## Abstract

This paper studies the problem of approximating a function *f* in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function *f*. These additional assumptions take the form of assuming that *f* is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in *K* which share these same measurements, the best approximation is the center of the smallest ball *B*, called the *Chebyshev ball*, which contains the set \(\bar{K}\) of all *f* in *K* with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for *K* the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and *V* a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called *approximation sets*, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.

### Keywords

Optimal recovery Reduced modeling Data assimilation Sampling### Mathematics Subject Classification

46B99 41A65 94A20## 1 Introduction

One of the most ubiquitous problems in science is to approximate an unknown function *f* from given data observations of *f*. Problems of this type go under the terminology of *optimal recovery*, *data assimilation* or the more colloquial terminology of *data fitting*. To prove quantitative results about the accuracy of such recovery requires not only the data observations, but also additional information about the sought after function *f*. Such additional information takes the form of assuming that *f* is in a prescribed model class \(K\subset \mathcal{X}\).

The classical setting for such problems (see, for example, [9, 25, 26, 35]) is that one has a bounded set *K* in a Banach space \(\mathcal{X}\) and a finite collection of linear functionals \(l_j\), \(j=1,\ldots ,m\), from \(\mathcal{X}^*\). Given a function which is known to be in *K* and to have known measurements \(l_j(f)=w_j\), \(j=1,\ldots ,m\), the *optimal recovery * problem is to construct the best approximation to *f* from this information. Since there are generally infinitely many functions in *K* which share these same measurements, the best approximation is the center of the smallest ball *B*, called the *Chebyshev ball*, which contains the set \(\bar{K}\) of all *f* in *K* with these measurements. The best error of approximation is then the radius of this Chebyshev ball.

Most results in optimal recovery study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for *K* the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in certain other model classes *K*, called *approximation sets*, that arise in various applications. As a motivating example, consider the analysis of complex physical systems from data observations. In such settings the sought after functions satisfy a (system of) parametric partial differential equation(s) with unknown parameters and hence lie in the solution manifold \(\mathcal{M}\) of the parametric model. There may also be uncertainty in the parametric model. Problems of this type fall into the general paradigm of uncertainty quantification. The solution manifold of a parametric partial differential equation (pde) is a complex object and information about the manifold is usually only known through approximation results on how well the elements in the manifold can be approximated by certain low dimensional linear spaces or by sparse Taylor (or other) polynomial expansions (see [11]). For this reason, the manifold \(\mathcal{M}\) is often replaced, as a model class, by the set \(\mathcal{K}(\varepsilon ,V)\) consisting of all elements in \(\mathcal{X}\) that can be approximated by the linear space \(V=V_n\) of dimension *n* to accuracy \(\varepsilon =\varepsilon _n\). We call these model classes \(\mathcal{K}(\varepsilon ,V)\)*approximation sets* and they are the main focus of this paper. Approximation sets also arise naturally as model classes in other settings such as signal processing where the problem is to construct an approximation to a signal from samples (see e.g. [1, 2, 3, 12, 36] and the papers cited therein as representative), although this terminology is not in common use in that setting.

Optimal recovery in this new setting of approximation sets as the model class was formulated and analyzed in [24] when \(\mathcal{X}\) is a Hilbert space, and further studied in [8]. In particular, it was shown in the latter paper that a certain numerical algorithm proposed in [24], based on least squares approximation, is optimal.

The purpose of the present paper is to provide a general theory for the optimal or near optimal recovery of approximation sets in a general Banach space \(\mathcal{X}\). While, as noted in the abstract, the optimal recovery has a simple theoretical description as the center of the Chebyshev ball and the optimal performance, i.e., the best error, is given by the radius of the Chebyshev ball, this is far from a satisfactory solution to the problem since it is not clear how to find the center and the radius of the Chebyshev ball. This leads to the two fundamental problems studied in the paper. The first centers on building numerically executable algorithms which are optimal or perhaps only near optimal, i.e., they either determine the Chebyshev center or approximate it sufficiently well. The second problem is to give sharp a priori bounds for the best error, i.e. the Chebyshev radius, in terms of easily computed quantities. We show how these two problems are connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Our main results determine a priori bounds for optimal algorithms and give numerical recipes for obtaining optimal or near optimal algorithms.

### 1.1 Notation and problem formulation

*f*from the information that \(f\in S\) and

*f*has the known measurements \(M(f):= M_m(f):=(l_1(f), \ldots ,l_m(f))=(w_1,\ldots ,w_m)\in \mathbb {R}^m\). In general, many functions \(f\in S\) may share the same data. We denote this collection by

*A*for this recovery problem is a mapping which when given the measurement data \(w=(w_1,\ldots ,w_m)\) assigns an element \( A(w)\in \mathcal{X}\) as the approximation to

*f*. Thus, an algorithm is a possibly nonlinear mapping

*pointwise optimal algorithm*\( A^*\) (if it exists) is one which minimizes the worst error for each

*w*:

*w*, we consider all balls \(B(a,r)\subset \mathcal{X}\) which contain \(S_w\). The smallest ball

*B*(

*a*(

*w*),

*r*(

*w*)), if it exists, is called the

*Chebyshev ball*and its radius is called the

*Chebyshev radius*of \(S_w\). We postpone the discussion of existence, uniqueness, and properties of the smallest ball to the next section. For now, we remark that when the Chebyshev ball

*B*(

*a*(

*w*),

*r*(

*w*)) exists for each measurement vector \(w\in \mathbb {R}^m\), then the pointwise optimal algorithm is the mapping \(A^*:\ w\rightarrow a(w)\) and the pointwise optimal error for this recovery problem is given by

**Pointwise near optimal algorithm**

*We say that an algorithm*

*A*

*is pointwise near optimal with constant*

*C*

*for the set*

*S*

*if*

*R*(

*S*) will generally not be optimal for each \(w\in \mathbb {R}^m\). We refer to the second type of estimates as global and a global optimal algorithm would be one that achieved the bound

*R*(

*S*).

**Global near optimal algorithm**

*We say that an algorithm A is a global near optimal algorithm with constant C for the set S, if*

Note that if an algorithm is near optimal for each of the sets \(S_w\) with a constant *C*, independent of *w*, then it is a global near optimal algorithm with the same constant *C*.

The above description in terms of \(\mathrm{rad}(S_w)\) provides a nice simple geometrical description of the optimal recovery problem. However, it is not a practical solution for a given set *S*, since the problem of finding the Chebyshev center and radius of \(S_w\) is essentially the same as the original optimal recovery problem, and moreover, is known, in general, to be NP hard (see [19]). Nevertheless, it provides some guide to the construction of optimal or near optimal algorithms.

*strictly convex*if

*uniform convexity*. To describe this property, we introduce the modulus of convexity of \(\mathcal{X}\) defined by

Uniform convexity implies strict convexity and it also implies that \(\mathcal{X}\) is reflexive (see [Prop. 1.e.3] in [23]), i.e. \(\mathcal{X}^{**}=\mathcal{X}\), where \(\mathcal{X}^*\) denotes the dual space of \(\mathcal{X}\). If \(\mathcal{X}\) is uniformly convex, then there is quite a similarity between the results we obtain and those in the Hilbert space case. This covers, for example, the case when \(\mathcal{X}\) is an \(L_p\) or \(\ell _p\) space for \(1<p<\infty \), or one of the Sobolev spaces for these \(L_p\). The cases \(p=1,\infty \), as well as the case of a general Banach space \(\mathcal{X}\), add some new wrinkles and the theory is not as complete.

Of central interest to us are special model classes, which we call *approximation sets*.

**Approximation set**

*We call the set*\(\mathcal{K}=\mathcal{K}(\varepsilon ,V)\)

*an approximation set if*

*where*\(V\subset \mathcal{X}\)

*is a known finite dimensional space, and*\(\mathrm{dist}(f, V)_\mathcal{X}:=\inf _{g\in V}\Vert f-g\Vert _{\mathcal{X}}\).

We denote the dependence of \(\mathcal{K}\) on \(\varepsilon \) and *V* only when this is not clear from the context.

### 1.2 Summary of results

The main contribution of the present paper is to describe near optimal algorithms for the recovery of approximation sets in a general Banach space \(\mathcal{X}\).

In the first part of this paper, namely 2 and 3, we use classical ideas and techniques of Banach space theory (see, for example, [6, 23, 31, 37]), to provide results on the optimal recovery of the sets \(S_w\) for any set \(S\subset \mathcal{X}\), \(w\in \mathbb {R}^m\). Much of the material in these two sections is known or easily derived from known results, but we recount this for the benefit of the reader and to ease the exposition of this paper. Not surprisingly, the form of these results depends very much on the structure of the Banach space. The determination of optimal or near optimal algorithms and their performance is connected to liftings (see §3), and the angle between the space *V* and the null space \(\mathcal{N}\) of the measurements (see §4). These concepts allow us to describe a general procedure for constructing recovery algorithms *A* for approximation sets which are pointwise near optimal (with constant 2) and hence are also globally near optimal (see §5).

*V*and the null space \(\mathcal{N}\) of the measurement map

*M*and

*C*is any constant larger than 4. We prove that this estimate is near optimal in the sense that, for any recovery algorithm

*A*, we have

*C*. Thus, the constant \(\mu (\mathcal{N},V)\), respectively the angle \(\theta (\mathcal{N},V)\), determines how well we can recover approximation sets and quantifies the compatibility between the measurements and

*V*. As we have already noted, this result is not new for the Hilbert space setting.

*A*that we exploit in this paper is that it does not depend on \(\varepsilon \) (and, in fact, does not require the knowledge of \(\varepsilon \)). This means that we have the performance bound

*f*and

*V*.

*A*. In Sect. §7, we give examples of how to implement our near optimal algorithm in concrete settings when \(\mathcal{X}=L_p\), or \(\mathcal{X}\) is the space of continuous functions on a domain \(D\subset \mathbb {R}^d\), \(\mathcal{X}=C(D)\). As a representative example of the results in that section, consider the case \(\mathcal{X}=C(D)\) with

*D*a domain in \(\mathbb {R}^d\), and suppose that the measurements functionals are \(l_j(f)=f(P_j)\), \(j=1,\ldots ,m\), where the \(P_j\) are points in

*D*. Then, we prove that

*V*. Results of this type are well-known, via Lebesgue constants, in the case of interpolation (when \(m=n\)).

In §8, we discuss how our results are related to generalized sampling in a Banach space and touch upon how several recent results in sampling can be obtained from our approach. Finally, in §9, we discuss good choices for where to take measurements if this option is available to us.

## 2 Preliminary remarks

In this section, we recall some standard concepts in Banach spaces and relate them to the optimal recovery problem of interest to us. We work in the setting that *S* is any set (not necessarily an approximation set). The results of this section are essentially known and are given only to orient the reader.

### 2.1 The Chebyshev ball

Note that, in general, the center of the Chebyshev ball may not come from *S*. This can even occur in finite dimensional setting (see Example 2.4 given below). However, it may be desired, or even required in certain applications, that the recovery for *S* be a point from *S*. The description of optimal algorithms with this requirement is connected with what we call the *restricted Chebyshev ball of S*. To explain this, we introduce some further geometrical concepts.

*S*in a Banach space \(\mathcal{X}\), we define the following quantities:

The

*diameter*of*S*is defined by \(\mathrm{diam}(S):= \displaystyle {\sup _{f,g\in S}\Vert f-g\Vert }\).- The
*restricted Chebyshev radius*of*S*is defined by$$\begin{aligned} {\mathrm{rad}}_{ R}(S):=\inf _{a\in S}\inf \{ r:\ S \subset B(a,r)\}=\inf _{a\in S}\sup _{f\in S} \Vert f-a\Vert . \end{aligned}$$ - The
*Chebyshev radius*of*S*was already defined as$$\begin{aligned} \mathrm{rad}(S):=\inf _{a\in \mathcal{X}}\inf \{ r:\ S \subset B(a,r)\}=\inf _{a\in \mathcal{X}}\sup _{f\in S} \Vert f-a\Vert . \end{aligned}$$

*S*if we can simply find a point \(a\in S\).

### Theorem 2.1

*S*be any subset of \(\mathcal{X}\). If \(a\in S\), then

*a*is, up to the constant 2, an optimal recovery of

*S*.

We say that an \(a\in \mathcal{X}\) which recovers *S* with the accuracy of (2.2) provides a *near optimal* recovery with constant 2. We shall use this theorem in our construction of algorithms for the recovery of the sets \(\mathcal{K}_w \), when \(\mathcal{K}\) is an approximation set. The relevance of the above theorem and (2.1) for our recovery problem is that if we determine the diameter or restricted Chebyshev radius of \(\mathcal{K}_w\), we will determine the optimal error \(\mathrm{rad}(\mathcal{K}_w)\) in the recovery problem, but only up to the factor two.

### 2.2 Is \(\mathrm{rad}(S)\) assumed?

In view of the discussion preceding (1.1), the best pointwise error we can achieve by any recovery algorithm for *S* is given by \(\mathrm{rad}(S)\). Unfortunately, in general, for arbitrary bounded sets *S* in a general infinite dimensional Banach space \(\mathcal{X}\), the radius \(\mathrm{rad}(S)\) may not be assumed. The first such example was given in [17], where \(\mathcal{X}=\{f\in C[-1,1] :\intop \limits _{-1}^1 f(t)\, dt =0\}\) with the uniform norm and \(S\subset \mathcal{X}\) is a set consisting of three functions. Another, simpler example of a set *S* in this same space was given in [33]. In [21], it is shown that each nonreflexive space admits an equivalent norm for which such examples also exist. If we place more structure on the Banach space \(\mathcal{X}\), then we can show that the radius of any bounded subset \(S\subset \mathcal{X}\) is assumed. We present the following special case of an old result of Garkavi (see [17, Th. II]).

### Lemma 2.2

If the Banach space \(\mathcal{X}\) is reflexive (in particular, if it is finite dimensional), then for any bounded set \(S\subset \mathcal{X}\), \(\mathrm{rad}(S)\) is assumed in the sense that there is a ball *B*(*a*, *r*) with \(r=\mathrm{rad}(S)\) which contains *S*. If, in addition, \(\mathcal{X}\) is uniformly convex, then this ball is unique.

### Proof

*S*and for which \(r_n\rightarrow \mathrm{rad}(S)=:r\). Since

*S*is bounded, the \(a_n\) are bounded and hence, without loss of generality, we can assume that \(a_n\) converges weakly to \(a\in \mathcal{X}\) (since every bounded sequence in a reflexive Banach space has a weakly converging subsequence). Now let

*f*be any element in

*S*. Then, there is a norming functional \(l\in \mathcal{X}^*\) of norm one for which \(l(f-a)=\Vert f-a\Vert \). Therefore

*B*(

*a*,

*r*) contains

*S*and so the radius is attained. If \(\mathcal{X}\) is uniformly convex and we assume that there are two balls, centered at

*a*and \(a'\), respectively, \(a\ne a'\), each of radius

*r*which contain

*S*. If \(\varepsilon :=\Vert a-a'\Vert >0\), since \(\mathcal{X}\) is uniformly convex, for every \(f\in S\) and for \(\bar{a}:=\frac{1}{2}(a+a')\), we have

### 2.3 Some examples

In this section, we will show that for centrally symmetric (i.e. if \(f\in S\) then also \(-f\in S\)), convex sets *S*, we have a very explicit relationship between the \(\mathrm{diam}(S)\), \(\mathrm{rad}(S)\), and \({\mathrm{rad}}_{ R}(S)\). We also give some examples showing that for general sets *S* the situation is more involved and the only relationship between the latter quantities is the one given in (2.1).

### Proposition 2.3

Let \(S\subset \mathcal{X}\) be a centrally symmetric, convex set in a Banach space \(\mathcal{X}\). Then, we have

*S*is centered at 0 and

### Proof

(i) We need only consider the case when \(r:=\sup _{f\in S} \Vert f\Vert <\infty \). Clearly \(0\in S\), \(S\subset B(0,r)\), and thus \({\mathrm{rad}}_{ R}(S)\le r\). In view of (2.1), \(\mathrm{diam}(S)\le 2\mathrm{rad}(S)\le 2{\mathrm{rad}}_{ R}(S)\le 2r\). Therefore, we need only show that \(\mathrm{diam}(S)\ge 2r\). For any \(\varepsilon >0\), let \(f_\varepsilon \in \mathcal{S}\) be such that \(\Vert f_\varepsilon \Vert \ge r-\varepsilon \). Since *S* is centrally symmetric \(-f_\varepsilon \in S\) and \(\mathrm{diam}(S)\ge \Vert f_\varepsilon -(-f_\varepsilon )\Vert \ge 2r-2\varepsilon .\) Since \(\varepsilon >0\) is arbitrary, \(\mathrm{diam}(S)\ge 2r\), as desired.

*S*, we know that \(\frac{1}{2}( g-h)\) and \(\frac{1}{2}( h-g)\) are both in \(S_0\). Therefore

*g*,

*h*were arbitrary, we get \(\mathrm{diam}(S_0)\ge \mathrm{diam}(S_w)\). \(\square \)

*x*, such that

For \(\mathcal{X}=c_0\), \(S:=\{x\ :\ x_j\ge 0 \text{ and } \sum _{j=1}^\infty x_j=1\}\), we have \(\mathrm{rad}(S)={\mathrm{rad}}_{R}(S)=\mathrm{diam}(S)=1.\)

For \(\mathcal{X}=\ell _p(\mathbb {N})\), \(1<p<\infty \), and \(S:=\{x\ :\ x_j\ge 0 \text{ and } \sum _{j=1}^\infty x_j=1\}\), one computes that \(\mathrm{diam}(S)=2^{1/p}\), and \({\mathrm{rad}}_{R}(S)=\mathrm{rad}(S)=1\).

For \(\mathcal{X}=L_1([0,1])\), \(S:=\{f\in L_1([0,1]) :\ f\ge 0 \ \text{ and } \ \intop \limits _0^1|f|=1\}\), we have \(\mathrm{diam}(S)={\mathrm{rad}}_{R}(S)=2\), but \(\mathrm{rad}(S)=1\).

### Example 2.4

*T*with vertices (1, 1, 0), (1, 0, 1), and (0, 1, 1),

Indeed, since *T* is the convex hull of its vertices, any point in *T* has coordinates in [0, 1], and hence the distance between any two such points is at most one. Since the vertices are at distance one from each other, we have that \(\mathrm{diam}(T)=1\). It follows from (2.1) that \(\mathrm{rad}(T)\ge 1/2\). Note that the ball with center \((\frac{1}{2},\frac{1}{2},\frac{1}{2})\) and radius 1 / 2 contains *T*, and so \(\mathrm{rad}(T)=1/2\). Given any point \(z\in T\) which is a potential center of the restricted Chebyshev ball for *T*, at least one of the coordinates of *z* is at least 2 / 3 (because \(z_1+z_2+z_3=2\)), and thus has distance at least 2 / 3 from one of the vertices of *T*. On the other hand, the ball with center \((\frac{2}{3}, \frac{2}{3}, \frac{2}{3})\in T\) and radius \(\frac{2}{3}\) contains *T*.

### 2.4 Connection to approximation sets and measurements

The discussion in this section is directed at showing that the behavior, observed in 2.3, can occur even when the sets *S* are described through measurements. The next example is a modification of Example 2.4, and the set under consideration is of the form \(\mathcal{K}_w\), where \(\mathcal{K}\) is an approximation set.

### Example 2.5

*T*is the set from Example 2.4. Thus, we have

The following theorem shows that any example for general sets *S* can be transferred to the setting of interest to us, where the sets are of the form \(\mathcal{K}_w\) with \(\mathcal{K}\) being an approximation set.

### Theorem 2.6

*X*is a Banach space and \(K\subset X\) is a non-empty, closed and convex subset of the closed unit ball

*U*of

*X*. Then, there exists a Banach space \(\mathcal{X}\), a finite dimensional subspace \(V\subset \mathcal{X}\), a measurement operator

*M*, and a measurement

*w*, such that for the approximation set \(\mathcal{K}:=\mathcal{K}(1,V)\), we have

### Proof

*X*, we first define \(Z:=X\oplus \mathbb {R}:=\{( f,\alpha ):\ f\in X, \ \alpha \in \mathbb {R}\}.\) Any norm on

*Z*is determined by describing its unit ball, which can be taken as any closed, bounded, centrally symmetric convex set. We take the set \(\Omega \) to be the convex hull of the set \((U,0)\cup (K,1)\cup (-K,-1)\). Since \(K\subset U\), it follows that a point of the form (

*f*, 0) is in \( \Omega \) if and only if \(\Vert f\Vert _X\le 1\). Therefore, for any \( f\in X\),

*f*, 1) is in \(\Omega \) if and only if \( f\in K\). We next define the space \(\mathcal{X}=Z \oplus \mathbb {R}:=\{(z,\beta ):\ z\in Z, \beta \in \mathbb {R}\}\), with the norm

## 3 A description of algorithms via liftings

*S*is any subset of \(\mathcal{X}\), and we wish to recover the elements in \(S_w\) for each measurement \(w\in \mathbb {R}^m\). That is, at this stage, we do not require that

*S*is an approximation set. Recall that given the measurement functionals \(l_1,\ldots ,l_m\) in \(\mathcal{X}^*\), the linear operator \(M:\mathcal{X}\rightarrow \mathbb {R}^m\) is defined as

*M*we have the null space

### Remark 3.1

Let us note that if in place of \(l_1,\ldots , l_m\), we use functionals \(l_1',\ldots ,l_m'\) which span the same space *L* in \(X^*\), then the information about *f* contained in *M*(*f*) and \(M'(f)\) is exactly the same, and so the recovery problem is identical. For this reason, we can choose any spanning set of linearly independent functionals in defining *M* and obtain exactly the same recovery problem. Note that, since these functionals are linearly independent, *M* is a linear mapping from \(\mathcal{X}\) onto \(\mathbb {R}^m\).

*M*. We introduce the following norm on \(\mathbb {R}^m\) induced by

*M*

*M*can be interpreted as mapping \(\mathcal{X}_w\rightarrow w\) and, in view of (3.2), is an isometry from \(\mathcal{X}/\mathcal{N}\) onto \(\mathbb {R}^m\) under the norm \(\Vert \cdot \Vert _M\).

**Lifting operator***A lifting operator*\(\Delta \)*is a mapping from*\(\mathbb {R}^m\)*to*\(\mathcal{X}\)*which assigns to each*\(w\in \mathbb {R}^m\)*an element from the coset*\(\mathcal{X}_w\), *i.e., a representer of the coset. *

Recall that any algorithm *A* is a mapping from \(\mathbb {R}^m\) into \(\mathcal{X}\). If \(S\subset \mathcal{X}\) is any subset of \(\mathcal{X}\), we would like the mapping *A* for our recovery problem to send *w* to an element of \(S_w\), provided \(S_w\ne \emptyset \), since then we would know that *A* is nearly optimal (see Theorem 2.1) up to the constant 2. So, in going further, we consider only algorithms *A* which take *w* to an element of \(\mathcal{X}_w\) for all \(w\in \mathbb {R}^m\). At this stage we are not yet invoking our desire that *A* actually maps into \(S_w\), only that it maps into \(\mathcal{X}_w\).

**Admissible algorithm***We say that an algorithm*\(A:\mathbb {R}^m \rightarrow \mathcal{X}\)*is admissible if, for each*\(w\in \mathbb {R}^m\), \(A(w)\in \mathcal{X}_w\).

Our interest in lifting operators is because any admissible algorithm *A* is a lifting \(\Delta \), and the performance of such an *A* is related to the norm of \(\Delta \). A natural lifting, and the one with minimal norm 1, would be one which maps *w* into an element of minimal norm in \(\mathcal{X}_w\). Unfortunately, in general, no such minimal norm element exists, as the following illustrative example shows.

### Example 3.2

*M*as

*M*that for every

*x*such that \(M(x)=w\), we have \(\Vert w\Vert _{\ell _2(2)}\le \Vert x\Vert _{\ell _1(\mathbb {N})}\), and thus \(\Vert w\Vert _{\ell _2(2)}\le \Vert w\Vert _M\). In particular, for every \(i=1, 2, \ldots \),

*i*-th coordinate vector in \(\ell _1(\mathbb {N})\). So, we have that \(\Vert h_i\Vert _{\ell _2(2)}=\Vert h_i\Vert _M=1\). Since the \(h_i\)’s are dense on the unit circle, every

*w*with Euclidean norm one satisfies \(\Vert w\Vert _M=1\). Next, we consider any \(w\in \mathbb {R}^2\), such that \(\Vert w\Vert _{\ell _2(2)}=1\), \(w\ne h_j\), \(j=1,2,\ldots \). If \(w=M(x)=\sum _{j=1}^\infty x_jh_j\), then

*x*in the coset \(\mathcal{X}_w\). This also shows there is no lifting \(\Delta \) from \(\mathbb {R}^2\) to \(\mathcal{X}/\mathcal{N}\) with norm one.

While the above example shows that norm one liftings may not exist for a general Banach space \(\mathcal{X}\), there is a classical theorem of Bartle-Graves which states that there are continuous liftings \(\Delta \) with norm \(\Vert \Delta \Vert \) as close to one as we wish (see [5, 6, 30]). In our setting, this theorem can be stated as follows.

### Theorem 3.3

\(\Delta \) is continuous.

\(\Delta (w)\in \mathcal{X}_w,\quad w\in \mathbb {R}^m\).

for every \(\lambda >0\), we have \(\Delta (\lambda w)=\lambda \Delta (w)\).

\(\Vert \Delta (w)\Vert _\mathcal{X}\le (1+\eta )\Vert w\Vert _M,\quad w\in \mathbb {R}^m\).

Unfortunately, a numerical recipe for constructing a lifting operator with norm as close to one as we wish, cannot be provided for a general Banach space. Note that if we put more structure on \(\mathcal{X}\), then we can guarantee the existence of a continuous lifting with norm one (see [6, Lemma 2.2.5]).

### Theorem 3.4

### Proof

*l*is a norming functional for

*f*, i.e. \(\Vert l\Vert _{\mathcal{X}^*}=1\) and \(l( f)=\Vert f\Vert \), then

*j*, with \(\alpha <1\) a fixed constant. Now, let \(l\in \mathcal{X}^*\) be a norm one functional, such that \(l( f)=\Vert f\Vert \). Then, we have

The latter theorem would not hold under the slightly milder assumptions on \(\mathcal{X}\) being strictly convex and reflexive (in place of uniform convexity), as shown in [10]. In that paper, the author gives an example of a strictly convex, reflexive Banach space \(\mathcal{X}\) and a measurement map \(M:\mathcal{X}\rightarrow \mathbb {R}^2\), for which there is no continuous norm one lifting \(\Delta \).

We conclude this section with the observation that linear liftings are closely related to projections. A linear lifting \(\Delta :\mathbb {R}^m\rightarrow \mathcal{X}\) with norm \(\le C\) exists if and only if there exists a linear projector *P* from \(\mathcal{X}\) onto a subspace \(Y\subset \mathcal{X}\) with \(\ker (P)=\mathcal{N}\) and \(\Vert P\Vert \le C\). Indeed, if \(\Delta \) is such a lifting then its range *Y* is a finite dimensional subspace and \(P({f}):=\Delta (M({f}))\) defines a projection from \(\mathcal{X}\) onto *Y* with the mentioned properties. On the other hand, given such a *P* and *Y*, notice that any two elements in \(M^{-1}(w)\) have the same image under *P*, since the kernel of *P* is \(\mathcal{N}\). Therefore, we can define the lifting \(\Delta (w):=P(M^{-1}(w))\), \(w\in \mathbb {R}^m\), which has norm at most *C*.

## 4 A priori estimates for the radius of \(\mathcal{K}_w \)

In this section, we discuss estimates for the radius of \(\mathcal{K}_w \) when \(\mathcal{K}=\mathcal{K}(\varepsilon ,V)\) is an approximation set. The main result we shall obtain is that the global optimal recovery error \(R(\mathcal{K})\) is determined a priori (up to a constant factor 2) by the angle between the null space \(\mathcal{N}\) of the measurement map *M* and the approximating space *V* [see (iii) of Theorem 4.4 below].

- (i)
If \( \mathcal{N}\cap V\ne \{0\}\), then for any \(0\ne \eta \in \mathcal{N}\cap V\), and any \(f\in \mathcal{K}_w\), the line \(f+t \eta \), \(t\in \mathbb {R}\), is contained in \(\mathcal{K}_w\), and therefore there is no finite ball

*B*(*a*,*r*) which contains \(\mathcal{K}_w\). Hence \(\mathrm{rad}(\mathcal{K}_w)=\infty \). - (ii)
If \(\mathcal{N}\cap V=\{0\}\), then \(n=\dim V \le \mathrm{codim}\mathcal{N}=\mathrm{rank}M=m\) and therefore \(n\le m\). In this case \(\mathrm{rad}(\mathcal{K}_w)\) is finite for all \(w\in \mathbb {R}^m\).

**Standing assumption**

*In view of this observation, the only interesting case is*(ii),

*and therefore we assume that*\(\mathcal{N}\cap V=\{0\}\)

*for the remainder of this paper.*

*X*and

*Y*of a given Banach space \(\mathcal{X}\), we recall the angle \(\Theta \) between

*X*and

*Y*, defined as

### Remark 4.1

Since *V* is a finite dimensional space and \(\mathcal{N}\cap V=\{0\}\), we have \(\Theta (\mathcal{N},V)>0\). Indeed, otherwise there exists a sequence \(\{\eta _k\}_{k\ge 1}\) from \(\mathcal{N}\) with \(\Vert \eta _k\Vert =1\) and a sequence \(\{v_k\}_{k\ge 1}\) from *V*, such that \(\Vert \eta _k-v_k\Vert \rightarrow 0\), \(k\rightarrow \infty \). We can assume \(v_k\) converges to \(v_\infty \), but then also \(\eta _k\) converges to \(v_\infty \), so \(v_\infty \in \mathcal{N}\cap V\) and \(\Vert v_\infty \Vert =1\), which is the desired contradiction to \(\mathcal{N}\cap V=\{0\}\).

Note that, in general, \(\mu \) is not symmetric, i.e., \(\mu (Y,X)\ne \mu (X,Y)\). However, we do have the following comparison.

### Lemma 4.2

*X*and

*Y*of a given Banach space \(\mathcal{X}\), such that \(X\cap Y=\{0\}\), we have

### Proof

The following lemma records some properties of \(\mu \) for our setting in which \(Y=V\) and \(X=\mathcal{N}\) is the null space of *M*.

### Lemma 4.3

*V*be any finite dimensional subspace of \( \mathcal{X}\) with \(\dim (V)\le m\), and \(M:\mathcal{X}\rightarrow \mathbb {R}^m\) be any measurement operator. Then, for the null space \(\mathcal{N}\) of

*M*, we have the following.

- (i)
\( \mu (V,\mathcal{N})=\Vert M_V^{-1}\Vert , \)

- (ii)
\( \mu (\mathcal{N},V) \le 1+\mu (V,\mathcal{N})=1+\Vert M_V^{-1}\Vert \le 2\Vert M_V^{-1}\Vert , \)

*M*on

*V*and \(M_V^{-1}\) is its inverse considered as a map from \(M(V)\subset \mathbb {R}^m\) onto

*V*.

### Proof

We have the following simple, but important theorem.

### Theorem 4.4

Let \(\mathcal{X}\) be any Banach space, *V* be any finite dimensional subspace of \( \mathcal{X}\), \(\varepsilon >0\), and \(M:\mathcal{X}\rightarrow \mathbb {R}^m\) be any measurement operator. Then, for the set \(\mathcal{K}=\mathcal{K}(\varepsilon ,V)\), we have the following

### Proof

Let us make a few comments about Theorem 4.4 viz a viz the results in [8] (see Theorem 2.8 and Remark 2.15 of that paper) for the case when \(\mathcal{X}\) is a Hilbert space. In the latter case, it was shown in [8] that the same result as (i) holds, but in the case of (ii), an exact computation of \(\mathrm{rad}(K_w)\) was given with the constant 2 replaced by a number (depending on *w*) which is less than one and for some *w* can be very small. It is probably impossible to have an exact formula for \(\mathrm{rad}(K_w)\) in the case of a general Banach space. However, we show in the appendix that when \(\mathcal{X}\) is uniformly convex and uniformly smooth, we can improve on the constant appearing in (ii) of Theorem 4.4.

## 5 Near optimal algorithms

*M*to the space

*V*. By our

*Standing Assumption*, \(M_V\) is invertible, and hence \(Z:=M(V)=M_V(V)\) is an

*n*-dimensional subspace of \(\mathbb {R}^m\). Given \(w\in \mathbb {R}^m\), we consider its error of best approximation from

*Z*in \(\Vert \cdot \Vert _M\), defined by

*w*, and it is unique when the norm is strictly convex, for a possible ease of numerical implementation, we consider other non-best approximation maps. We say a mapping \(\Lambda :\mathbb {R}^m\mapsto Z\) is

*near best*with constant \(\lambda \ge 1\), if

*w*into a best approximation of

*w*from

*Z*.

*A*maps \(\mathbb {R}^m\) into \(\mathcal{X}\), so that it is an algorithm. It also has the property that \(A(w)\in \mathcal{X}_w\), which means that it is an admissible algorithm. Finally, by our construction, whenever \(w=M(v)\) for some \(v\in V\), then \(\Lambda (w)=w\), and so \(A(w)=v\). Let us note some important properties of such an algorithm

*A*.

### Theorem 5.1

Let \(\mathcal{X}\) be a Banach space, *V* be any finite dimensional subspace of \( \mathcal{X}\), \(\varepsilon >0\), \(M:\mathcal{X}\rightarrow \mathbb {R}^m\) be any measurement operator with a null space \(\mathcal{N}\), and \(\mathcal{K}=\mathcal{K}(\varepsilon ,V)\) be an approximation set. Then, for any lifting \(\Delta \) and any near best approximation map \(\Lambda \) with constant \(\lambda \ge 1\), the algorithm *A*, defined in (5.4), has the following properties:

(i) \(A(w)\in \mathcal{X}_w,\quad w\in \mathbb {R}^m\), i.e. *A* is admissible.

(ii) \(\mathrm{dist}(A(M( f)),V)_{ \mathcal{X}}\le \lambda \Vert \Delta \Vert \mathrm{dist}( f,V)_\mathcal{X},\quad f\in \mathcal{X}.\)

*A*has the a priori performance bound

*A*is pointwise near optimal with constant 2, i.e. for any \(w\in \mathbb {R}^m\),

*A*is also global near optimal with constant 2 i.e.

### Proof

*f*be any element in \(\mathcal{X}\). Then \(M_V^{-1}(\Lambda (M( f)))\in V\), and therefore

It follows from the above theorem, the best choice for *A*, from a theoretical point of view, is to choose \(\Delta \) with \(\Vert \Delta \Vert =1\) and \(\Lambda \) with constant \(\lambda =1\). When \(\mathcal{X}\) is uniformly convex, we can always accomplish this theoretically, but there may be issues in the numerical implementation. If \(\mathcal{X}\) is a general Banach space, we can choose \(\lambda =1\) and \( \Vert \Delta \Vert \) arbitrarily close to one, but as in the latter case, problems in the numerical implementation may also arise. In the next section, we discuss some of the numerical considerations in implementing an algorithm *A* of the form (5.4). In the case that \(\lambda \Vert \Delta \Vert >1\), we only know (5.5) which is only slightly worse than the a priori bound \(4\varepsilon \mu (\mathcal{N},V)\) which we obtain when we know that *A*(*w*) is in \(\mathcal{K}_w(\varepsilon ,V)\). In this case, the algorithm *A* is globally near optimal with the constant \(4\lambda \Vert \Delta \Vert \). Note that we do not have pointwise near optimality because when \(\lambda \Vert \Delta \Vert >1\), \(\mathrm{rad}(\mathcal{K}_w(\lambda \Vert \Delta \Vert \varepsilon ,V))\) may be much bigger than \(\mathrm{rad}(\mathcal{K}_w(\varepsilon ,V))\); this can be easily seen in the Hilbert space case or from Proposition 10.4.

### 5.1 Noisy measurements

In this section, we discuss the issue of noisy measurements. Suppose that in place of the exact measurements \(w_i=l_i(f)\), where \(M(x)=(w_1,\ldots ,w_m)\), we receive the noisy data \(w_i+\delta _i\), \(i=1,\ldots ,m\). Without loss of generality, we can assume that \(\Vert l_i\Vert =1\), \(i=1,\ldots ,m\). Let us denote by \(\delta :=(\delta _1,\ldots ,\delta _m)\). Then, the following theorem holds.

### Theorem 5.2

*A*, defined in (5.4), to the noisy measurements \(w+\delta \), gives the error bound for \(f=M(w)\),

### Proof

## 6 Numerical issues in implementing the algorithms *A*

How to compute \(\Vert \cdot \Vert _M\) on \(\mathbb {R}^m\)?

How to numerically construct near best approximation maps \(\Lambda \) for approximating the elements in \(\mathbb {R}^m\) by the elements of \(Z=M(V)\) in the norm \(\Vert \cdot \Vert _M\)?

How to numerically construct lifting operators \(\Delta \) with a controllable norm \(\Vert \Delta \Vert \)?

*V*, and the measurement functionals \(l_j\), \(j=1,\ldots ,m\). In this section, we will consider general principles and see how these principles are implemented in three examples.

### Example 1

\(\mathcal{X}=C(D)\), where *D* is a domain in \(\mathbb {R}^d\), *V* is any *n* dimensional subspace of \(\mathcal{X}\), and \(M=(l_1,\ldots ,l_m)\) consists of *m* point evaluation functionals at distinct points \(P_1,\ldots ,P_m\in D\), i.e., \(M(f)=(f(P_1),\ldots , f(P_m))\).

### Example 2

*D*is a domain in \(\mathbb {R}^d\),

*V*is any

*n*dimensional subspace of \(\mathcal{X}\) and

*M*consists of the

*m*functionals

### Example 3

*V*is any

*n*dimensional subspace of \(\mathcal{X}\), and

*M*consists of the

*m*functionals

### 6.1 Computing \(\Vert \cdot \Vert _M\)

*s*is an integer. On the other hand, on such an interval, \(r_{k+1}\) takes on both of the values 1 and \(-1\). Therefore, by induction on

*k*, we get \(\Vert \alpha \Vert _M^*= \sum _{j=1}^m|\alpha _j|\). Hence, we have

### 6.2 Approximation maps

Once the norm \(\Vert \cdot \Vert _M\) is numerically computable, the problem of finding a best or near best approximation map \(\Lambda (w)\) to *w* in this norm becomes a standard problem in convex minimization. For instance, in the examples from the previous subsection, the minimization is done in \(\Vert \cdot \Vert _{\ell _p(m)}\). Of course, in general, the performance of algorithms for such minimization depend on the properties of the unit ball of \(\Vert \cdot \Vert _M\). This ball is always convex, but in some cases it is uniformly convex and this leads to faster convergence of the iterative minimization algorithms and guarantees a unique minimum.

### 6.3 Numerical liftings

*m*which has \(\mathcal{N}\) as its kernel. We can find all

*Y*that can be used in this fashion as follows. We take elements \(\psi _1,\ldots ,\psi _m\) from \(\mathcal{X}\), such that

It follows from the Kadec-Snobar theorem that we can always choose a *Y* such that \(\Vert P_Y\Vert \le \sqrt{m}\). In general, the \(\sqrt{m}\) cannot be replaced by a smaller power of *m*. However, if \(\mathcal{X}=L_p\), then \(\sqrt{m}\) can be replaced by \(m^{|1/2-1/p|}\). We refer the reader to Chapter III.B of [37] for a discussion of these facts.

In many settings, the situation is more favorable. In the case of Example 1, we can take for *Y* the span of any norm one functions \(\psi _j\), \( j=1,\ldots , m\), such that \(l_i(\psi _j)=\delta _{i,j}\), \(1\le i,j\le m\). We can always take the \(\psi _j\) to have disjoint supports, and thereby get that \(\Vert P_Y\Vert =1\). Thus, we get a linear lifting \(\Delta \) with \(\Vert \Delta \Vert =1\) (see (6.4)). This same discussion also applies to Example 2.

*w*, we define

*A*for which the integral of the left hand side of (6.7) is nonzero is when \(A=\{k\}\). This observation, together with (6.5) gives

The above situation, in which a nonlinear lifting can have better norm that any linear lifting is not restricted only to Example 3, as the following remark shows.

### Remark 6.1

*C*. If \(\Delta :\mathbb {R}^{m+1}\rightarrow C([-\pi ,\pi ])\) is any linear lifting, using [37, III.B.5 and III.B.16], we obtain

## 7 Performance estimates for the examples

In this section, we consider the examples from 6. In particular, we determine \(\mu (\mathcal{N},V)\), which allows us to give the global performance error for near optimal algorithms for these examples. We begin with the optimal algorithms in a Hilbert space, which is not one of our three examples, but is easy to describe.

### 7.1 The case when \(\mathcal{X}\) is a Hilbert space \(\mathcal{H}\)

*m*dimensional subspace of \(\mathcal{H}\). We can always perform a Gram-Schmidt orthogonalization and assume therefore that \(\phi _1,\ldots ,\phi _m\in \mathcal{H}\) is an orthonormal basis for

*W*(see Remark 3.1). We have \(\mathcal{N}=W^\perp \). From (6.2) and (6.3) we infer that \(\Vert \cdot \Vert _M \) on \(\mathbb {R}^m\) is the \(\ell _2(m)\) norm. Therefore, the approximation map is simple least squares fitting. Namely, to our data

*w*, we find the element \( z^*(w)\in Z\), where \(Z:=M(V)\), such that

*w*in \(\Vert \cdot \Vert _M\) by the elements of

*Z*, and hence \(\lambda =1\). The lifting \(\Delta (w_1,\ldots ,w_m):=\sum _{j=1}^m w_j \phi _j\) is linear and \(\Vert \Delta \Vert =1\). Hence, we have the algorithm

Note that our general theory states that the above algorithm is near optimal with constant 2 for recovering \(\mathcal{K}_w\). Moreover, it is shown in [7] that it is actually an optimal algorithm. The reason for this is that the sets \(\mathcal{K}_w\) in this Hilbert space setting have a center of symmetry, so Proposition 2.3 can be applied. Furthermore, it was shown in [8] that the calculations can be streamlined by choosing at the beginning certain favorable bases for *V* and *W*. In particular, the quantity \(\mu (\mathcal{N},V)\) can be immediately computed from the cross-Grammian of the favorable bases.

### 7.2 Example 1

*w*from the space \(Z:=M(V)\subset \mathbb {R}^m\). In other words, we find

*M*(

*V*) the point \(z^*(w)\) is not necessarily unique. For certain

*V*, however, we have uniqueness.

*V*is a Chebyshev space on

*D*, i.e., for any

*n*points \(Q_1,\ldots ,Q_n\in D\), and any data \(y_1,\ldots ,y_n\), there is a unique function \(v\in V\) which satisfies \(v(Q_i)=y_i\), \(1\le i\le n\). In this case, when \(m=n\), problem (7.2) has a unique solution

*V*to the point set \(\Omega :=\{P_1,\ldots ,P_m\}\). Clearly, \(V_m\) is a Chebyshev space on \(C(\Omega )\) as well, and therefore there is a unique point \(z^*(w):=(\tilde{v}(P_1), \ldots ,\tilde{v}(P_m))\in V_m\), coming from the evaluation of a unique \(\tilde{v}\in V\), which is the best approximant from \(V_m\) to

*f*on \(\Omega \). The point \(z^*(w)\) is characterized by an oscillation property. Various algorithms for finding \(\tilde{v}\) are known and go under the name Remez algorithms.

*V*is not necessarily a Chebyshev space, a minimizer \(z^*(w)\) can still be found by convex minimization, and the approximation mapping \(\Lambda \) maps

*w*to a \(z^*(w)\). Moreover, \(z^*(w)=M(v^*(w))\) for some \(v^*(w)\in V\), where \(v^*(w) \) is characterized by solving the minimization

*A*is given by

To give an a priori bound for the performance of this algorithm, we need to compute \(\mu (\mathcal{N},V)\).

### Lemma 7.1

*V*be a subspace of

*C*(

*D*), and \(M(f)=(f(P_1),\ldots ,f(P_m))\), where \(P_j\in D\), \(j=1,\ldots ,m\) are

*m*distinct points in \(D\subset \mathbb {R}^d\). Then, for \(\mathcal{N}\) the null space of

*M*, we have

### Proof

### Remark 7.2

*f*with the data \(w=(f(P_1),\ldots ,f(P_m))\) the performance bound

*V*of trigonometric polynomials of degree \(\le n\) on \(D:=[-\pi ,\pi ]\), which is a Chebyshev system of dimension \(2n+1\). We take \(\mathcal{X}\) to be the space of continuous functions on

*D*which are periodic, i.e., \( f(-\pi )=f(\pi )\). If the data consists of the values of

*f*at \(2n+1\) distinct points \(\{P_i\}\), then the min-max approximation is simply the interpolation projection \(\mathcal{P}_nf\) of

*f*at these points and \(A(M(f)))=\mathcal{P}_nf\). The error estimate for this case is

### 7.3 Example 2

### 7.4 Example 3

As mentioned earlier, our interest in Example 3 is because it illustrates certain theoretical features. In this example, the norm \(\Vert \cdot \Vert _M\) is the \(\ell _\infty (m)\) norm, and approximation in this norm was already discussed in Example 1. The interesting aspect of this example centers around liftings. We know that any linear lifting must have norm \(\ge \sqrt{m/2}\). On the other hand, we have given in (6.8) an explicit formula for a (nonlinear) lifting with norm one. So, using this lifting, the algorithm *A* given in (5.4) will be near optimal with constant 2 for each of the classes \(\mathcal{K}_w\).

## 8 Relation to sampling theory

The results we have put forward, when restricted to problems of sampling, have some overlap with recent results, see, for example [1, 2, 3] . In this section, we point out these connections and what new light our general theory sheds on sampling problems. The main point to be made is that our results give a general framework for sampling in Banach spaces that includes many of the specific examples studied in the literature.

*f*. For \(m=1,2,\ldots \) we define mappings \(M_m:\mathcal{X}\rightarrow \mathbb {R}^m\) as

*Example PS: Point samples of continuous functions* Consider the space \(\mathcal{X}=C(D)\) for a domain \(D\subset \mathbb {R}^d\) and a sequence of points \(P_j\) from *D*. Then, the point evaluation functionals \(l_j(f)=f(P_j)\), \(j=1,2,\ldots \), are point samples of *f*. Given a compact subset \(K\subset \mathcal{X}\), we are interested in how well we can recover \(f\in K\) from the information \(l_j(f)\), \(j=1,2,\ldots \).

*Example FS: Fourier samples*Consider the space \(\mathcal{X}=L_2(\Omega )\), \(\Omega =[-\pi ,\pi ]\), and the linear functionals

*f*. Given a compact subset \(K\subset \mathcal{X}\), we are interested in how well we can recover \(f\in K\) in the norm of \(L_2(\Omega )\) from the information \(l_j(f)\), \(j=1,2,\ldots \). The main problem in sampling is to build reconstruction operators \(A_m:\mathbb {R}^m\mapsto \mathcal{X}\) such that the reconstruction mapping \(R_m( f):=A_m(M_m( f))\) provide a good approximation to

*f*. Typical questions are:

- (i)
Do there exist such mappings such that \(R_m( f)\) converges to

*f*as \(m\rightarrow \infty \), for each \( f\in \mathcal{X}\)? - (ii)
What is the best performance in terms of rate of approximation on specific compact sets

*K*? - (iii)
Can we guarantee the stability of these maps in the sense of how they perform with respect to noisy observations?

*K*typically considered are either directly defined by approximation or can be equivalently described by such approximation. That is, associated to

*K*is a sequence of spaces \(V_n\), \(n\ge 0\), each of dimension

*n*, and \(f\in K\) is equivalent to

*K*are often provided by the theory of approximation. For example, a periodic function \(f\in C[-\pi ,\pi ]\) is in Lip \(\alpha \), \(0<\alpha <1\) if and only if

*n*, and moreover, the Lip \(\alpha \) semi-norm is equivalent to the smallest constant

*C*(

*f*) for which (8.3) holds, see [13]. Similarly, a function

*f*defined on \([-1,1]\) has an analytic extension to the region in the plane with boundary given by Bernstein ellipse \(E_\rho \) with parameter \(\rho >1\) and belongs to the unit ball \(U(E_\rho )\) if and only if

*n*in one variable, see [29]. Recall that the Bernstein ellipse \(E_\rho \) is the open region in the complex plane bounded by the ellipse with foci \(\pm 1\) and semiminor and semimajor axis lengths summing to \(\rho \).

*K*is given by

### Remark 8.1

It is more convenient in this section to use the quantity \(\mu (V,\mathcal{N})\) rather than \(\mu (\mathcal{N},V)\). Recall that \(\frac{1}{2}\mu (V,\mathcal{N})\le \mu ( \mathcal{N},V)\le 2\mu (V,\mathcal{N})\) and therefore this switch only effects constants by a factor of at most 2.

*m*samples \(M_m( f)\), \(f\in \mathcal{X}\), for any \(n\le m\), the mapping \(A_{n,m}\) from \(\mathbb {R}^m\mapsto \mathcal{X}\), given by (5.4), provides an approximation \(A_{n,m}(M_m( f))\) to \( f\in K\) with the accuracy, see (5.5)

*C*will generally be a known constant but larger than 8. In the two above examples, one can take \(C=8\) both theoretically and numerically.

### Remark 8.2

Let \(A_{n,m}^*:\mathbb {R}^m\mapsto V_n\) be the mapping defined by (5.4) with the second term \(\Delta (w-M(w))\) on the right deleted. Fom (5.2), it follows that the term that is dropped has norm not exceeding \(\Vert \Delta \Vert \varepsilon _n\) whenever \( f\in K\), and since we can take \(\Vert \Delta \Vert \) as close to one as we wish, the resulting operators satisfy (8.6), with a new value of *C*, but now they map into \(V_n\).

*m*and the information map \(M_m\), we are allowed to choose

*n*, i.e., we can choose the space \(V_n\). Since \(\varepsilon _n\) is known, from the point of view of the error bound (8.6), given the

*m*samples, the best choice of

*n*is

*n*(

*m*). Consider the case of point samples. Then, the results of Example 1 in the previous section show that in this case, we have

*Example FS*above), we have for any space \(V_n\) of dimension

*n*

*m*-th partial sum of the Fourier series of

*v*. The right side of (8.10), in the case \(V_n=\mathcal{P}_{n-1}\) is studied in [3] (where it is denoted by \(B_{n,m}\)). Giving bounds for quotients, analogous to those in (8.9), has been a central topic in sampling theory (see [1, 2, 3, 29]) and such bounds have been obtained in specific settings, such as the case of equally spaced point samples on \([-1,1]\) or Fourier samples. The present paper does not contribute to the problem of estimating such ratios of continuous to discrete norms.

The results of the present paper give a general framework for the analysis of sampling. Our construction of the operators \(A_{n,m}\) (or their modification \(A_{n,m}^*\) given by Remark 8.2), give performance bounds that match those given in the literature in specific settings such as the two examples given at the beginning of this section. It is interesting to ask in what sense these bounds are optimal. Theorem 5.1 proves optimality of the bound (8.6) if the assumption that \(f\in K\) is replaced by the less demanding assumption that \(f\in \mathcal{K}(\varepsilon _n,V_n)\), for this one fixed value of *n*. The knowledge that \(K=\{ f\in \mathcal{X}: \mathrm{dist}( f,V_n)_\mathcal{X}\le \varepsilon _n\}=\bigcap _{n\ge 0}\mathcal{K}(\varepsilon _n,V_n)\), could allow an improved performance, since it is a more demanding assumption. In the case of a Hilbert space, this was shown to be the case in [7] where, in some instances, much improved bounds were obtained from this additional knowledge. However, the examples in [7] are not for classical settings such as polynomial or trigonometric polynomial approximation. In these cases, there is no known improvement over the estimate (8.8).

*V*. Notice that the spaces \(\mathcal{N}_m\) satisfy \(\mathcal{N}_{m+1}\subset \mathcal{N}_m\) and hence the sequence \((\mu (V,\mathcal{N}_m))\) is non-increasing.

In what follows we give some conditions on the sequence of functionals \((l_j)_{j\ge 1}\), that guarantee estimate (8.12). Then, we proceed with discussing what that would mean for our algorithm, defined in (5.4), and in particular, what that would mean for Example PS and Example FS.

*V*, we have that (8.12) holds with a constant \(C_V\) depending on

*V*. Indeed, if (8.12) fails then \(\mu (V,\mathcal{N}_m)=+\infty \) for all

*m*so by our previous remark \(V\cap \mathcal{N}_m\ne \{0\}\). The linear spaces \( V\cap \mathcal{N}_m\), \(m\ge 1\), are nested and contained in

*V*. If \(V\cap \mathcal{N}_m\ne \{0\}\) for all

*m*, then there is a \(v\ne 0\) and \(v\in \bigcap _{m\ge 1} \mathcal{N}_m\). This contradicts (8.13).

*V*. To derive such a uniform bound, we introduce the following notation. Given the sequence \((l_j)_{j\ge 1}\), we let \(L_m:=\mathrm{span}\ \{l_j\}_{j\le m} \) and \(L:=\ \mathrm{span}\{l_j\}_{j\ge 1}\) which are closed linear subspaces of \(\mathcal{X}^*\). We denote by \(U(L_m)\) and

*U*(

*L*) the unit ball of these spaces with the \(\mathcal{X}^*\) norm. For any \(0<\gamma \le 1\), we say that the sequence \((l_j)_{j\ge 1}\) is \(\gamma \)-norming, if we have

### Theorem 8.3

### Proof

*U*(

*V*) of

*V*in \(\mathcal{X}\) is compact, for any \(\delta >0\), there exists an

*m*such that

*m*, then for any \(v\in U(V)\) and any \(\eta \in \mathcal{N}_m\) we have

*V*to be \(V:=\mathrm{span}\{e_1\}\). We denote by \((e_j^*)_{j=1}^\infty \) the coordinate functionals, fix any number \(0<\gamma \le 1\) and define the linear functionals

### Remark 8.4

### Corollary 8.5

Let \(\mathcal{X}\) be any separable Banach space and let \((l_j)_{j\ge 1}\) be any sequence of functionals from \(\mathcal{X}^*\) which are \(\gamma \)-norming for some \(0<\gamma \le 1\). Then, we have the following results:

*m*, large enough, there is a choice \(\tilde{n}(m)\) such that

*C*an absolute constant, and the right side of (8.18) tends to zero as \(m\rightarrow \infty \).

In particular, both (i) and (ii) hold whenever \(\mathcal{X}\) is reflexive and \((l_j)_{j\ge 1}\) is total.

### Proof

*n*, there is an

*N*(

*n*) such that for \(m>N(n)\), we have \(\mu (V_{n},\mathcal{N}_m)\le 2\gamma ^{-1}\). Without loss of generality, we can assume that \(N(1)<N(2)<\ldots \), in which case we have \(\mu (V_{k},\mathcal{N}_m)\le 2\gamma ^{-1}\) for all

*k*, \(1\le k\le n\) provided \(m\ge N(n)\). We set \(\tilde{n}(m)=n\) for \(N(n)<m\le N(n+1)\), \(n=1,2,\ldots \). Note that \(\tilde{n}(m)\rightarrow \infty \) as \(m\rightarrow \infty \). Then, (i) follows from (8.6) since

*n*whose closure is dense in \(\mathcal{X}\). \(\square \)

While the spaces *C* and \(L_1\), are not reflexive, our two examples are still covered by Theorem 8.3 and Corollary 8.5.

**Recovery for Example PS** If the points \(P_j\) are dense in *D*, then the sequence of functionals \(l_j(f)=f(P_j)\), \(j\ge 1\) is 1-norming and Theorem 8.3 and Corollary 8.5 hold for this sampling.

**Recovery for Example FS** The sequence \((l_j)_{j\ge 1}\) of Fourier samples is 1-norming for each of the spaces \(L_p(\Omega )\), \(1\le p<\infty \), or \(C(\Omega )\) and hence Corollary 8.5 holds for this sampling.

We leave the simple details of these last two statements to the reader.

*C*if

*m*is sufficiently large.

*m*

*H*, and \(V_n=\mathrm{span}\{\phi _1,\ldots ,\phi _n\}\) (this is the space \(T_n\) according to the notation in [1]) for some given linearly independent elements \(\phi _j\in \mathcal{X}\). The Algorithm \(\tilde{A}_{n.m}:\mathbb {R}^m\rightarrow V_n\), considered in [1], is given by the formula

*w*from

*Z*with respect to the \(\Vert \cdot \Vert _M\) norm. Note that both norms, the Euclidean norm in \(\mathbb {R}^m\) and the \(\Vert \cdot \Vert _M\) norm are hilbertian and they are equivalent but different, so the recovery is also different.

*n*. Since \(C_{n,m}\) are converging to one when

*n*is fixed and \(m\rightarrow \infty \), they end up with the estimate

*n*. One can make

*C*arbitrarily close to one by choosing \(m=m(n)\) large enough.

*n*and choosing

*m*large that one should for each

*m*choose \(n=\tilde{n}(m)\) to establish convergence of \(A_{\tilde{n}(m),m}f \rightarrow f\), \(m\rightarrow \infty \), for each

*f*. Hence, in the case that \(\mathcal{X}\) is a Hilbert space the results in [1] and those in the present paper are essentially the same, although stated differently. However, the main point of the present paper is to handle the case of a general Banach space rather than only Hilbert spaces which is an issue not previously discussed in the sampling literature.

## 9 Choosing measurements

In some settings, one knows the space *V*, but is allowed to choose the measurement functionals \(l_j\), \(j=1,\ldots ,m\). In this section, we discuss how our results can be a guide in such a selection. The main issue is to keep \(\mu (\mathcal{N},V)\) as small as possible, and so we concentrate on this.

*M*so as to keep

**Case 1**Let us first consider the case when \(m=n\). Given any linear functionals \(l_1,\ldots ,l_n\), which are linearly independent over

*V*(our candidates for measurements), we can choose a basis for

*V*which is dual to the \(l_j\)’s, that is, we can choose \(\psi _j\in V\), \(j=1,\ldots ,n\), such that

*V*, and any projector onto

*V*is of this form. If we take \(M(v)=(l_1(v),\ldots ,l_n(v))\), we have

*V*with smallest norm, then take any basis \(\psi _1,\ldots ,\psi _n\) for

*V*and represent the projection in terms of this basis as in (9.1). The dual functionals in this representation are the measurement functionals.

*V*of a Banach space \(\mathcal{X}\) is a well-studied problem in functional analysis. A famous theorem of Kadec-Snobar [20] says that there always exists such a projection with

*V*of dimension

*n*, where (9.3) cannot be improved in the sense that for any projection onto

*V*we have \(\Vert P_V\Vert \ge c\sqrt{n}\) with an absolute constant \(c>0\). If we translate this result to our setting of recovery, we see that given

*V*and \(\mathcal{X}\) we can always choose measurement functionals \(l_1,\ldots ,l_n\), such that \(\mu (\mathcal{N},V)\le 2\sqrt{n}\), and this is the best we can say in general.

For a general Banach space \(\mathcal{X}\) and a finite dimensional subspace \(V\subset \mathcal{X}\) of dimension *n*, finding a minimal norm projection or even a near minimal norm projection onto *V* is not constructive. There are related procedures such as Auerbach’s theorem [37, II.E.11], which give the poorer estimate *Cn* for the norm of \(\Vert P_V\Vert \). These constructions are easier to describe but they also are not computationally feasible.

If \(\mathcal{X}\) is an \(L_p\) space, \(1<p<\infty \), then the best bound in (9.3) can be replaced by \(n^{|1/2-1/p|}\), and this is again known to be optimal, save for multiplicative constants. When \(p=1\) or \(p=\infty \) (corresponding to *C*(*D*)), we obtain the best bound \(\sqrt{n}\) and this cannot be improved for general *V*. Of course, for specific *V* the situation may be much better. Consider \(\mathcal{X}=L_p([-1,1])\), and \(V=\mathcal{P}_{n-1}\) the space of polynomials of degree at most \(n-1\). In this case, there are projections with norm \(C_p\), depending only on *p*. For example, the projection given by the Legendre polynomial expansion has this property. For \(\mathcal{X}=C([-1,1])\), the projection given by interpolation at the zeros of the Chebyshev polynomial of first kind has norm \(C\log n\), and this is again optimal save for the constant *C*.

**Case 2**Now, consider the case when the number of measurement functionals \(m>n\). One may think that one can drastically improve on the results for \(m=n\). We have already remarked that this is possible in some settings by simply doubling the number of data functionals [see (7.6)]. While adding additional measurement functionals does decrease \(\mu \), generally speaking, we must have

*m*exponential in

*n*to guarantee that \(\mu \) is independent of

*n*. To see this, let us discuss one special case of Example 1. We fix the domain

*D*to be the unit sphere in \(\mathbb {R}^n\), namely \(D=\{x\in \mathbb {R}^{n}\, :\ \sum _{j=1}^{n} x_j^2 = 1\}\) and the subspace \(V\subset C(D)\) of all linear functions restricted to

*D*, i.e., \(f\in V\) if and only if

*V*is an

*n*dimensional subspace. Since for \(f\in V\), we have \(\Vert f\Vert _{C(D)}=\Vert a\Vert _{\ell _2(n)}\), the map \(a\rightarrow f_a(x)\) establishes a linear isometry between

*V*with the supremum norm and \(\ell _2(n)\). Let

*M*be the measurement map given by the linear functionals corresponding to point evaluation at any set \(\{P_j\}_{j=1}^m\) of

*m*points from

*D*. Then

*M*maps

*C*(

*D*) into \(\ell _\infty (m)\) and \(\Vert M_V\Vert =1\). It follows from (7.4) that \(\mu (\mathcal{N},V)\approx \Vert M_V\Vert \cdot \Vert M_V^{-1}\Vert \). This means that

*A*is a \(\delta \)-net for a set

*S*(\(A\subset S\subset \mathcal{X}\) and \(\delta >0\)) if for every \( f\in S\) there exists \(a\in A\), such that \(\Vert f-a\Vert \le \delta \). For a given

*n*-dimensional subspace \(V\subset \mathcal{X}\) and \(\delta >0\), let us fix a \(\delta \)-net \(\{v_j\}_{j=1}^{ m}\) for \( \{v\in V\,:\,\Vert v\Vert =1\}\) with \( m\le (1+2/\delta )^n\). It is well known that such a net exists (see [15, Lemma 2.4] or [27, Lemma 2.6]). Let \(l_j\in \mathcal{X}^*\) be norm one functionals, such that \(1=l_j(v_j)\), \(j=1,2,\ldots , m\). We define our measurement

*M*as \(M:=(l_1,\ldots ,l_{ m})\), so \(\mathcal{N}=\bigcap _{j=1}^{ m} \ker l_j\). When \( \eta \in \mathcal{N}\), \(v\in V\) with \(\Vert v\Vert =1\), and \(v_j\) is such that \(\Vert v-v_j\Vert \le \delta \), we have

*M*, we have

### References

- 1.Adcock, B., Hansen, A.C.: Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon. Appl. Comput. Harm. Anal.
**32**, 357–388 (2012)MathSciNetCrossRefMATHGoogle Scholar - 2.Adcock, B., Hansen, A.C., Poon, C.: Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem. SIAM J. Math. Anal.
**45**, 3132–3167 (2013)MathSciNetCrossRefMATHGoogle Scholar - 3.Adcock, B., Hansen, A.C., Shadrin, A.: A stability barrier for reconstructions from Fourier samples. SIAM J. Numer. Anal.
**52**, 125–139 (2014)MathSciNetCrossRefMATHGoogle Scholar - 4.Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings and Applications. Springer, Berlin (2009)Google Scholar
- 5.Bartle, R., Graves, L.: Mappings between function spaces. Trans. Am. Math. Soc.
**72**, 400–413 (1952)MathSciNetCrossRefMATHGoogle Scholar - 6.Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 48. American Mathematical Society Colloquium Publications, Providence, RI (2000)MATHGoogle Scholar
- 7.Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for Greedy Algorithms in reduced basis methods. SIAM J. Math. Anal.
**43**, 1457–1472 (2011)MathSciNetCrossRefMATHGoogle Scholar - 8.Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.:
*Data Assimilation in Reduced Modeling*, SIAM UQ, to appear; arXiv: 1506.04770v1 (2015) - 9., B.: Optimal Recovery of Functions and Integrals, First European Congress of Mathematics, Vol. I, pp. 371–390 (1992), Progr. Math., Birkhauser, Basel,
**119**(1994)Google Scholar - 10.Brown, A.: A rotund space have a subspace of codimension 2 with discontinuous metric projection. Mich. Math. J.
**21**, 145–151 (1974)CrossRefMATHGoogle Scholar - 11.Cohen, A., DeVore, R.: Approximation of high dimensional parametric PDEs. Acta Numer.
**24**, 1–159 (2016)MathSciNetCrossRefMATHGoogle Scholar - 12.Demanet, L., Townsend, A.: Stable extrapolation of analytic functions, preprintGoogle Scholar
- 13.DeVore, R., Lorentz, G.: Constructive Approximation, vol. 303. Springer, Grundlehren (1993)MATHGoogle Scholar
- 14.Figiel, T.: On the moduli of convexity and smoothness. Stud. Math.
**56**, 121–155 (1976)MathSciNetMATHGoogle Scholar - 15.Figiel, T., Lindenstrauss, J., Milman, V.: The dimension of almost spherical sections of convex bodies. Acta Math.
**139**, 53–94 (1977)MathSciNetCrossRefMATHGoogle Scholar - 16.Fournier, J.: An interpolation problem for coefficients of \(H^\infty \) functions. Proc. Am. Math. Soc.
**42**, 402–407 (1974)MathSciNetMATHGoogle Scholar - 17.Garkavi, A.: The Best Possible Net and the Best Possible Cross Section of a Set in a Normed Space, Vol. 39, pp. 111–132. Translations Series 2. American Mathematical Society, Providence, RI (1964)Google Scholar
- 18.Hanner, O.: On the uniform convexity of \(L^p\) and \(\ell ^p\). Ark. Matematik
**3**, 239–244 (1956)MathSciNetCrossRefMATHGoogle Scholar - 19.Henrion, D., Tarbouriech, S., Arzelier, D.: LMI approximations for the radius of the intersection of ellipsoids: survey. J. Optim. Theory Appl.
**108**, 1–28 (2001)MathSciNetCrossRefMATHGoogle Scholar - 20.Kadec, M., Snobar, M.: Certain functionals on the Minkowski compactum. Mat. Zamet.
**10**, 453–457 (1971). (Russian)MathSciNetGoogle Scholar - 21.Konyagin, S.: A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center. Mosc. Univ. Math. Bull.
**43**, 55–56 (1988)MathSciNetMATHGoogle Scholar - 22.Lewis, J., Lakshmivarahan, S., Dhall, S.: Dynamic Data Assimilation: A Least Squares Approach, Encyclopedia of Mathematics and its Applications, vol. 104. Cambridge University Press, Cambridge (2006)CrossRefMATHGoogle Scholar
- 23.Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
- 24.Maday, Y., Patera, A., Penn, J., Yano, M.: A parametrized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics. Int. J. Numer. Method Eng.
**102**, 933–965 (2015)CrossRefMATHGoogle Scholar - 25.Micchelli, C., Rivlin, T.: Lectures on optimal recovery, numerical analysis, Lancaster 1984 (Lancaster, 1984), 21–93. Lecture Notes in Math, vol. 1129. Springer, Berlin (1985)Google Scholar
- 26.Micchelli, C., Rivlin, T., Winograd, S.: The optimal recovery of smooth functions. Numerische Mathematik
**26**, 191–200 (1976)MathSciNetCrossRefMATHGoogle Scholar - 27.Milman, V., Schechtman, G.: Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin (1986)Google Scholar
- 28.Powell, M.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)MATHGoogle Scholar
- 29.Platte, R., Trefethen, L., Kuijlaars, A.: Impossibility of fast stable approximation of analytic functions from equispaced samples. SIAM Rev.
**53**, 308–318 (2011)MathSciNetCrossRefMATHGoogle Scholar - 30.Repovš, D., Semenov, P.: Continuous Selections of Multivalued Mappings. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
- 31.Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Die Grundlehren der mathematischen Wissenschaften, vol. 171. Springer, Berlin (1970)CrossRefGoogle Scholar
- 32.Schönhage, A.: Fehlerfort pflantzung bei interpolation. Numer. Math.
**3**, 62–71 (1961)MathSciNetCrossRefMATHGoogle Scholar - 33.Smith, P., Ward, J.: Restricted centers in \(C(\Omega )\). Proc. Am. Math. Soc.
**48**, 165–172 (1975)MathSciNetMATHGoogle Scholar - 34.Szarek, S.: On the best constants in the Khinchin inequality. Stud. Math.
**58**, 197–208 (1976)MathSciNetMATHGoogle Scholar - 35.Traub, J., Wozniakowski, H.: A General Theory of Optimal Algorithms. Academic Press, London (1980)MATHGoogle Scholar
- 36.Trefethen, L.N., Weideman, J.A.C.: Two results on polynomial interpolation in equally spaced points. J. Approx. Theory
**65**, 247–260 (1991)MathSciNetCrossRefMATHGoogle Scholar - 37.Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
- 38.Zygmund, A.: Trigonometric Series. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar