider first P+; note, by construction each Φj ⊂ S⊥ vanishes S. Thus
P+ ∈ PS(X,V+) and so by (3.3), we have
P0 ⊕ P+ ∈ PS(X,V0 ⊕ V+). (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⊕ V+ and P0⊕P+ 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 ∈ V− we have
P+v =
d∑
i=d−(k+1)
〈v, Ui〉vi =
d∑
i=d−(k+1)
〈v, ui + Φi〉vi =
=
d∑
i=d−(k+1)
〈v, ui − ui〉vi = 0
by the definition of each Φi. So P+ 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,V+) 6=
6= ∅ is equivalent to S]|V+
simplicial, which we now establish. Recall that S] ⊂ S∗ is a pointed, weak*
closed cone and, as such, is exactly the dual cone of
S1 :=
{
x ∈ X
∣∣ 〈x, ψ〉 ≥ 0 ∀ψ ∈ S]
}
.
Note that S1 contains the cone S. By Theorem 3.2,
S]|V+
is simplicial ⇐⇒ PS1(X,V+) 6= ∅
and thus we need only show
PS(X,V+) 6= ∅ ⇐⇒ PS1(X,V+) 6= ∅. (3.6)
Let P ∈ PS(X,V+); we claim 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 〈f, P ∗u〉 = 〈Pf, u〉 for f ∈ X
and u ∈ X∗). 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 ψ ∈ S], non-zero P ∗ψ does not vanish against S. But
P ∗ψ =
∑k
j=1〈vj , ψ〉uj , where (via relabeling) {v1, . . . , vk} ⊂ S is our fixed basis for V+. And so
P ∗ψ 6= 0 implies 〈vi, ψ〉 6= 0 for some i. Therefore P ∗ψ ∈ S], which establishes P (S1) ⊂ S1. Thus
P ∈ PS1(X,V+). To complete the proof, let P ∈ PS1(X,V+). Arguing as above, it follows that
P ∗S∗ ⊂ S∗ and thus P ∈ PS(X,V+), which establishes (3.6).
Theorem 3.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE 681
4. Application: the q-monotone case. In this section we consider the preservation of q-
monotonicity (defined below) by a projection from X = (Cq[−1.1], ‖·‖) onto V = Πn (the subspace
of algebraic polynomials of degree less than or equal to n), where
‖f‖ := max
j=0,...,q
{
‖f (j)‖∞
}
.
For s ∈ N, let Ys denote the collection of s distinct points Y = {yi}si=1 where y0 = −1 <
< y1 < . . . < ys < 1 = ys+1. For q ∈ N and Y ∈ Ys, define
SqY =
{
f ∈ X
∣∣ (−1)jf (q)(t) ≥ 0 whenever t ∈ [yj , yj+1], j = 0, . . . , s
}
.
We say f ∈ X is q-monotone (with respect to Y ∈ Ys) exactly when f ∈ SqY . We denote by PSq
Y
the set of q-monotone preserving projections from X onto Πn.
The main point of this section is the following characterization. The proof of this theorem con-
siders the (topological) consequence of restricting a dual cone to subspace V = Πn. For purposes of
illustration, we include (in Subsection 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 ∈ N. Then, for Y ∈ Ys,
PSq
Y
6= ∅ ⇐⇒ n− s− q ≤ 1.
Proof. We prove this result through induction on q. The q = 1 case is verified (for all s and n)
in the following section (see Lemma 4.1). We now proceed with the inductive step; for fixed q0, we
assume
PSq0
Y
6= ∅ ⇐⇒ n− s− q0 ≤ 1 (4.1)
and show
P
S
q0+1
Y
6= ∅ ⇐⇒ n− s− (q0 + 1) ≤ 1. (4.2)
Suppose n − s − (q0 + 1) ≤ 1; then we have (n − 1) − s − q0 ≤ 1 and so by (4.1) there exists
P ∈ PSq0
Y
(X,Πn−1). Using the notation from Subsection 3.2, we may write P =
∑n−1
k=1
uk ⊗ vk
where Pf =
∑n−1
k=1
〈f, uk〉vk ∈ Πn−1. Define P̂ :=
∑n
k=0
ûk ⊗ v̂k where û0 ⊗ v̂0 := δ−1 ⊗ 1 and,
for k > 0, ûk := uk ◦Dt (Dt is the differential operator), v̂k := It ◦ vk (It is the integral operator).
Thus
(P̂ f)(t) =
n∑
k=0
〈f, ûk〉v̂k(t) = f(−1) +
n∑
k=1
〈f ′, uk〉It(vk) =
= f(−1) +
t∫
−1
n∑
k=1
〈f ′, uk〉vk(x) dx = f(−1) +
t∫
−1
(Pf ′)(x) dx.
Note that P̂ : Cq0+1[−1, 1]→ Πn. Moreover, since P is a projection (onto Πn−1), so is P̂ (onto Πn).
And finally, if f ∈ Sq0+1
Y then f ′ ∈ Sq0Y which implies Pf ′ ∈ Sq0Y . Therefore, since (P̂ f)(q0+1) =
= (Pf ′)(q0), we have P̂ f ∈ P
S
q0+1
Y
. Thus P
S
q0+1
Y
6= ∅. To establish the other direction of (4.2),
consider n−s−(q0+1) > 1; we show that this implies P
S
q0+1
Y
= ∅. Suppose there exists P ∈ P
S
q0+1
Y
.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
682 M. P. PROPHET, I. A. SHEVCHUK
Arguing as above, express P as P =
∑n
k=0
uk ⊗ vk, where vk := xk. Define P̂ :=
∑n−1
k=0
ûk ⊗ v̂k
where ûk = uk ◦ It and v̂k = Dt ◦ vk. Then(
P̂ f
)
(t) =
n∑
k=0
〈f, ûk〉v̂k(t) = Dt
(
n∑
k=1
〈Itf, uk〉vk
)
= Dt (P (Itf)) .
Evidently P̂ is a projection from Cq0 onto Πn−1. If f ∈ Sq0Y then P̂ f ∈ Sq0Y since P (Itf) ∈ Sq0+1
Y
and this implies P̂ ∈ PSq0
Y
(X,Πn−1). But from our supposition, we have (n − 1) − s − q0 > 1,
which, from (4.1), implies PSq0
Y
= ∅. This contradiction has resulted from assuming P ∈ P
S
q0+1
Y
and therefore we must have P
S
q0+1
Y
= ∅. This establishes (4.2).
Theorem 4.1 is proved.
4.1. The q = 1 case. In this subsection we verify the q0 = 1 case via the following lemma.
Lemma 4.1. PS1
Y
(X,Πn) 6= ∅ ⇐⇒ n− s ≤ 2.
To begin, denote S1
Y 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 = V− ⊕ V0 ⊕ V+. Note that V0 is 1-dimensional
and V0 = [1]. As we will see below, dim (V+) = n − s; recall from above that we may assume
S∗|V+
⊂ Rn−s. For fixed Y, put
∆ = ∆(x) :=
s∏
i=1
(yi − x).
Proposition 4.1. dim (V+) = max{0, n− s}. If n− s > 0 then, for i = 1, . . . , n− s,
vi(x) :=
x∫
−1
(1− ti)∆(t) dt ∈ SY
and {v1, . . . , vn−s} forms a basis for V+.
Let v ∈ V ∩SY ; then for i = 1, . . . , s we have v′(yi) = 0. Thus if n− s ≤ 0 then dim (V+) = 0.
Assume n − s > 0; then by definition of SY we can write v′(x) = p(x)∆(x) for some polynomial
p. But deg(∆) = s and so p ∈ Πn−(s+1). Therefore dim (V+) ≤ n − s. Finally, note that for
i = 1, . . . , n− s,
vi =
x∫
−1
(1− ti)∆(t) dt ∈ SY
and are independent. Thus V+ = [v1, . . . , vn−s].
Note that in this application we have have labeled the basis elements for V+ 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 n− s > 2. Then S∗|V+
⊂ Rn−s is not closed and thus PSY
(X,Πn) = ∅.
Proof. Fix yj for some j ∈ {1, . . . , s}. Since n−s ≥ 3, it is clear from Proposition 4.1 that a basis
for V+ can be chosen as prescribed to include elements v1 :=
∫ x
−1
∆(t) and v2 :=
∫ x
−1
(1− t2)∆(t).
Without loss, assume ∆(t) ≥ 0 for t ∈ (yj−1, yj). And so, since S∗|V+
is a cone, it must contain, for
each such t, the point (or vector)
(δ′t)|V+
∆(t)
. Thus by Proposition 4.1 there exists a vector
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
SHAPE-PRESERVING PROJECTIONS IN LOW-DIMENSIONAL SETTINGS AND THE q-MONOTONE CASE 683
z = [1, 1, z3, . . . , zn−s] := lim
t→y−j
(δ′t)|V+
∆(t)
belonging to the closure of S∗|V+
. Now, by way of contradiction, let us suppose there exists ϕ ∈ S∗
such that ϕ|V+ = z. Note that
1 = ϕ(v1) = ϕ
x∫
−1
∆(t)
= ϕ(v2) = ϕ
x∫
−1
(1− t2)∆(t)
(4.3)
which implies
ϕ
x∫
−1
t2∆(t)
= 0.
Moreover, for every even integer ν ≥ 2 we have
x∫
−1
tν∆(t) ∈ S and
x∫
−1
(t2 − tν)∆(t) ∈ S
since t2 − tν ≥ 0 on [−1, 1]. And thus for every ν
ϕ
x∫
−1
tν∆(t)
= 0. (4.4)
For convenience, assume yj = 0. Define ∆̂(x) by ∆(x) = x∆̂(x). Let TO(x) be an odd Tchebyshev
polynomial of (arbitrary odd) degree d. Consider the polynomial p(x) :=
x∫
−1
TO∆̂ ∈ X; the norm
‖p‖ is clearly bounded independent of d. But by (4.3) and (4.4) we find
|ϕ(p)| =
∣∣∣∣∣∣∣ϕ
x∫
−1
d∑
i=1
i odd
cit
i
∆̂(t)
∣∣∣∣∣∣∣ =
∣∣∣∣∣∣∣ϕ
d∑
i=1
i odd
x∫
−1
cit
i−1∆
∣∣∣∣∣∣∣ = d
since |c1| = d. This implies that ϕ is unbounded and thus cannot be an element of S∗. Therefore
S∗|V+
is not closed. Consequently, by Lemma 4.2 and Corollary 3.4, we have PSY
(X,V+) = ∅ and
thus PSY
(X,V ) = ∅ by Theorem 3.1.
Lemma 4.2 is proved.
Lemma 4.3. Suppose n− s ≤ 2. Then PSY
(X,V ) 6= ∅.
Proof (Version 1). Set ys+2 := y0 = −1. Fix n ∈ N, n− s ≤ 2. For each g ∈ C[−1, 1] denote
by Ln−1(x, g) := L(x, g; y1, . . . , yn) — the Lagrange polynomial of degree < n, that interpolates g
at yj’s, j = 1, . . . , n. First we remark, that the operator P ∈ L(C1[−1, 1],Πn), defined by
(Pg)(x) := g(0) +
x∫
0
Ln−1(t, g
′)dt,
is a projection, that is P ∈ P(C1[−1, 1],Πn). This readily follows from the fact, that for each
pn−1 ∈ Πn−1 we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
684 M. P. PROPHET, I. A. SHEVCHUK
Ln−1(x, pn−1) ≡ pn−1(x).
So, to end the proof we have to check, that if f ∈ SY , then (Pf) ∈ SY as well, or, which is the
same,
Ln−1(x, f
′)∆(x) ≥ 0, x ∈ [−1, 1], (4.5)
where ∆(x) :=
∏s
j=1
(yj − x). Indeed, if n ≤ s, then Ln−1(x, f
′) ≡ 0, that yields (4.5). If
n = s + 1, then Ln−1(x, f ′) = A∆(x), where A ≥ 0, that yields (4.5). Finally, if n = s + 2, then
Ln−1(x, f
′) = (ax+ b)∆(x). Let us show, that
ax+ b ≥ 0, x ∈ [−1, 1]. (4.6)
If x = −1, then
−a+ b =
Ln−1(−1, f ′)
∆(−1)
=
f ′(−1)
∆(−1)
≥ 0.
Similarly a+ b ≥ 0. Thus (4.6) holds, that 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 := dim (V−) = min{s, n} and V− = [x, x2, . . . , xr]. And since
{δ′yi}
s
i=1 ⊂ S⊥ we have that S⊥ is total over V−. Now in the case n− s ≤ 0 we have dim (V+) = 0
and so trivially PS(X,V+) 6= ∅ since the zero-operator belongs to this set. Suppose n − s > 0; by
Proposition 4.1, n− s is exactly the dimension of S∗|V+
. We claim, in the cases n− s = 1, 2, the cone
S∗|V+
is simplicial. This is clear in the n − s = 1 case, since every 1-dimensional pointed cone is
(trivially) simplicial. For n− s = 2, note that a 2-dimensional pointed cone is simplicial if and only
if it is closed. We now show S∗|V+
⊂ R2 is closed. Recall that S∗|V+
belongs to the positive quadrant
of R2. And it will suffice to show that for some basis for V+, there exist functionals ϕ1, ϕ2 ∈ S∗
such that (ϕi)|V+ belongs to the ray determined by ei (the standard basis element) for i = 1, 2. To
this end, note that
v1 :=
∫ x
−1
−(t− 1)∆(t) and v2 :=
∫ x
−1
(t+ 1)∆(t)
are elements of S and form a basis for V+. Moreover (δ′−1)|V+ = [a, 0] and (δ′1)|V+ = [0, b] for some
a, b > 0. Therefore S∗|V+
is exactly the positive quadrant of R2. Thus, in the cases n − s = 1, 2 we
have S∗|V+
simplicial, which implies PS(X,V+) 6= ∅ by Theorem 3.3. By Theorem 3.1 we conclude
PS(X,V ) 6= ∅.
Lemma 4.3 is proved.
1. Chalmers B., Mupasiri D., Prophet M. P. A characterization and equations for minimal shape-preserving projections
// J. Approxim. Theory. – 2006. – 138. – P. 184 – 196.
2. Cheney E. W. Introduction to approximation theory. – New York: Chelsea Publ., 1982.
3. Donner K. Extension of positive operators and Korovkin theorems. – Berlin: Springer, 1982.
4. Lewicki G., Prophet M. P. Minimal multi-convex projections // Stud. Math. – 2007. – 178, № 2. – P. 99 – 124.
5. Mupasiri D., Prophet M. P. A note on the existence of shape-preserving projections // Rocky Mountain J. Math. –
2007. – 37, № 2. – P. 573 – 585.
6. Mupasiri D., Prophet M. P. On the difficulty of preserving monotonicity via projections and related results // Jaen J.
Approxim. – 2010. – 2, № 1. – P. 1 – 12.
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Received 08.12.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 5
|
| id | umjimathkievua-article-2607 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:45Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4c/205e01a918318754b7305f7455c1d04c.pdf |
| spelling | umjimathkievua-article-26072020-03-18T19:30:33Z Shape-preserving projections in low-dimensional settings and the q -monotone case Формозберiгаючi проекцiї у маловимiрнiй постановцi та q -монотонний випадок Prophet, M. P. Shevchuk, I. A. Профет, М. П. Шевчук, І. О. Let $P: X \rightarrow V$ be a projection from a real Banach space $X$ onto a subspace $V$ and let $S \subset X$. In this setting, one can ask if $S$ is left invariant under $P$, i.e., if $PS \subset S$. If $V$ is finite-dimensional and $S$ is a cone with particular structure, then the occurrence of the imbedding $PS \subset S$ 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, шє remove the structural assumptions on $S^{*}$ and characterize the cases where $PS \subset S$. We note that the (so-called) $q$-monotone shape forms a cone which (lacks structure and thus) serves as an application for our characterization. Нехай $P: X \rightarrow V$ — проекцiя дiйсного банахового простору $X$ на пiдпростiр $V$ i, крiм того, $S \subset X$. У цiй постановцi виникає питання: чи є $S$ лiвоiнварiантним пiд дiєю $P$, тобто чи має мiсце вкладення $PS \subset S$? Якщо пiдпростiр $V$ є скiнченновимiрним, а $S$ є конусом iз певною структурою, то вкладення $PS \subset S$ може бути охарактеризовано шляхом геометричного опису. Ця характеризацiя iстотно залежить вiд структури $S$, або, точнiше, вiд структури конуса $S^{*}$, спряженого до $S$. У цiй роботi усунено структурнi припущення щодо $S^{*}$ i охарактеризовано випадки, у яких $PS \subset S$. Вiдзначено, що (так звана) $q$-монотонна форма утворює конус, який (не має структури i тому) може бути використаний для застосування нашої характеризацiї. Institute of Mathematics, NAS of Ukraine 2012-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2607 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 5 (2012); 674-684 Український математичний журнал; Том 64 № 5 (2012); 674-684 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2607/1973 https://umj.imath.kiev.ua/index.php/umj/article/view/2607/1974 Copyright (c) 2012 Prophet M. P.; Shevchuk I. A. |
| spellingShingle | Prophet, M. P. Shevchuk, I. A. Профет, М. П. Шевчук, І. О. Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title | Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title_alt | Формозберiгаючi проекцiї у маловимiрнiй постановцi та q -монотонний випадок |
| title_full | Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title_fullStr | Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title_full_unstemmed | Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title_short | Shape-preserving projections in low-dimensional settings and the q -monotone case |
| title_sort | shape-preserving projections in low-dimensional settings and the q -monotone case |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2607 |
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