Minimal multi-convex projections

Grzegorz Lewicki^{1}, Michael Prophet^{2}

April 20, 2005

IMUJ Preprint 2005/08

Abstract

We say that a function from X = C^{L}[a, b] is k-convex (for k ≤ L) if the kth derivative of the function is nonnegative. Let P denote a projection from X onto V = Π_{n} ⊂ X, where Π_{n} denotes the space of algebraic polynomials of degree less than equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed only for k = n−1 and k = n does such a projection exist. So let us consider instead a more general ’shape’ to preserve. Let σ = (σ_{0}, σ_{1}, …, σ_{n}) be an (n + 1)-tuple with σ_{i} ∈ {0, 1}; we say f ∈ X is multi-convex if f^{(i)} ≥ 0 for i such that σ_{i} = 1. In this paper we characterize those σ for which there exists a projection onto V preserving the multi-convex shape. For those shapes able to be preserved via a projection, we construct (in all but one case) a minimal norm multi-convex preserving projection. Out of necessity, we include some results concerning the geometrical structure of C^{L}[a, b]

1Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland

2Department of Mathematics, University of Northern Iowa, Cedar Falls Iowa, USA

1 Introduction

When X is a Banach space and V ⊂ X a subspace, we denote by P(X, V) the set of all projections from X onto V; in the cases where there no ambiguity, we will simply write P. We say that a projection P_{0} is a minimal projection if ∥P_{0}∥ ≤ ∥P ∥ for all P ∈ P(X, V).

It is worth noting that there exists a large number of papers concerning minimal projections. Mainly the problems concern existence ([15], [18]), uniqueness ([14], [16], [27], [39], [40]), characterization of one-complemented subspaces ([1], [2], [29], [36], [37], citemu) concrete formulas for minimal projections ([3], [4], [5], [6], [7], [13], [15], [23], [24], [26], [35], [41]), estimates of the relative projection constants ([5], [17], [21], [25], [33], [38], [42]), construction of spaces with large relative projection constants ([4], [5], [20] [22]). For basic information concerning this topic the reader is referred to [32]

While a minimal projection will, in general, provide good approximations, it may fail to preserve particular properties of elements, as illustrated below. As such, we are motivated to look for projections which leave invariant (or preserve) a particular functional characteristic (or ’shape’). These characteristics are often described using cones.

More precisely, a cone in X is convex set closed under nonnegative scalar multiplication. Assuming P ≠ ∅, we may fix cone S ⊂ X and ask if any element from P leaves S invariant; i.e, let

P_{S} = P_{S}(X, V) = {P ∈ P | P S ⊂ S}

and determine if P_{S} ≠ ∅. When P ∈ P_{S} we say P is shape-preserving (in the sense of S). Some basic results on the existence of shape-preserving projections can be found in [10], [31], [12] and [34]. Not surprisingly, for given X, V and S, the problem of determining if P_{S} ≠ ∅ is nontrivial in general.

In this paper we first characterize, for a large collection of X, V and S, when P_{S} ≠ ∅; then, for each setting in which P_{S} ≠ ∅, we calculate inf_{P ∈P_{S}} ∥P ∥. Moreover we construct a minimal shape-preserving projection.

Specifically, for positive integer L let X denote the L-th continuously differentiable functions on [a, b], C^{L}[a, b], normed by

∥f∥_{L} = max_{i=0..L}{∥f^{(i)}∥_{∞}}.

In this case we simply write X = (C^{L}[a, b], ∥ · ∥_{L}). We denote by X^{∗} the dual space of X. In this setting, note that δ_{t}^{k}, k-th derivative evaluation at t, belongs to sphere of X^{∗} for k = 0 … L and t ∈ [a, b]. For fixed k, consider the cone S ⊂ X of all f ∈ X with nonnegative k-th derivative on [a, b]. We refer to this set as the cone of k-convex functions. With V = Π_{n}, the n-th degree algebraic polynomials, it was shown in [11] that

P_{S} ≠ ∅ ⟺ k ≥ n − 1.  (1)

For example, with X = (C^{1}[0, 1], ∥ · ∥_{1}) and k = 1 we see that there is no monotonicity-preserving (1-convex preserving) projection from X onto V = Π_{3}. There is however a projection preserving convexity (or 2-convexity) onto V. Moreover, Theorem 4.2 in [11] constructs a minimal norm element of P_{S} for k = n − 1 (with norm 3/2 for every n) using techniques from minimal projection theory found in [7].

As we will see in the Section 3, the existence of a projection preserving k-convexity onto Π_{n} can be determined via a geometric consideration; in the case k = n or k = n − 1, this geometric approach reduces (respectively) to a 1-dimensional or 2-dimensional problem and, as such, is relatively easy to solve. That is, the geometric approach quickly reveals the result in (1).

We now look to generalize k-convexity. Using notation similar to that of [30], for fixed positive integer n let σ = (σ_{0}, σ_{1}, …, σ_{n}) be an (n + 1)-tuple with σ_{i} ∈ {0, 1}; let M = max_{σ_{i}=1} i. With X = C^{L}[a, b], (L ≥ M) define

Sσ := {f ∈ X | σ_{i}f^{(i)} ≥ 0, i = 0, …, n}.

We say f ∈ X is multi-convex if f belongs to the cone Sσ. In this paper we fix V = Π_{n} and consider projections from X onto V leaving invariant a cone of multi-convex functions - so-called multi-convex projections. We denote this set of projections by P_{Sσ} and look to construct minimal norm elements from this set.

This paper is organized into five sections. Following these introductory remarks, the main content of this paper is described in Section 2. Here we characterize those σ for which P_{Sσ} ≠ ∅, where P_{Sσ} ⊂ X = (C^{L}[a, b], ∥ · ∥_{L}). Furthermore, we develop an iterative, norm-preserving construction of multiconvex projections from X onto (n + 1)-dimensional subspaces V, where the iteration is with respect to n. This construction yields minimal norm multiconvex projections in the case V = Π_{n}. Sections 3 and 5 provide proofs of the results of Section 2. The proofs in Section 5 require basic, non-trivial facts about the unit ball of X = (C^{L}[a, b], ∥ · ∥_{L}). For sake of completeness, we prove the needed results in Section 4 (indeed we found no single source which described the geometry of this ball and thus hope that Section 4 may be of independent utility to others).

As a summary of notation used in following, the dual space of Banach space X is denoted by X^{∗}; we denote by B(X) and S(X), respectively, the unit ball of X and the unit sphere of X. For convex set K ⊂ X, we denote the set of extreme points of K by ext (K). The convex hul of subset A ⊂ X is denoted by co A while the conex cone generated by A is denoted and defined as cone (A) = {ρa | ρ ∈ [0, ∞) and a ∈ A}.

2 Main Results

Let L and n denote positive integers such that L ≥ n − 1 (the reason for this inequality will be made clear). Let σ = (σ_{0}, σ_{1}, …, σ_{n}) with σ_{i} ∈ {0, 1}; let M = max_{σ_{i}=1} i and m = min_{σ_{i}=1} i. We say that σ is 1-connected if whenever σ_{i} = σ_{j} = 1 for i < j, we have σ_{k} = 1 for all k = i, i + 1, …, j.

THEOREM 2.1 Let X = (C^{L}[0, 1], ∥ · ∥_{L}). P_{Sσ}(X, Π_{n}) ≠ ∅ iff M ≥ n − 1 and σ is 1-connected.

The next theorems describe minimal norm multi-convex projections. But first a few comments are in order. By defintion we always have m ≤ M. Whenever M = n, we automatically assume L ≥ n. Theorem 2.1 indicates that there two possible situations (of interest to us) in which m = M: they are m = M = n − 1 and m = M = n. These cases are actually ’k-convex’ shapes (regarded as specific multi-convex shapes); moreover these situations constitute somewhat extreme cases in the multi-convex realm. The case m = M = n − 1 has been handled in [11]. For sake of completeness, we give the result on minimality for this cases in Theorem 2.2.

In the case m = M = n, the minimal shape-preserving projection problem is completely unsolved for n≥ 2 (the projection given in [8] partially solves the problem in the n = 2 case). Indeed, it is conjectured in [11] that a minimal norm projection from X = C^{L}[a, b] onto V = Π_{n} preserves n-convexity for every L = 0, 1, . . .. That is, in the case of n-convexity, the minimal shape-preserving projection problem is perhaps equivalent to the minimal projection problem. As such, this paper does not address this case.

There is one other exceptional case: m = n − 1 and M = n. It turns out that the results concerning minimal shape-preserving projections in this case are similar to those in the case where m = M = n−1 but the method of proof differs substantially from the approach in [11] as well as here. Consequently, the m = n − 1, M = n case is handled in [28]. Howerver, Theorem 2.2 below states the result for this case.

Through the remainder of this paper we will assume n ≥ 2. In the n = 1 case, there is a projection of norm one in P_{Sσ}(X, Π_{1}).

THEOREM 2.2 (see [11] and [28]) Let X = (C^{L}[a, b], ∥ · ∥_{L}). For fixed n let m = n − 1 ≤ M. Then there exists P_{m} ∈ P_{Sσ}(X, Π_{n}) such that ∥P_{m}∥ = 3/2 and ∥P_{m}∥ ≤ ∥P ∥ for every P ∈ P_{Sσ}(X, Π_{n}).

THEOREM 2.3 Let X = (C^{L}[a, b], ∥ · ∥_{L}). For fixed n, assume M ≥ n − 1 and σ is 1-connected. Suppose m = 0 and define P_{0,n} = ∑_{i=0}^{n} u_{i} ⊗ v_{i} where

u_{i} = δ_{0}^{i} for i ≠ n,

u_{n} = δ_{1}^{n−1},

v_{i} = \frac{x^{i}}{i!} for i ≠ n − 1

and

v_{n−1} = \frac{x^{n−1}}{(n − 1)!} − \frac{x^{n}}{n!} ;

i.e.,

P_{0,n} = δ_{0} ⊗ 1 + δ_{0}^{1} ⊗ \frac{x}{1!} + ⋯ + δ_{0}^{n−1} ⊗ (\frac{x^{n−1}}{(n − 1)!} − \frac{x^{n}}{n!} ) + δ_{1}^{n−1} ⊗ \frac{x^{n}}{n!} . (2)

Then P_{0,n} has minimal norm in P_{Sσ}(X, Π_{n}) and

∥P_{0,n}∥ = ∑_{k=0}^{n−1} \frac{1}{k!}.  (3)

Moreover, in the case that M = n − 1, we have {P_{0,n}} = P_{Sσ}(X, Π_{n}).

THEOREM 2.4 Let X = (C^{L}[a, b], ∥ · ∥_{L}) and Let X_{1} = (C^{L+1}[a, b], ∥ · ∥_{L+1}). For fixed integer n, assume 0 < m < n − 1 ≤ M and σ is 1connected. Let Y ⊂ X denote an (n + 1)-dimensional subspace, spanned by {w_{0}, w_{1}, …, w_{n}}; i.e., Y = [w_{0}, …, w_{n}]. Let P_{m,n} = ∑_{i=0}^{n} q_{i} ⊗ w_{i} denote a projection from X onto Y such that P_{m,n} preserves Sσ; i.e.,

P_{m,n} ∈ P_{Sσ}(X, Y).

Define the operator P_{m+1,n+1} on X_{1} by

\begin{aligned}(P_{m+1,n+1}f)(x) & = \frac{f(0) + f(1)}{2} + ∫_{0}^{x}(P_{m,n}f′)(t) dt \\ & − \frac{1}{2} ∫_{0}^{1}(P_{m,n}f′)(t) dt \end{aligned}  (4)

where f′ denotes the derivative of f. Then

P_{m+1,n+1} ∈ P_{Sσ̂}(X_{1}, [1, W_{0}, W_{1}, …, W_{n}])

where

W_{i}(t) = ∫_{0}^{t} w_{i}(s) ds

and σ̂ is the 1-connected (n + 2)-tuple such that max_{σ_{i}=1} i = M + 1 and min_{σ_{i}=1} i = m + 1. Moreover, if ∥P_{m,n}∥ ≥ 2 then

∥P_{m+1,n+1}∥ = ∥P_{m,n}∥  (5)

THEOREM 2.5 Let k be a nonnegative integer. Let X = (C^{L+k}[a, b], ∥ · ∥) where ∥ · ∥ is any norm such that

∥ · ∥_{2,L+k} ≤ ∥ · ∥ ≤ ∥ · ∥_{L+k}

where

∥f∥_{2,L+k} = max { max_{j=0,...,L+k−1}{|f^{(j)}(0)|, |f^{(j)}(1)|}, ∥f^{L+k}∥_{∞}}

and

∥f∥_{L+k} = max_{i=0..L+k}{∥f^{(i)}∥_{∞}}.

Let P_{k,n+k} denote the specific operator obtained by k applications of (4) beginning with P_{0,n} given in Theorem 2.3. Then P_{k,n+k} is a minimal norm element of P_{Sσ}(X, Π_{n+k}) where σ is the 1-connected (n + k + 1)-tuple such that max_{σ_{i}=1} i ≥ n + k − 1 and min_{σ_{i}=1} i = k.

In general, given two norms that are equivalent (but not proportional), we should not expect a projection that has minimal operator norm with respect to first norm to be minimal in the operator norm determined by the second. From this viewpoint, we note that Theorem 2.5 is quite surprising.

The proofs of Theorems 2.1, 2.3 and 2.4 and Theorem 2.5 are contained in the sections that follow. We first verify existence in Section 3. Then, in Section 4, we show how to calculuate the norms of functionals from a particular family. This calculation will play a crucial role in Section 5, where we verify shape-preserving properties and norm minimality of constructed projections.

3 Proof of Existence

We employ results from [31] to establish existence. The relevant material from this paper is included below.

A cone K in a Banach space is defined to be a convex set which is closed under nonnegative scalar multiplication. K is said to be pointed if K contains no lines through 0.

For ϕ ∈ K, let [ϕ]^{+} := {αϕ | α ≥ 0}. We say [ϕ]^{+} is an extreme ray of K if ϕ = ϕ_{1} + ϕ_{2} implies ϕ_{1}, ϕ_{2} ∈ [ϕ]^{+} whenever ϕ_{1}, ϕ_{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 = co(E(K)).

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 3.1 Let X be a Hausdorff topological vector space over ℝ and let X^{∗} be the topological dual of X. We say that a pointed closed cone K ⊂ X^{∗} is simplicial if K can be recovered from its extreme rays, (i.e., K = \overline{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 iff K has exactly d extreme rays.

For given Sσ, we define its dual cone as

S^{∗} = {u ∈ X^{∗} | u(f) ≥ 0 ∀f ∈ Sσ}.

Note that for each Sσ, the cone dual S^{∗} is simplicial with [ϕ]^{+} ∈ E(S^{∗}) iff ϕ = ρδ_{t}^{k} where ρ > 0, t ∈ [0, 1] and integer k ∈ [0, 1, …, L]. The result we will need is the following.

THEOREM 3.1 Let X = (C^{L}[0, 1], ∥ · ∥_{L}). P_{Sσ}(X, Π_{n}) ≠ ∅ if and only if the cone S_{|_{V}}^{∗} is simplicial.

(Proof of Theorem 2.1) Throughout this proof we denote the dual cone of Sσ by S^{∗}. (⇐) We verify that S_{|_{V}}^{∗} is simplicial. Note that for each t ∈ [0, 1] and each integer j ∈ [m, M] the functional δ_{t}^{j} belongs to an extreme ray of S^{∗}. As such, we need only demostrate that a simplicial sub-cone of S_{|_{V}}^{∗} captures all restrictions (δ_{t}^{j})_{|_{V}}; this will then imply that S_{|_{V}}^{∗} is itself simplicial. To this end, we fix for V the basis {v_{i}}_{i=0}^{n} where v_{i} = x^{i}/i! and embed S_{|_{V}}^{∗} into (the positive orthant of) ℝ^{n+1} via the identification

ϕ_{|_{V}} ≡ ⟨v, ϕ⟩ = (⟨v_{0}, ϕ⟩ ; ⟨v_{1}, ϕ⟩ ; ⋮ ; ⟨v_{n}, ϕ⟩);

with this understanding we will regard S_{|_{V}}^{∗} ⊂ ℝ^{n+1}. Let us now consider the case M = n−1 (we will see that the M = n case follows in an identical way). Notice that, for integer j ∈ [m.n − 1] and t ∈ [0, 1], we have

(δ_{t}^{j})_{|_{V}} = ( 0_{1} ; ⋮ ; 0_{j} ; 1 ; \frac{t}{1!} ; ⋮ ; \frac{t^{k−j}}{(k−j)!} ; ⋮ ; \frac{t^{n−j}}{(n−j)!} ) .  (6)

Denote by e_{i} the vector (0_{1}, …, 0_{i−1}, 1, 0_{i+1}, …, 0_{n+1})^{T} ∈ R^{n+1}; from (6) it is clear that, for every integer j ∈ [m, n−1], e_{j+1} ∈ S_{|_{V}}^{∗} (given by (δ_{0}^{j})_{|_{V}}) as well as e_{n} + e_{n+1} ∈ S_{|_{V}}^{∗} (given by (δ_{1}^{n−1})_{|_{V}}). Moreover, for integer j ∈ [m, n − 1] we have

(δ_{t}^{j})_{|_{V}} = ∑_{k=j}^{n−2} \frac{t^{k−j}}{(k − j)!}e_{k+1}+\frac{t^{n−j−1}}{(n − j − 1)!} (1 − \frac{t}{n − j} ) e_{n}+\frac{t^{n−j}}{(n − j)!} (e_{n} + e_{n+1}) .

Since the coefficient functions of e_{n} + e_{n+1} and each e_{k}, k = j, …, n are nonnegative, we have that S_{|_{V}}^{∗} is simplicial. In the case that M = n, we again note from (6) that e_{j} ∈ S_{|_{V}}^{∗} for every integer j = m + 1, …, n + 1 and thus S_{|_{V}}^{∗} is simplicial. (⇒) Assume P_{Sσ} ≠ ∅. By Theorem 3.1, we know S_{|_{V}}^{∗} is simplicial. Let E = {[x_{1}]^{+}, …, [x_{d}]^{+}} be the set of extreme rays of S_{|_{V}}^{∗}. We first show that M must be at least n − 1. Suppose, to contrary, that M ≤ n − 2. For convenience, fix for V the basis {v_{i}}_{i=0}^{n} where

v_{i} = \begin{cases}\frac{x^{i}}{i!} & i < M \\ \frac{(i−M)!}{i!}x^{i} & i ≥ M \end{cases} .  (7)

For t ∈ [0, 1], let Δ_{t} := (δ_{t}^{M})_{|_{V}} and, via the embedding into ℝ^{n+1} described above, notice that Δ_{t} = (0_{1}, …, 0_{M}, 1, t, t^{2}, …, t^{n−M}). Consider the subcone K of S_{|_{V}}^{∗} ⊂ R^{n+1} generated by rays [Δ_{t}]^{+}; i.e., let

K = \overline{co} (\{[Δ_{t}]^{+} | t ∈ [0, 1]\}).  (8)

K has infinitely many extreme rays since each Δ_{t} is an extreme point of C = \overline{co} (\{Δ_{t} | t ∈ [0, 1]\}) (C is a translate of the convex hull of the moment curve (t, t^{2}, …, t^{n−M})). If m = M then we have an immediate contraction; assume then that 0 ≤ m < M. Consequently, E cannot belong entirely to K; without loss assume {[x_{1}]^{+}, …, [x_{k}]^{+}} = E − (E ∩ K). Since each such ray is extreme, we must have for each i = 1, ..., k, [x_{i}]^{+} = [(δ_{t}^{j})_{|_{V}}]^{+} for some t ∈ [0, 1] and some integer j ∈ [m, M − 1]. But because j < M, we see that every (non-zero) element (a_{1}, a_{2}, …, a_{n+1})^{T} from co ({[x_{1}]^{+}, …, [x_{k}]^{+}}) contains at least one non-zero entry in the first M coordinates; i.e., there exists integer s ∈ [0, M] such that a_{s} ≠ 0. This implies co ({[x_{1}]^{+}, …, [x_{k}]^{+}}) ∩ K = ∅. Thus K ⊂ S_{|_{V}}^{∗} cannot be captured in a simplicial subcone of S_{|_{V}}^{∗}, and this contradicts the fact that S_{|_{V}}^{∗} simplicial. Therefore M ≥ n − 1. We now show σ is 1-connected. Suppose it is not; let Z := max{i | σ_{i+1} = 0 and i < M} (Z marks the location of the last 1 in σ before the last break of the sequence of 1’s). For convenience fix for V the basis in (7) using Z rather than M. Similar to the above, define Δ_{t} := (δ_{t}^{Z})_{|_{V}} and K as in (8). K has infinitely many extreme rays and thus, as before, the set of extreme rays of S_{|_{V}}^{∗}, E = {[x_{1}]^{+}, …, [x_{d}]^{+}}, cannot belong entirely to K. Every x_{i} must be of the form (δ_{t}^{j})_{|_{V}} for some t ∈ [0, 1] and some integer j as prescribed by σ. The convex hull of the set {[x_{i}]^{+} | x_{i} = (δ_{t}^{j})_{|_{V}} for j ≥ Z + 2} misses every (non-zero) element of K since every element (a_{1}, … a_{n+1}) of this convex hull is such that a_{Z+1} = 0. Similarly, the convex hull of the set {[x_{i}]^{+} | x_{i} = (δ_{t}^{j})_{|_{V}} for j ≤ Z − 1} misses every (non-zero) element of K since every element (a_{1}, … a_{n+1}) of this convex hull is such that a_{s} ≠ 0 for some integer s ∈ [m, Z − 1]. If σ is not 1-connected then we have exhausted all possible choices for x_{i} (in particular, there is not x_{i} of the form (δ_{t}^{Z+1})_{|_{V}}) and we find that S_{|_{V}}^{∗} cannot be simplicial. This contradiction forces us to conclude that σ is 1-connected.

4 Results on the Geometry of B(C^{L}[a, b], ∥·∥_{L})

We start with two well-known lemmas, which straightforward proofs will be omitted.

LEMMA 4.1 Let (X, ∥ · ∥) be a normed space. Suppose that (∥ · ∥_{k}) is a sequence of equivalent norms on X such that

∥x∥_{k}(1 − a_{k}) ≤ ∥x∥ ≤ (1 + a_{k})∥x∥_{k}  (9)

for any x ∈ X. Assume a_{k} → 0. Let L_{k}(X) denote the space of linear, continuous with respect to ∥·∥ operators defined on X with the norm induced by ∥ · ∥_{k}. Then for any T ∈ L(X),

∥T ∥_{k} → ∥T ∥,

where ∥T ∥ denotes the operator norm of T induced by ∥ · ∥.

LEMMA 4.2 Let (X, ∥ · ∥) be a normed space and let ∥ · ∥_{k} be a sequence of norms on X satisfying (9) such that a_{k} → 0. Let X_{k}^{∗} denote the dual space X^{∗} equipped with the norm induced by ∥ · ∥_{k}. Then for any f ∈ X^{∗},

∥f∥_{k} → ∥f∥,

where ∥f∥ denotes the norm of f in X^{∗}.

DEFINITION 4.1 Let {t_{i}} be a countable, dense subset of [0, 1] such that t_{0} = 0 and t_{1} = 1. Let us define for k ∈ ℕ a norm ∥ · ∥_{k,L} on C^{L}[0, 1] by:

∥f∥_{k,L} = max_{i=0,...,L}A_{ik}(f),

where for i = 0, ..., L − 1

A_{ik} = max_{j=1,...,k}|f^{(i)}(t_{j})|

and

A_{L,k} = ∥f^{(L)}∥_{∞}.

LEMMA 4.3 Let ∥ · ∥_{k,L} be as in Definition 4.1. Then for any ϵ > 0 there exists k_{o} ∈ ℕ such that for any f ∈ C^{L}[0, 1] and k ≥ k_{o}

∥f∥_{k,L} ≤ ∥f∥_{L} ≤ (1 + ϵ)∥f∥_{k,L},  (10)

where

∥f∥_{L} = max_{i=0..L}{∥f^{(i)}∥_{∞}}.

Proof. Fix k ∈ ℕ, k ≥ 3. Without loss of generality, we can assume that

1 = t_{o} < t_{2} < ... < t_{k} < t_{1} = 1.

Set

Δ_{k} = max{t_{2} − t_{o}, t_{1} − t_{k}, t_{j} − t_{j−1}, j = 3, ..., k}.

By the density of {t_{j}}, Δ_{k} → 0. Fix k_{o} ∈ ℕ such that

(1 + Δ_{k})^{L+1} < 1 + ϵ

for k ≥ k_{o}. Take any k ≥ k_{o}. Fix f ∈ X. First we show that

∥f^{(L−1)}∥_{∞} ≤ (1 + Δ_{k})∥f∥_{k,L}.  (11)

Let t ∈ [0, 1] be so chosen that ∥f^{(L−1)}∥_{∞} = |f^{(L−1)}(t)|. Then t ∈ [t_{o}, t_{2}] or t ∈ [t_{k}, t_{1}] or t ∈ [t_{i}, t_{i+1}] for some i = 2, ..., k. Hence by the definition of Δ_{k} and the mean value theorem, for a properly chosen t_{i} i=0,...,k,

|f^{(L−1)}(t) − f^{(L−1)}(t_{i})| ≤ ∥f^{(L)}∥_{∞}|t − t_{i}|.

Hence

∥f^{(L−1)}∥_{∞} = |f^{(L−1)}(t)| ≤ |f^{(L−1)}(t_{i})| + ∥f^{(L)}∥_{∞}Δ_{k}
≤ ∥f∥_{k,L}(1 + Δ_{k}),

which proves (11). Analogously, by the mean value theorem,

∥f^{(L−j)}∥_{∞} ≤ (1 + Δ_{k})^{j}∥f∥_{k,L}

for j = 0, ..., L. Hence

max_{i=0,...,L}{∥f^{(i)}∥_{∞}} ≤ (1 + Δ_{k})^{L+1}∥f∥_{k,L}

which gives

∥f∥ ≤ (1 + ϵ)∥f∥_{k,L}

for k ≥ k_{o}, as required.

THEOREM 4.1 Let X = (C^{L}[0, 1], ∥ · ∥_{L}) and X_{k} = (C^{L}[0, 1], ∥ · ∥_{k,L}). Then

ext (B(X^{∗})) ⊂ {±δ_{t}^{i} | i = 0, .., L, t ∈ [0, 1]}

and

ext (B(X_{k}^{∗})) ⊂ {±δ_{t_{j}}^{i} | i = 0, .., L − 1, j = 0, ..., k} ∪ {δ_{t}^{L} | t ∈ [0, 1]}.

Proof. Note that X can be isometrically embeded in Z = (C[0, 1])^{L+1} with a norm

∥(f_{1}, ..., f_{L+1})∥ = max_{i=1,...,L+1}∥f_{i}∥_{∞}.

The embedding is given by a formula

T (f) = (f, f^{(1)}, ..., f^{(L)}).

Note that

ext (B(Z^{∗})) = {(0, ..., ±δ_{t}, 0, ..., 0) : t ∈ [0, 1]}.

To show our claim we prove that X is a weakly separating subspace of Z. Recall that a linear subspace V of a Banach space W is called weakly separating if any point from ext (B(V^{∗})) has only one Hahn-Banach extension in B(W^{∗}). So assume x^{∗} ∈ ext B(X^{∗}). Set

K = {f ∈ B(Z^{∗}) : f|_{X} = x^{∗}}.

We show that K consists of exaclty one element from ext (B(Z^{∗})). It is easy to see that K is a convex, weak* closed subset of B(Z^{∗}). By the Banach-Alaoglu theorem K is weakly^{∗} compact. By the Krein-Milman theorem ext (K) ≠ ∅. First we show that ext (K) ⊂ ext (B(Z^{∗})). Take any g ∈ ext (K) and assume g = (g_{1} + g_{2})/2, where g_{1}, g_{2} ∈ B(Z^{∗}). Then

f = g_{X} = \frac{(g_{1})|_{X} + (g_{2})|_{X})}{2}.

Since f ∈ ext (B(X^{∗})), ((g_{1})|_{X} = f and (g_{2})|_{X} = f. Hence g_{1}, g_{2} ∈ K. Since g ∈ ext (K), g_{1} = g_{2}, as required. Now assume on the contrary that K consinsts of more than one element. Then we can find at least two different points from ext(B(Z^{∗}))

z_{1} = ±(0, , , .(δ_{s})_{i}, 0, ...0)
z_{2} = ±(0, , , .(δ_{t})_{j}, 0, ...0)

belonging to ext (K). Hence (z_{1})|_{X} = (z_{2})_{X}. But if i < j taking f_{i}(t) = t^{i} we get (via the isometric embedding) z_{1}(f_{i}) = 1 and z_{2}(f_{i}) = 0. Hence i = j. If s ≠ t taking f_{i}(t) = t^{i+1}, we also get z_{1}(f_{i}) ≠ z_{2}(f_{i}). Finally if

z_{1} = (0, , , .(δ_{t})_{i}, 0, ...0)
z_{2} = −(0, , , .(δ_{t})_{i}, 0, ...0)

then also z_{1}(f_{i}) ≠ z_{2}(f_{i}), where f_{i}(t) = t^{i}. Consequently, z_{1} = z_{2}; a contradiction. Hence K consists of exactly one element z. By the previous reasoning z ∈ ext(B(Z^{∗}). Consequently, (via isometric embedding) x^{∗} = ±δ_{t}^{i}, for some t ∈ [0, 1] and i = 0, ..., L, as required.

Now we consider the case of X_{k}. Note that X_{k} can be isometrically embeded into

C = ℝ^{(k+1)L} × C[0, 1]

equipped with a norm

∥(r_{1}, ..., r_{(k+1)L}, f)∥ = max{|r_{i}|, i = 1, ..., (k + 1)L, ∥f∥_{∞}}.

The embedding is given by

T (f) = (f(t_{j})^{(i)}, i = 0, ..., L − 1, j = 0, ..., k, f^{(L)}).

Reasoning in the same way as in the case of X we get our result.

THEOREM 4.2 Let X_{k} = (C^{L}[0, 1], ∥ · ∥_{k,L}), L ≥ 2. Set

g = \frac{∑_{i=0}^{L−1} δ_{0}^{i}}{L} ∈ (X_{k})^{∗}

Then ∥g∥ = 1.

Proof. Without loss of generality, we can assume that

0 = t_{o} < t_{1} < ... < t_{k} = 1.

It is clear that ∥g∥ ≤ 1. Assume that ∥g∥ < 1. Then ∥bg∥ = 1 for some b > 1, since g ≠ 0. By Theorem 4.1 and the Krein -Milman Theorem

bg = ∑_{i=0}^{L−1} ∑_{j=0}^{k} a_{ij}δ_{t_{j}}^{i} + u.  (12)

Here u is a Radon measure on [0,1] acting as a functional on X_{k} as

û(f) = ∫_{[0,1]} f^{(L)}(t)du(t)

and

∑_{i=0}^{L−1} ∑_{j=0}^{k} |a_{ij}| + ∥u∥ = 1,  (13)

where ∥u∥ denotes the total variation of u. First we show that u = 0. Assume on the contrary that u ≠ 0. Then

u = bg − ∑_{i=0}^{L−1} ∑_{j=0}^{k} a_{ij}δ_{t_{j}}^{i}.

Note that u = ∑_{l=0}^{k−1} u_{l}, where u_{l} is a Radon measure defined by u_{l}(A) = u(A∩(E_{l} = [t_{l}, t_{l+1})) for l = 0, ..., k−2 and u_{k−1}(A) = u(A∩(E_{k−1} = [t_{k−1}, 1]). Let us first assume that for every l = 0, ..., k − 1 u_{l} = c_{l}m_{l}, where m_{l} is the Lebesgue measure on E_{l} and c_{l} ∈ ℝ. By the Fundamental Theorem of Calculus

m_{l}(f) = c_{l} ∫_{E_{l}} f^{(L)}(s)ds = c_{l} (f^{(L−1)}(t_{l+1}) − f^{(L−1)}(t_{l}))

for l = 0, ..., k − 1. By the Hermite interpolation theorem there exists a polynomial p such that

p^{(i)}(0) = 0 for i = 0, ..., L − 1,
p^{(i)}(t_{j}) = sgn(a_{ij}) for i = 0, ..., L − 2, j = 1, ..., k,

and

p^{(L−1)}(t_{j}) = sgn(a_{L−1,j} − c_{j−1} + c_{j})

for j = 1, ..., k. Observe that

0 = (bg)(p) = ∑_{i=0}^{L−2} ∑_{j=1}^{k} |a_{ij}| + ∑_{j=1}^{k} |a_{L−1,j} + c_{j−1} − c_{j}|.

Hence all coefficients in the above sum are equal to 0. But this implies that

bg = ∑_{i=0}^{L−2} a_{i,0}δ_{0}^{i} + (a_{L−1,0} − c_{0})δ_{0}^{(L−1)}.

Since b > 1 and the set {δ_{0}^{i} : i = 0, ..., L−1} is linearly independent this leads to a contradiction with (13). So assume that there exists l ∈ {0, ..., k − 1} such that u_{l} ≠ 0 and u_{l} is not a constant multiple of m_{l}. By (12)

u(f) = ∫_{[0,1]} f^{(L)}(t)du(t) = 0  (14)

for any f ∈ X_{k} satisfying

(f)^{(i)}(t_{j}) = 0 for i = 0, ..., L − 1, j = 0, ..., k.  (15)

Fix any f ∈ X_{k} satisfying (14). Suppose there exists D_{1} ⊂ E_{l} and D_{2} ⊂ E_{l} such that

u(D_{1}) · u(D_{2}) < 0.  (16)

Assume u(D_{1}) > 0. Modifying D_{1} and D_{2}, if necessary, we can assume that D_{1} ∩ D_{2} = ∅. By the properties of Radon measures there exists two disjoint subintervals (let us also denote them by D_{1}, D_{2}) of E_{l} of the same lengths c > 0 satisfying (16). Set

h_{L}(t) = χ_{D_{1}} − χ_{D_{2}}.  (17)

We now modify h_{L} to a continuous function h_{L}^{l} on [0,1] with support containing in D_{1} ∪ D_{2}. To do this fix l ∈ ℕ such that c − 2/l > 0. S Assume that D_{1} = (s_{1}, s_{2}), D_{2} = (w_{1}, w_{2}) and s_{2} < w_{1}. Set h_{L}^{l}(t) = h_{L}(t) if t ∉ D_{1} ∪ D_{2}, h_{L}^{l}(t) = 1 if t ∈ (s_{1} + 1/l, s_{2} − 1/l), h_{L}^{l}(t) = −1 if t ∈ (w_{1} + 1/l, w_{2} − 1/l) and define it in the linear way for other t. Note that, for Lebesgue measure m,

∫_{[0,1]} h_{L}^{l}(t)dm(t) = 0,  (18)

and l can be increased so large that

∫_{[0,1]} h_{L}^{l}(t)du(t) > 0.  (19)

Set H_{L}(t) = h_{L}^{l}(t), where l is so chosen that (16) and (19) are satisfied. Set

H_{L−1}(t) = ∫_{[0,t]} H_{L}(s)dm(s)

and

H_{j−1}(t) = ∫_{[0,t]} H_{j}(s)dm(s)

for j = L − 1, ..., 1. Define

G = f + H_{0}.

By the construction of H_{L}

∫_{E_{l}} H_{j}(s)dm(s) = 0,

for j = 1, ..., L. Consequently,

(G)^{(i)}(t_{j}) = 0 for i = 0, ..., L − 1, j = 0, ..., k

and thus G satisfies (15). As such G should satisfy (14) - but this is in contradiction with (19). To end the proof that u = 0, assume that u_{l} does not satisfy (16). Hence u_{l} or −u_{l} is a nonzero measure on E_{l} which is not a constant multiple of the Lebesgue measure of E_{l}. Without loss of generality we can assume that u_{l} is a measure. By the above conditon u_{l} is not a Haar measure of E_{l}. Hence exists an open interval D_{1} ⊂ E_{l} and t > 0 such that

u_{l}(D_{1}) ≠ u_{l}(D_{1} + t).

Hence there exists two open disjoint intervals D_{1}, D_{2} ⊂ E_{l} of the same lengths c > 0 satisfying

u_{l}(D_{1}) ≠ u_{l}(D_{2}).  (20)

Since obviously m(D_{1}) = m(D_{2}) = c, reasoning as above and replacing (16) by (20) we get a contradiction with (19). This finally shows that u = 0. Hence (12) reduces to

bg = ∑_{i=0}^{L−1} ∑_{j=0}^{k} a_{ij}δ_{t_{j}}^{i}.  (21)

By the Hermite interpolation theorem there exists a polynomial p such that

p^{(i)}(0) = 0 for i = 0, ..., L − 1,
p^{(i)}(t_{j}) = sgn(a_{ij}) for i = 0, ..., L − 1, j = 1, ..., k.

Observe that

0 = (bg)(p) = ∑_{i=0}^{L−1} ∑_{j=1}^{k} |a_{ij}|.

Hence

a_{ij} = 0

for j = 1, ..., k and i = 0, ..., L − 1. By (21),

∑_{i=0}^{L−1}(b/n − a_{i,0})δ_{0}^{i} = 0.

By the linear independence of functionals δ_{0}^{i}, i = 0, ..., L−1, we get a_{i,0} = b/n for i = 0, .., L − 1. Hence

∑_{i=0}^{L−1} a_{i,0} = b > 1;

a contradiction with (13). The proof is complete.

COROLLARY 4.1 Let X = (C^{L}[0, 1], ∥ · ∥_{L}). Set

g = \frac{∑_{i=0}^{L} δ_{0}^{i}}{(L + 1)} ∈ X^{∗}.

Then ∥g∥ = 1.

Proof. Applying Theorem 4.2 to Y_{k} = (C^{L+1}[0, 1], ∥ · ∥_{k,L+1}) and g we get ∥g∥_{Y_{k}^{∗}} = 1, where ∥ · ∥_{Y_{k}^{∗}} denotes the norm on (Y_{k})^{∗}. By Lemma 4.2, Lemma 4.3 and Theorem 4.2,

∥g∥_{Y^{∗}} = 1,

where ∥·∥_{Y^{∗}} denotes the norm on Y^{∗} where Y = (C^{L+1}[0, 1], ∥·∥_{L+1}). Hence, using the weak* density of Y in Y^{∗∗}, there exists a sequence {f_{k}} ⊂ Y with

∥f_{k}∥_{L+1} = max_{i=0,...,L+1} ∥f_{k}^{(i)}∥_{∞} = 1,

such that g(f_{k}) → 1. Note that f_{k} ∈ X and

∥f_{k}∥_{L} = max_{i=0,...,L} ∥f^{(i)}∥_{∞} ≤ ∥f_{k}∥_{L+1} = 1.

Hence ∥g∥ = 1, as required.

LEMMA 4.4 Let X = (C^{L}[0, 1], ∥ · ∥_{L}). Fix integer k ∈ [0, L] and set

g = ∑_{i=0}^{k} δ_{0}^{(i)} + δ_{1}^{(k)} ∈ X^{∗}.

Set

W = {F ∈ X^{∗∗} | F (g) = k + 2 and ∥F ∥ = 1}.

Then W ≠ ∅.

Proof. By Corollary 4.1, there exists F ∈ (X^{L})^{∗∗}, ∥F ∥ = 1 such that F (δ_{0}^{(i)}) = 1 for i = 0, ..., k. By the weak* density of X in X^{∗∗}, there exists a sequence {f_{j}} ⊂ X, ∥f_{j}∥_{L} ≤ 1 such that f_{j}^{(i)}(0) → 1 for i = 0, ..., k. Let us define a sequence of continuous functions {g_{j}^{(k)}} by g_{j}^{(k)}(t) = f_{j}^{(k)}(t) if t ∈ [0, 1 − 1/j] g_{j}^{(k)}(1) = 1, and in the linear way on the interval [1 − 1/j, 1]. Note that for any j ∈ ℕ,

g_{j}^{(k)} = f_{j}^{(k)} + h_{j}^{(k)}

where h_{j}^{(k)}(t) = 0 for t ∈ [0, 1 − 1/j] and ∥h_{j}^{(k)}∥_{∞} ≤ 2. Let

h_{j}^{(k−1)}(t) = ∫_{0}^{t} h_{j}^{(k)}(s)ds

and

h_{j}^{(i−1)}(t) = ∫_{0}^{t} h_{j}^{(i)}(s)ds

for i = k − 2, ..., 1. Set

g_{j} = f_{j} + h_{j}^{(0)}.

By the Mean Value Theorem

∥g_{j}^{(i)}∥_{∞} ≤ 1 + 2/j

for i = 0, ..., k − 1 and

∥g_{j}^{(k)}∥_{∞} ≤ 1

by definition. Hence ∥g_{j}∥ ≤ 1 + 2/j for j ∈ ℕ. By the Banach Alaoglu Theorem {g_{j}} has a cluster point F_{1} ∈ X^{∗∗}, ∥F_{1}∥ = 1. It is clear by definition of g_{j} that F_{1} ∈ W.

THEOREM 4.3 Let X = C^{L}[0, 1], ∥ · ∥_{L}) and fix integer k ∈ [0, L − 1]. Set

W = {F ∈ X^{∗∗} : F (δ_{0}^{(k)}) = 1, ∥F ∥ = 1}.

Assume u is a Borel measure on [0,1]. Define u^{k} ∈ X^{∗} by

u^{k}(f) = ∫_{[0,1]} f^{(k)}(t)du(t).  (22)

Then for any F ∈ W and for any Borel measure u on [0,1],

F (u^{k}) ≥ 0.

Proof. Fix F ∈ W and a Borel measure u. By the Goldstine Theorem there exists a sequence {f_{l}} ⊂ X, ∥f_{l}∥_{L} ≤ 1, such that

f_{l}(δ_{o}^{(k)}) = f_{l}^{(k)}(0) → F (δ_{o}^{(k)}) = 1  (23)

and

f_{l}(u^{(k)}) = ∫_{[0,1]} f_{l}^{(k)}(t)du(t) → F (u^{(k)}).  (24)

In particular, ∥f_{l}^{(k+1)}∥_{∞} ≤ 1. Hence by the Mean Value Theorem for any s, t ∈ [0, 1] l ∈ ℕ,

|f_{l}^{(k)}(s) − f_{l}^{(k)}(t)| ≤ |t − s|.

Also ∥f_{j}^{(k)}∥_{∞} ≤ 1. By the Ascoli-Arzela Theorem, passing to a subsequence, if necessary, we can assume that there exists f ∈ C[0, 1] such that

∥f_{l}^{(k)} − f∥_{∞} → 0.  (25)

Now we show that f(t) ≥ 0 for any t ∈ [0, 1]. By (23), f(0) = 1. Assume on the contrary, that there exists t_{0} ∈ (0, 1] such that f(t_{o}) < 0. By (23) there exists δ > 0 such that

|f_{l}^{(k)}(0) − f_{l}^{(k)}(t_{0})| > 1 + δ.

for l ≥ l_{o}. By The Mean Value Theorem,

1 + δ < |f_{l}^{(k)}(0) − f_{l}^{(k)}(t_{o})| ≤ ∥f_{l}^{(k+1)}∥_{∞}t_{0} ≤ 1;

a contradiction. Hence f(t) ≥ 0 for any t ≥ 0. By (25)

f_{l}(u^{(k)}) = ∫_{[0,1]} f_{l}^{(k)}(t)du(t) → ∫_{[0,1]} f(t)du(t) ≥ 0,

since f is nonegative and u is a measure. By (24),

F (u^{(k)}) = ∫_{[0,1]} f(t)du(t) ≥ 0,

which completes the proof.

THEOREM 4.4 Let X = (C^{L}[0, 1], ∥ · ∥_{L}) and fix integer k ∈ [0, L]. Let g = ∑_{i=0}^{k} δ_{o}^{i} and

W_{1} = {F ∈ X^{∗∗} : F (g) = k + 1, ∥F ∥ = 1}.

Assume u is a Borel measure on [0,1]. Define u^{k} ∈ X^{∗} by

u^{k}(f) = ∫_{[0,1]} f^{(k)}(t)du(t).

Then there exists F ∈ W_{1} such that

F (u^{k}) ≥ 0

for any Borel measure u on [0,1]. Moreover F (m_{t}^{k}) = 0, for any t ∈ [0, 1], where

m_{t}^{k}(f) = ∫_{[0,t]} f^{(k)}(t)dm(t)

and m is the Lebesgue measure on [0, 1].

Proof. Let Z = (C^{L+1}[0, 1], ∥ · ∥). Set g = ∑_{i=0}^{k+1} δ_{0}^{i} and

W_{2} = {F ∈ Z^{∗∗} : F (g) = k + 2, ∥F ∥ = 1}.

By Corollary 4.1, W_{2} ≠ ∅. Take any G ∈ W_{2}. Since W_{2} ⊂ W, by Theorem 4.7, for any Borel measure u, G(u^{k}) ≥ 0, where u^{k} is a functional defined on Z by (22). By the Goldstine Theorem applied to B(Z^{∗∗}), there exists a net {f_{β}} ⊂ Z, ∥f_{β}∥ ≤ 1 for any β such that f_{β} → G weak-∗ in Z^{∗∗}. Since Z ⊂ X (as sets) {f_{β}} ⊂ X. Moreover, each f_{β} has norm one in X, since its norm in Z is at most one. By the Banach-Alaoglu Theorem applied to B(X^{∗∗}) the set {f_{β}} has an accumulation point F ∈ B(X^{∗∗}). Since G ∈ W_{2},

1 = G(δ_{0}^{i}) = lim_{β} f_{β}^{(i)}

for i = 0, ..., k + 1. Hence obviously F (δ_{o}^{(i)}) = 1, for i = 0, ..., k. Since ∥F ∥ ≥ 1, F ∈ W_{1}. Moreover by Theorem 4.7, for any Borel measure u on [0, 1]

F (u^{k}) = lim_{β}(∫_{[0,1]} f_{β}^{(k)}(t)du(t)) = G(u^{k}) ≥ 0,

which proves our claim. In particular, for any t ∈ [0, 1] F (m_{t}^{k}) ≥ 0. By the Fundamental Theorem of Calculus, for any f ∈ X,

m_{t}^{k}(f) = ∫_{[0,t]} f^{(k)}(t)dm(t) = f^{(k−1)}(t) − f^{(k−1)}(0).

Since F ∈ W_{1},

0 ≤ F (m_{t}^{k}) = F (δ_{t}^{(k−1)} − δ_{0}^{(k−1)}) = F (δ_{t}^{(k−1)}) − 1.

Hence F (δ_{t}^{(k−1)}) ≥ 1. Since ∥F ∥ = 1,

F (δ_{t}^{(k−1)}) = 1.

Consequently, F (m_{t}^{k}) = 0, which completes the proof.

5 Proofs of Minimality

Proof of Theorem 2.3

The fact that P_{0,n} = ∑_{i=0}^{n} u_{i}⊗v_{i} is a projection follows from the definition of each u_{i} and v_{i}; it is easy to check that ⟨v_{i}, u_{j}⟩ = 0 unless i = j, in which case the result is 1.

The verification that P_{0,n} preserves the multi-convex shape described by σ (i.e., P_{0,n}Sσ ⊂ Sσ) consists of a direct calculation; let f ∈ Sσ, t ∈ [0, 1], integer j ≤ M and consider

⟨P_{0,n}f, δ_{t}^{j}⟩ = ⟨∑_{k=0}^{n−1} f^{(k)}(0)v_{k} + f^{(L−1)}(1)v_{n}, δ_{t}^{j}⟩

= ∑_{k=j}^{n−1} f^{(k)}(0)v_{k}^{(j)}(t) + f^{(L−1)}(1)v_{n}^{(j)}(t)
≥ 0

since every term in the sum is nonnegtive. Thus P_{0,n} ∈ P_{Sσ}(X, Π_{n})

To verify (3) of Theorem 2.3, note that, from the form of P_{0,n} in (2), we have ∥P_{0,n}∥ ≤ ∑_{k=0}^{n−1} \frac{1}{k!}. If L = n − 1 then by Lemma 4.4 then ∥P_{0,n}∥ = ∑_{k=0}^{n−1} \frac{1}{k!}. If L ≥ n Theorem 4.4 guarantees the existence of F ∈ B(X^{∗∗}) such that F (δ_{0}^{i}) = 1 for i = 0, …, n − 1 and F (m) = 0, where m denotes the Lebesgue measure. But we see from the proof of Theorem 4.4 that F vanishing on m implies F (δ_{1}^{n−1}) = 1, which implies ∥P_{0,n}∥ ≥ ∑_{k=0}^{n−1} \frac{1}{k!} and thus (3) follows.

To show P_{0,n} has minimal norm in P_{Sσ}(X, Π_{n}), we consider two cases: M = n − 1 and M = n. We handle the M = n − 1 case first, using the following uniqueness argument.

We begin with a corollary given in [9]; it describes how the functionals that define a projection must be chosen in order for the projection to preserve shape.

COROLLARY 5.1 (see [9]) Suppose P ∈ P_{S}. If S_{|_{V}}^{∗} is k-dimensional then there exists a basis v = (v_{1}, …, v_{n})^{T} for V such that whenever P = u⊗v ∈ P_{S}, where u = (u_{1}, …, u_{n}) ∈ (X^{∗})^{n}, we have, for i = n−k+1, …, n, u_{i} ∈ S^{∗}. Moreover, each such u_{i} restricts to a distinct extreme ray of S_{|_{V}}^{∗}.

To utilize this result, we note that the proof of Theorem 2.1 demonstrates that the simplicial cone S_{|_{V}}^{∗} has easily-described extreme rays; they are generated (via nonnegative scalar multiplication) by

{(δ_{0})_{|_{V}} , (δ_{0}^{1})_{|_{V}} , …, (δ_{0}^{n−1})_{|_{V}} , (δ_{1}^{n−1})_{|_{V}} }.  (26)

Thus, by Corollary 5.1, every projection P = ∑_{i=0}^{n} u_{i} ⊗ v̂_{i} ∈ P_{Sσ}(X, Π_{n}) must be such that u_{i} ∈ S^{∗} and (u_{i})_{|_{V}} = (δ_{0}^{j})_{|_{V}} for some j or (u_{i})_{|_{V}} = (δ_{1}^{n−1})_{|_{V}}. However, it is easy to check that for every j there exists a unique element of S^{∗} whose restriction to V is (δ_{0}^{j})_{|_{V}} - namely δ_{0}^{j}. Similarly, δ_{1}^{(n−1)} is the unique element of S^{∗} with restriction to V given by (δ_{1}^{n−1})_{|_{V}}. Consequently, the form of P_{0,n} given in (2) implies that P_{0,n} is the unique element of P_{Sσ}(X, Π_{n}) and therefore of minimal norm.

We consider now the case M = n. Unlike the previous case, the projection P_{0,n} is not unique in P_{Sσ}(X, Π_{n}); indeed consider P_{0,n} written in the following way:

P_{0,n} = δ_{0} ⊗ 1 + δ_{0}^{1} ⊗ \frac{x}{1!} + ⋯ + δ_{0}^{n−1} ⊗ \frac{x^{n−1}}{(n − 1)!} + (δ_{1}^{n−1} − δ_{0}^{n−1}) ⊗ \frac{x^{n}}{n!} .

Replacing the functional (δ_{1}^{n−1} − δ_{0}^{n−1}) with (a positive scalar multiple of) any nonzero element from the weak* closure of cone{δ_{t}^{n}}_{t∈[0,1]} will result in a element of P_{Sσ}(X, Π_{n}). In fact, by Corollary 5.1 every element of P_{Sσ}(X, Π_{n}) can be constructed in this way. And it is because of this that we are unable to appeal to standard theory of minimal projections, (described for example in [9]) which relies on best approximations from a linear space (and not from a cone). Thus we proceed in the following way: we show that replacing (δ_{1}^{n−1} − δ_{0}^{n−1}) in P_{0,n} with any other allowable functional from S^{∗} results in an element of P_{Sσ}(X, Π_{n}) with norm at least as large as ∥P_{0,n}∥. The following summarizes the form of an element from P_{Sσ}(X, Π_{n}) in the M = n case.

LEMMA 5.1 Let Q ∈ P_{Sσ}(X, Π_{n}). Then there exists u ∈ X^{∗} such that

Q = δ_{0} ⊗ 1 + δ_{0}^{1} ⊗ \frac{x}{1!} + ⋯ + δ_{0}^{n−1} ⊗ \frac{x^{n−1}}{(n − 1)!} + u ⊗ \frac{x^{n}}{n!} .

Moreover, there exists a probabilistic Borel measure μ such that for every f ∈ X we have

u(f) = ∫_{0}^{1} f^{(n)}(t) dμ(t)  (27)

Proof. Fix the basis {1, x, x/2!, …, x^{n}/n!}; then Corollary 5.1 guarantees this representation of Q and implies that u ∈ S^{∗} ⊂ X^{∗}. Furthermore E(S^{∗}), the set of extreme rays of S^{∗}, is (strictly) contained in the set of rays

{[δ_{t}^{j}]^{+} | t ∈ [0, 1], j = 0, …, n}.

Let

C = \overline{co} (E(S^{∗}) ∩ S(X^{∗}))

where the closure is taken with respect to the weak* topology. Note that ext(C) ⊂ {δ_{t}^{j} | t ∈ [0, 1], j = 0, …, n}. Then by Proposition 2 from [31], for every non-zero ϕ ∈ S^{∗}, there exists a positive scalar c ∈ R and a probabilistic Borel measure μ supported on ext(C) such that μ represents cϕ (in the sense of Choquet); i.e.,

ϕ(f) = ∫_{ext(C)} f(δ) dμ(δ)

for every f ∈ X. Consider now our u above; the fact that ⟨x^{i}, u⟩ = 0 for every i = 0, …, n − 1 implies that a representing measure μ for u cannot have any positive support on the set of extreme points of C of the form {δ_{t}^{j} | t ∈ [0, 1], j = 0, …, n − 1}. And thus the representation in (27) is the only choice.

Now from Theorem 4.4, we have the existence of an F ∈ B(X^{∗∗}) such that F (δ_{0}^{i}) = 1 for i = 0, …, n − 1 and F (u) ≥ 0. Therefore ∥Q∥ ≥ ∑_{k=0}^{n−1} \frac{1}{k!} which implies P_{0,n} is of minimal norm in the M = n case. This completes the proof of Theorem 2.3.

REMARK 5.1 Let X = (C^{L}[0, 1], ∥ · ∥_{2,L}), where

∥f∥_{2,L} = max { max_{j=0,...,L−1}{|f^{(j)}(0)|, |f^{(j)}(1)|}, ∥f^{(L)}∥_{∞}} .

Note that ∥ · ∥_{L} and ∥ · ∥_{2,L} are equivalent since

(\frac{2}{3})^{L} ∥ · ∥_{L} ≤ ∥ · ∥_{2,L} ≤ ∥ · ∥_{L}.

And so P_{0,n} ∈ P_{Sσ}(X, Π_{n}). Moreover, an arguement identical to the above shows P_{0,n} is minimal in P_{Sσ}(X, Π_{n}) (for either M = n − 1 or M = n) and and (by Lemma 4.4 and Theorem 4.4) ∥P_{0,n}∥ = ∑_{k=0}^{n−1} \frac{1}{k!}.

Proof of Theorem 2.4

To simplify notation, let P denote the operator P_{m+1,n+1} defined in (4). Also, for positive integer k, we will denote the Banach space C^{k}[0, 1] as simply C^{k}.

We begin by verifying that P is a projection onto [1, W_{0}, W_{1}, …, W_{n}]. Let k(x) denote a (non-zero) constant function and note that

(P k)(x) = (k(0) + k(1))/2 = k(x)

since k′ ≡ 0; thus (P 1)(x) = 1. Moreover, using the fact that P_{m,n} is a projection onto Y, we have for each integer j ∈ [0, n]

(P W_{j})(x) = (W_{j}(0) + W_{j}(1))/2 + ∫_{0}^{x} w_{j}(t) dt − W_{j}(1)/2
= W_{j}(x)

since W_{j}(0) = 0. Thus P is a projection onto [1, W_{0}, W_{1}, …, W_{n}]. Note the following (derivative) relationships between projections P and P_{m,n}: for any f ∈ C^{L+1}, integer j ∈ [1, L + 1] and x ∈ [0, 1] we have

(P f)^{j}(x) = (P_{m,n}f′)^{(j−1)}(x).  (28)

To see that P preserves shape, let f ∈ Sσ̂ ⊂ C^{L+1} and fix integer j ∈ [m + 1, M + 1]. Then (28) implies (P f)^{j}(x) ≥ 0 since f′ ∈ Sσ ⊂ C^{L} and P_{m,n} ∈ P_{Sσ}(C^{L}, W). Hence

P ∈ P_{Sσ̂}(C^{L+1}, [1, W_{0}, W_{1}, …, W_{n}].

We now verify (5). Let A = {f′ | f ∈ B(C^{L+1})}. We claim

A = B(C^{L});  (29)

clearly A ⊂ B(C^{L}). Let g ∈ B(C^{L}) and define f(x) = ∫_{0}^{x} g(t) dt. Note that f ∈ C^{L+1}, since for each integer j ∈ [1, L + 1] we have

f^{(j)}(x) = g^{(j−1)}(x).  (30)

Furthermore, from (30) if follows that ∥f^{(j)}∥_{∞} ≤ 1 for each integer j ∈ [1, L + 1]. And finally using the definition of f we have ∥f∥_{∞} ≤ ∥g∥_{∞} ≤ 1 and thus f ∈ B(C^{L+1}). This establishes our claim in (29) since g = f′. We are now ready to compare ∥P ∥ and ∥P_{m,n}∥. Recall that

∥P ∥ = sup_{f∈B(C^{L+1})} ∥P f∥ = sup_{f∈B(C^{L+1})} max_{j=0,...,L+1} ∥(P f)^{(j)}∥_{∞}

= max_{j=0,...,L+1} sup_{f∈B(C^{L+1})} ∥(P f)^{(j)}∥_{∞}.

Consider first the case in which j ≥ 1; for each such j we have

sup_{f∈B(C^{L+1})} ∥(P f)^{(j)}∥_{∞} = sup_{f∈B(C^{L+1})} sup_{x∈[0,1]} |(P f)^{(j)}(x)|
= sup_{f∈B(C^{L+1})} sup_{x∈[0,1]} |(P_{m,n}f′)^{(j−1)}(x)| by (30)

= sup_{f∈B(C^{L+1})} ∥(P_{m,n}f′)^{(j−1)}∥_{∞}

= sup_{f∈A} ∥(P_{m,n}f)^{(j−1)}∥_{∞}

= sup_{f∈B(C^{L})} ∥(P_{m,n}f)^{(j−1)}∥_{∞}  by (29).

Consequently we have

max_{j=1,...,M+1} sup_{f∈B(C^{L+1})} ∥(P f)^{(j)}∥_{∞} = max_{k=0,...,M} sup_{f∈B(C^{L})} ∥(P_{m,n}f)^{(k)}∥_{∞}
= ∥P_{m,n}∥.

(31)

To finish the comparison, we must check the j = 0 case. Recalling the form of P (or equivalenly P_{m+1,n+1}) given in (4), we find

sup_{f∈B(C^{L+1})} ∥(P f)∥_{∞} ≤ 1+ sup_{f∈B(C^{L+1})} ∥∫_{0}^{x}(P_{m,n}f′)(t) dt − \frac{1}{2} ∫_{0}^{1}(P_{m,n}f′)(t) dt∥ .

(32)

However, the right-hand side of (32) becomes

1+ sup_{f∈B(C^{L+1})} ∥\frac{1}{2} ∫_{0}^{x}(P_{m,n}f′)(t) dt − \frac{1}{2} (∫_{0}^{1}(P_{m,n}f′)(t) dt − ∫_{0}^{x}(P_{m,n}f′)(t) dt)∥_{∞}

= 1 + \frac{1}{2} sup_{f∈B(C^{L+1})} ∥∫_{0}^{x}(P_{m,n}f′)(t) dt − ∫_{x}^{1}(P_{m,n}f′)(t) dt∥_{∞}

= 1 + \frac{1}{2} sup_{f∈B(C^{L+1})} sup_{x∈[0,1]} |∫_{0}^{x}(P_{m,n}f′)(t) dt − ∫_{x}^{1}(P_{m,n}f′)(t) dt|

≤ 1 + \frac{1}{2} sup_{f∈B(C^{L+1})} ∫_{0}^{1} |P_{m,n}f′)(t)| dt

= 1 + \frac{1}{2} sup_{f∈B(C^{L})} ∫_{0}^{1} |P_{m,n}f)(t)| dt  by (29)

≤ 1 + \frac{1}{2}∥P_{m,n}∥.

Thus

sup_{f∈B(C^{L+1})} ∥(P f)∥_{∞} ≤ 1 + \frac{1}{2}∥P_{m,n}∥ ≤ ∥P_{m,n}∥

since, by assumption, ∥P_{m,n}∥ ≥ 2. This result, in combination with (31), establishes (5) and completes the proof of Theorem 2.4.

REMARK 5.2 This proof demonstrates that when C^{L}[0, 1] is normed by ∥ · ∥_{L}, the construction given in (4) is (operator) norm-preserving. It is a staightforward verification that this proof can be repeated when ∥ · ∥_{L} is replaced by ∥ · ∥_{2,L} and thus we have norm preservation in this case as well.

Proof of Theorem 2.5 We begin by verifying this theorem in the (L + k)-norm case. For notation sake, let X_{L+k} = (C^{L+k}[0, 1], ∥·∥_{L+k}). From our assumption on the construction of P_{k,n+k} and Theorem 2.4 we have that P_{k,n+k} ∈ P_{Sσ}(X_{L+k}, Π_{n+k}) and

∥P_{k,n+k}∥ = ∥P_{0,n}∥.  (33)

In fact, we can say more; a straightforward generalization of (29) gives

{f^{(k)} | f ∈ B(X_{L+k})} = B(X_{L})

and so

∥P_{0,n}∥ = sup_{f∈B(X_{L})} ∥P_{0,n}f∥_{∞}

= sup_{f∈B(X_{L+k})} ∥P_{0,n}(f^{(k)})∥_{∞}

= sup_{f∈B(X_{L+k})} ∥(P_{k,n+k}f)^{(k)})∥_{∞}.

This result, together with (33), yields

∥P_{k,n+k}∥ = sup_{f∈B(X_{L+k})} ∥(P_{k,n+k}f)^{(k)})∥_{∞}.  (34)

Let Q ∈ P_{Sσ}(X_{L+k}, Π_{n+k}). Recall M = max_{σ_{i}=1} i. Then by Corollary 5.1 there exists {v̂_{0}, v̂_{1}, …, v̂_{n+k}} for Π_{n+k} such that Q may be represented as

Q = ∑_{i=0}^{k−1} ϕ_{i} ⊗ v̂_{i} + ∑_{i=k}^{n+k−1} δ_{0}^{i} ⊗ v̂_{i} + Δ ⊗ v̂_{n+k}  (35)

where, in the case M = n + k − 1, Δ = δ_{1}^{n+k−1} and otherwise (M = n + k) Δ is any non-zero element of the weak* closure of the cone generated by the set {δ_{t}^{n+k}}_{t∈[0,1]}. We claim that for i = 0, …, k −1, the degree of v̂_{i} is strictly less than k. To the contrary, suppose for some i we have k ≤ deg(v̂_{i}) ≤ n+k. If deg(v̂_{i}) < n + k then ⟨v̂_{i}, δ_{0}^{deg(v̂_{i})}⟩ ≠ 0 which is a contradiction; a similar conclusion is obtained if deg(v̂_{i}) = n + k since ⟨v̂_{i}, Δ⟩ ≠ 0. Thus deg(v̂_{i}) < k for i = 0, …, k − 1. For i = k, …, n + k, write v̂_{i} = a_{i}v_{i} + p_{i} where a_{i} ∈ ℝ, v_{i} is as in Theorem 2.3, and p_{i} ∈ Π_{n+k}. Using an orthogonality argument identical to that above (e.g., ⟨v̂_{i}, δ_{0}^{j}⟩ = 0 whenever i ≠ j), we conclude that a_{i} = 1 and deg(p_{i}) < k. Thus

(Qf)^{(k)} = (P_{k,n+k}f)^{(k)}  (36)

and so by (34) we find

∥Q∥ ≥ sup_{f∈B(X_{L+k})} ∥(Qf)^{(k)}∥_{∞} = sup_{f∈B(X_{L+k})} ∥(P_{k,n+k}f)^{(k)}∥_{∞} = ∥P_{k,n+k}∥.

Therefore P_{k,n+k} has minimal norm in P_{Sσ}(X_{L+k}, Π_{n+k}).

Consider now the case X_{2,L+k} = (C^{L+k}[0, 1], ∥ · ∥_{2,L+k}). From our assumption on the construction of P_{k,n+k} and Remarks 5.1 and 5.2 we have that P_{k,n+k} ∈ P_{Sσ}(X_{2,L+k}, Π_{n+k}) and

∥P_{k,n+k}∥_{2,L+k} = ∥P_{0,n}∥_{2,L},  (37)

where ∥Q∥_{2,L+k} denotes the operator norm of Q defined on X_{2,L+k}. From Remark 5.1 we have that ∥P_{0,n}∥_{2,L} has minimal norm (in the context of Theorem 2.3 and Remark 5.1) and therefore, from an arguement identical to that above in the L-norm case, we can conclude that P_{k,n+k} is a minimal norm element from P_{Sσ}(X_{2,L+k}, Π_{n+k}).

We now make the following observation; note that

P_{k,n+k} : X_{L+k} = (C^{L+k}[0, 1], ∥ · ∥_{L+k}) → Y = (Π_{n+k}, ∥ · ∥_{2,L+k})

is an operator which preserves the multi-convex shape Sσ. From the form of P_{k,n+k}, it follows that the operator norm of P_{k,n+k} : X_{L+k} → Y is equal to the operator norm of P_{k,n+k} ∈ P_{Sσ}(X_{2,L+k}, Π_{n+k}). Let ∥P_{k,n+k}∥ denote this common value. Again using the form of P_{k,n+k} we find

∥P_{k,n+k}∥ = sup_{f∈B(X_{2,L+k})} sup_{t∈{0,1}} |(P_{k,n+k}f)^{(k)}(t)|.

We claim that P_{k,n+k} has minimal (operator) norm among all operators between spaces X_{L+k} and Y preserving Sσ. Indeed, let Q be any such operator. From Remark 5.1 we recall that norms ∥ · ∥_{L+k} and ∥ · ∥_{2,L+k} are equivalent and therefore we may consider Q as an element of P_{Sσ}(X_{L+k}, Π_{n+k}) - i.e., a projection from X_{L+k} onto subspace Π_{n+k} such that QSσ ⊂ Sσ. This implies that Q : X_{L+k} → Y has the form described in (35) and therefore we have the relation in (36). The minimal of P_{k,n+k} : X_{L+k} → Y follows since

∥Q∥ = sup_{f∈B(X_{L+k})} ∥Qf∥_{2,L+k}

≥ sup_{f∈B(X_{L+k})} sup_{t∈{0,1}} |(Qf)^{(k)}(t)|

= sup_{f∈B(X_{L+k})} sup_{t∈{0,1}} |(P_{k,n+k}f)^{(k)}(t)|

= sup_{f∈B(X_{2,L+k})} sup_{t∈{0,1}} |(P_{k,n+k}f)^{(k)}(t)|

= ∥P_{k,n+k}∥.

Finally, let X = (C^{L+k}[0, 1], ∥ · ∥) such that

∥ · ∥_{2,L+k} ≤ ∥ · ∥ ≤ ∥ · ∥_{L+k}.

From the definitions of the L- and (2, L)-norms, we have

∥P_{k,n+k}∥_{2,L+k} ≤ ∥P_{k,n+k}∥_{X} ≤ ∥P_{k,n+k}∥_{L+k}

where ∥P_{k,n+k}∥_{X} is the operator norm of P_{k,n+k} defined on X. But from (33),

(37) and Remark 5.1 we find

∥P_{k,n+k}∥_{X} = ∥P_{k,n+k}∥_{2,L+k} = ∥P_{k,n+k}∥_{L+k}.

As done above, let ∥P_{k,n+k}∥ denote this common value. To show P_{k,n+k} has minimal norm in P_{Sσ}(X, Π_{n+k}) let Q ∈ P_{Sσ}(X, Π_{n+k}). Note that we have Q : X_{L+k} → Y such that QSσ ⊂ Sσ and so

∥Q∥_{X} = sup_{f∈B(X)} ∥Qf∥ ≥ sup_{f∈B(X_{L+k})} ∥Qf∥_{2,L+k} ≥ ∥P_{k,n+k}∥

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