Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.


Introduction
Diagonalization of operators is deeply tied to the linear character of the operators themselves. Indeed, interest in obtaining the eigenvectors of a given linear map implies that we can use them, together with the corresponding eigenvalues, to obtain an easy representation of the operator. If we think of the finite dimensional case, all symmetric operators are diagonalizable, so that for all of them one can find, after a change of basis, a diagonal form that gives the simplest representation of the map. Thus, a fruitful diagonalization theory has to provide a way to compute the eigenvectors, as well as a structural rule to extend the representation of the map (or at least an approximation) from the eigenvectors to the whole space. This is the point of view from which we develop the research presented in this paper: we consider the set of eigenvalues of a given map and use a given rule to approximate the original map starting from them.
What is fundamental in the present work is that we do not assume linearity, at least in its "full version". Instead, we will use metric notions, approximating the value of the original endomorphism T : E → E in the Banach space E over complex field by using the "diagonal behavior" of functions at other points. The proximity of any other vectors in E to such points will then be used to find suitable approximate expressions. This is why the Lipschitz character of the functions involved is required, since we have to perform the extension/approximation formulas from the set of eigenvectors ev(T) of T to the whole normed space E. On the other hand, to be Lipschitz continuous seems to be a natural requirement for talking on the spectrum of a non-linear operator (see [1], Introduction, p. 3). Much research has been done for finding results on the spectrum of a Lipschitz operator that could be similar to those that hold in the linear case [1].
Of course, this topic is not new [1,2]. A lot of research has been done, from the abstract and the applied points of view, to understand what diagonalization would be for non-linear maps, and a lot of applied procedures to solve concrete issues appearing in relation to this have also being published (which are normally centered on "almost" rule", and in Section 5 we explain what we call the "convexity rule". We finish the paper by providing some hints (Section 6) on how effective computations can be performed for finding approximations to eigenvectors and eigenvalues of Lipschitz maps.
As a main theoretical framework, we adopt the usual mathematical context of the non-linear spectral theory; we use [1] as reference book for this. Standard definitions and arguments of measure and integration theory, as well as Banach spaces, will be used [22][23][24]. Our main reference on Lipschitz maps is [25]. For the aim of simplicity, we will consider only the real case. If E is a Banach space with norm · , we write E * for its dual space, which in the (finite dimensional) Euclidean space and in the (real) Hilbert space can be identified with E. We will write x for the vectors in E and x * for the elements of E * . The unit ball of the dual space B E * is a convex set that is compact with respect to the weak* topology (Alaoglu's Theorem; see, for example, [22]). Then, we can consider the space M(B E * ) of regular Borel measures on B E * , which will play a relevant role in the present paper. Let us recall that, for any x * ∈ E * , the Dirac's delta δ x * belongs to this space. In the same way, if E is a dual space or E is reflexive, we will write B(B E ) for the corresponding Borel sets. This notion can be extended to any norm bounded subset of E instead of B E by including it in a larger ball. As usual, we denote by S E to the unit sphere of E.

Approximation Results for Linear and Non-Linear Maps and First Results for Lipschitz Functions
The aim of this section is to explain the motivation of our investigation and to provide the first general results. Let us start by presenting how we can rewrite the well-known context of diagonalization or linear operators in terms of integral expressions, under the formalism provided by the recently introduced eigenmeasures [26]. We expect to find the best diagonal approximation to a given (linear and continuous) operator T : E → E. Following the mentioned paper, we introduce first a measure on B E * . We start by considering a (regular Borel) probability measure ν ∈ P (B E * ). Roughly speaking, this gives formal support for those elements of the dual space that we want to keep "active", i.e., those whose actions on the elements of E we are interested in "keeping running". In the linear case, this set can actually be quite small; in the case of dual Banach spaces with unconditional basis (or finite-dimensional spaces), we choose the simplest: take an unconditional normalized basis {e * k } +∞ k=1 of the dual space and define ν as Since case E is a (real) separable Hilbert space or a finite dimensional Euclidean space, we choose the same measures, but the basis are assumed to be also orthonormal.
In the general case, as will be shown later on, this formalism provides a useful tool for choosing renormings for the Hilbert space that is considered by means of an inclusion x E → x, · ∈ L 2 (ν), and so every element x is identified with the function x, · acting on the elements of the unit ball of the dual space. It is clearly seen that this map is always continuous: we also need it to be injective to obtain an alternate representation of the original space. Throughout the paper, we will write · ν for the norm induced on E by the norm of L 2 (ν), that is, Let us fix now the measure structure on the measurable space (B E , B(B E )) that is needed to complete the integral diagonal representation of the approximation to the operator T. Let us recall the definition of weak eigenmeasure (Definition 2 in [26]), adapted for the case of linear maps. Definition 1. Let E be a reflexive Banach space and let T : E → E be an operator. Fix a Borel regular measure ν on B E * , and let µ be a Borel regular measure on a bounded subset of E. We say that µ is a weak eigenmeasure for T if for each N and the functions λ N (x) are uniformly bounded in N and x ∈ E.
In this paper, we will use a stronger version of this notion, which has been also introduced in [26]. We say that a probability measure satisfying the requirement above is an eigenmeasure if λ N (x) does not depend on N, that is, it has a constant value once x is fixed. Note that the linearity requirement for T is not necessary for this definition. The same notion can be defined in the same way for the case of Lipschitz maps, and in general for any function that satisfies the conditions necessary for all integrals to make sense.
It can be seen that the existence of an eigenmeasure µ implies a diagonal representation µ-a.e. That is, for µ-a.e., all elements x ∈ B E , once the set N is fixed, we have a diagonal representation of the operator. Indeed, consider the set ev(T) of all the vectors that satisfy that there is a scalar value λ such that T(x) = λx : we have that, if µ is an eigenmeasure, µ(A \ ev(T)) = 0 for every measurable set A ⊂ E. In this paper, we intend to show to what extent this tool can be used to obtain quasi-diagonal representations of T for the case of linear operators. We will tackle this question in the following subsection for the canonical case, i.e., the Euclidean case in which v is given by the average of Dirac's deltas of the elements of an orthonormal basis.

Motivation: Minimization Procedure and Calculation of the Best Diagonal Approximation for a Linear Operator in a Euclidean Space
In this section, we show the canonical case, which will allow us to explain the geometric meaning of the diagonal approximation of an operator. Let us take the real n-dimensional Euclidean space E. Its dual space can be identified with E, so we do not write the superscript * for any space or any element of the orthonormal basis. Thus, for a given orthogonal basis {e 1 , . . . , e n } fix the measure Note that, for this case, we have that for every x ∈ E, where x k , k = 1, . . . , n, are the coordinates of x in the fixed basis. Let us compute the optimal value for every element x 0 of the sphere S E , which is represented by the Dirac's measure x 0 → δ x 0 . The stochastic eigenvalue equation is then written, for x 0 ∈ S E and every N ∈ B(B E ), as For example, for N = {e k }, k = 1, . . . , n, we obtain the equations x 0 ,e k if x 0 , e k = 0, and λ k = 0 otherwise. Let us show an estimate of the best λ for which the equation T(x) = λ(x) x holds for a given x ∈ E in the integral setting that has been fixed above. In other words, we want to minimize the error The result has a clear geometric meaning: once the point x ∈ S E is fixed, the best diagonal approximation to T(x) is the projection of T(x) on the subspace generated by x. It is given by a direct minimization argument, summarized in the next result. Proposition 1. Consider the measure ν fixed at the beginning of this section for the n-dimensional Euclidean space. Let x ∈ E. Then, the following assertions hold.
(i) The best diagonal approximation to T in x is given by T(x), e k x k .
(ii) The minimal integral error ε(x) is given by Proof.
(i) Since the basis {e k } n k=1 is orthonormal, we have that the error ε(x) can be written as Therefore, we can rewrite the error as Recall that the vector x is fixed. The critical points equation involving the derivative with respect to λ are x / x 2 . Since d 2 ε(x) dλ 2 = 2 x 2 /n > 0, this value of λ gives a minimum. Therefore, the best diagonal approximation for (ii) Just writing this value of λ in the expression of the error, we find that the minimal value of the error at x is given by The best approximation to a diagonal form for any vector x in the space given by λ(x) provides a direct way of representation of an operator. Of course, the only vectors for which the error ε equals 0 are the eigenvectors of the operator T; clearly, we have that a linear operator T : E → E is diagonalizable if and only if there is a basis {x k } n k=1 such that ε(x k ) = 0 for all k = 1, . . . , n.
Let us remark that, under the assumption of the existence of an orthonormal basis of eigenvectors for T, an integral representation formula is provided: take such a basis {e k } n k=1 and consider an associated eigenmeasure µ that coincides with the measure ν that have been considered above. Then, for every x 0 ∈ E we have that where µ(·) = 1 n ∑ n i=1 δ e k (·). Therefore, we can write a representation of a diagonalizable operator T as a Bochner integral average associated with the diagonal function x → λ(x) · x. This is the linear version of the main idea that we use in the subsequent sections of the paper: even for non-linear cases, an integral approximation to a function can be obtained as an integral average, once a(n) (approximated) eigenmeasure is known.

General Finite Dimensional Case: Local Diagonal Approximation for Functions T : E → E
In this section, we show the general equations for the finite dimensional case, without linearity assumptions: no restriction is imposed on the function T, apart from some weak measurability properties. For the sake of clarity, we revert to the notation E for the original finite dimensional space and E * for its dual, which is isomorphic to it because it is finite dimensional.
Let us fix again the measures for getting an integral representation of an equivalent norm for E. Take regular Borel probability measures ν and µ on the spaces S E * and S E , respectively. Suppose that the probability measure ν separates the points of S E , that is, it satisfies that if x ∈ S E we have B E * | x, x * | 2 dν(x * ) = 0. This requirement implies the injectivity of the representation provided by the norm in L 2 (ν) that was explained at the beginning of this section, and it is enough to get that the norm gives an equivalent norm for the space (E, · ). Let us write (E ν , · ν ) for the resulting normed space, which inherits a Hilbert space structure from the inclusion E → L 2 (ν) given by x → x, · . This is the key for the introduction of stochastic arguments for the estimate of diagonal approximations. However, note that the dual space (and thus its unit ball) that we consider in this section is endowed with the natural norm of (E, · ) * , that is, Let us start as in the linear case, considering the diagonal error, that is, the measure of how far the operator is at point x of being diagonal. Given a vector x ∈ S E ν , the error committed when T(x) is approximated as a line following the direction of x is This provides a simplified error formula, which generalizes the geometric idea based on the projection on x of T(x) explained in the previous section. The computations that give the proof are similar to the ones that prove Proposition 1 but consider the duality relation in L 2 (ν) instead of that in the original space E. Proposition 2. Let x ∈ E be fixed. Then, the error ε(x) attains its minimum value for and has the minimum value for this λ(x) given by We are ready to give the local version of the diagonal approximation for T. Recall that we are searching for an approximation provided by an integral average involving measures on the Borel subsets of E, mainly eigenmeasures. This makes it relevant to obtain integral formulas for subsets A that, in the next sections, will be neighborhoods of the point on which we focus our attention.
Thus, given a measurable subset A ⊂ E, the error committed when the function T is approximated as a diagonal integral average is given by The proof of the next result also follows the same arguments than the ones of Propositions 1 and 2.

Proposition 3. Let A ⊆ E be a non-µ-null measurable set. Then, the minimal average error ε(A)
for the diagonal approximation is given by the formula and it is attained for the value Note that, for A ⊂ S(E), this formula can be rewritten as an integral average, This expression suggests the arguments used in the next sections.

Approximation Formulas Based on Eigenmeasures: Bochner Integration for Lipschitz Maps
Let us present the integral approximation procedure that can be constructed by adapting the previous framework for the case of Lipschitz functions. Thus, let us describe now how to get an approximation based on eigenmeasures in this case. In [26], it is shown, for the case of bilinear maps, how the calculus of eigenmeasures can be used to deal with non-linear diagonalization. Here, we assume that such an eigenmeasure µ is given for a certain non-linear map f : E → E, where E is a (finite dimensional) Euclidean space, and we show how we can use it to obtain an approximation to f . In order to do this, we change the context of scalar integration used before, which has been useful for the computation of the error and to obtain the best diagonal pointwise approximation for a given map, by the vector-valued Bochner integration.
The idea is to consider f as a Bochner (locally) integrable function (see, for example, [23], Chapter II). In order to do this, if f : E → E is a strongly measurable function (see [23], Chapter II), we consider the vector-valued function A ⊂ E → E given by x → f (x) as a Bochner integrable function with respect to the measure µ. Integration of such a function provides a (countably additive) vector measure, which assures convergence of the limits of the integrals on sequences of disjoint sets (see [23,24]). In order to get the desired integral representation, we only need f to be bounded in the sets A, which will be a certain class of neighborhoods of the points in the Euclidean space E.
Definition 2. Let f : E → E be a Lipschitz function, and let µ be an eigenmeasure for f . We define the eigenmeasure approximation of a strongly measurable Lipschitz function for a given > 0, under the requirement µ(B (x 0 )) > 0, by the formula Let us write λ f (·) and λ f (·) for the best approximated eigenvalue fixed for the functions f and f , respectively. We need a previous lemma for the computation of the error committed when we approximate the value of f (x) by its projection on the subspace defined by other vector x 0 . It is similar to Proposition 2, but in this case we are considering two different points, x 0 and x. Lemma 1. Let x 0 , x ∈ E and f : E → E satisfying all the requirements of integrability explained above. The best value of λ that minimizes the error committed when we approximate f (x) by its projection on the subspace generated by x 0 is given by Proof. The computations are again similar to the ones that prove Proposition 1. We have that and so, by computing the value of the derivative with respect to λ to be equal to 0, we obtain Remark 1. Lemma 1 allows one to introduce a new concept, in the line of the integral means that the eigenmeasures formalism provides. If x 0 ∈ E, we can consider the integral average λ x 0 of the pointwise diagonal approximation to f , which is given by where the function λ x 0 (x) is given in Lemma 1.
This value does not allow one to find a new eigenvector but provides the natural way of controlling how far a given vector x 0 is from being an eigenvector for the function f . Next result covers the relevant information about the behavior of the eigenmeasure approximation introduced in this section: expected error, both in the pointwise approximation to the function f and in the corresponding eigenvalues, and bounds for the distance among the value of the approximating at a given point x 0 and its optimal diagonal form. Note that even for a linear and continuous operator, it may happen that the set of eigenvectors is not sufficient to represent the operator by eigenmeasures: a rotation in R 2 does not have eigenvectors. Theorem 1. Let ν be a regular Borel probability measure on (B E , B(B E * )). Consider a Lipschitz map f : E → E with Lipschitz constant Lip( f ) and assume that ev( f ) is non-trivial. Consider an eigenmeasure µ for f , and suppose that f is a strongly measurable function with respect to µ.
Fix a vector x 0 ∈ E, and let ε > 0 such that µ(B (x 0 )) > 0. Then, Proof. Again a rewriting of the arguments around Theorem 1 gives the result. Let us show them step by step.
(1) It is just a consequence of the integral domination provided by the Lipschitz inequality for f . Indeed, (2) Let us recall that for any function f , and at any point x 0 , the optimal value for λ We have to use the property of the Bochner integration that assures that, if φ is a Bochner integrable function with respect to a measure η, φdη, x * = φ, x * dη (see, for example, [27] (Lemma 11.44), or [23] (Chapter II)). Consequently, Therefore, using Fubini's Theorem, we get (3) Using Hölder inequality, we have that Thus, applying the equation for the best approximation provided by Lemma 1 and doing some standard calculations, we obtain that the minimal expression for this bound is given by

Remark 2.
The eigenmeasure approximation gives a direct formula to compute diagonal-based representation of operators. Note that, the smaller is, the better the approximation, but cannot in general converge to 0. Indeed, in this case it may happen that ev( f ) ∩ B (x 0 ) is empty, and so by the definition of eigenmeasure, µ(ev( f ) ∩ B (x 0 )) = 0, which implies that the integral formula that gives f (x 0 ) equals 0.
Notice that this provides a way to approximate a Lipschitz function f once the set of eigenvectors of f is known, at least partially. Since we are thinking about computational applications supported by these ideas, we also need a numerical way of finding an approximation to the set ev( f ). One of the most straightforward methods would be the minimization of the functions | f (x) − λ(x)x|, which can be performed by sampling methods and Monte Carlo approximations of the integrals involved. Although explanations of these procedures and the algorithms involved are not part of the content of this paper, some hints are given in the last section.

Diagonalizable Non-Linear Operators: Examples and Situations
Let E be a Banach space. Suppose that T : E → E is a (non-necessarily linear) map such that the set of its eigenvectors ev(T) is not empty. For every non-empty subset S ⊂ ev(T), consider the restriction T S : S → E. We will say that a procedure for extending the value of the function T from ev(T) to the whole space E is an extension rule (ER) if for every non-empty subset S ⊂ ev(T), (ER) produces a unique extension T S ER : Application of the usual linear extension rule for linear operators gives that such equality can be achieved exactly. However, in general we cannot expect to find a complete representation and only approximate expressions can be obtained. We will look for such approximations in the subsequent sections, after showing some basic examples in the present one.

Linear and Quasi-Linear Rules
Let us explain a general setting for using the usual framework of spectral representation of linear operators. In the finite dimensional case, diagonalization of a linear map is essentially given by linearity, once a basis of eigenvectors is obtained. If the operator allows a diagonal representation, the extension rule is then provided by the linear combination of eigenvectors, which define a basis of the space. Let us see how this can be extended to the nonlinear case.
Let T : E → E, where E is a Banach space with an unconditional (or finite) normalized basis B := {b 1 , . . . , b k , . . . : k = 1, . . . , n}, where n ∈ N ∪ {+∞}. Assume that {τb k : τ ∈ R} ⊂ ev(T) for every k. Consider the projections P b k (v) = α k (v) b k , v ∈ E, and take a probability measure as µ = ∑ +∞ i=1 β k δ b k , for β k ≥ 0. Then, T is diagonalizable under a linear rule conditioned to the basis B, if we can write by means of the integral formula Note that µ({b k }) = β k , and this term does not appear in the integral if β k = 0. This expression can be rewritten, for the finite dimensional case, as

Example 1.
Let us see a particular example. Take E = R 2 and the canonical basis If we write the eigenvalue equations we directly get that (x 1 , 0) and (0, x 2 ) are eigenvectors for all x 1 , x 2 ∈ R, and the corresponding eigenvalues are |x 1 | and |x 2 |, respectively. That is, all the vectors in the subspaces {τb 1 : τ ∈ R} and {τb 2 : τ ∈ R} are eigenvectors of T. Moreover, note that for we obtain that the vectors as (x, x) and (x, −x) for x ∈ R are also eigenvectors with eigenvalues |x|. Figure 1 provides a representation of the eigenvectors of T.

Consider the orthonormal projections
Then, T is diagonalizable under the linear rule since According to the given definition, this map is diagonalizable under a linear rule, the one given by this equation. Let us note that there are other eigenvectors that are not used in the representation, with which we could also define a basis of the space.

Example 2.
Linearization techniques for nonlinear operators are interesting tools used in many applied contexts. A relevant example is given by the so-called Koopman nodes, which allow one to treat nonlinear systems by means of linear procedures. This method is used to find spectral representations of observables in dynamical systems (see, for example, [28]). In [29], Koopman nodes are used to describe the behavior of complex nonlinear flows, in the context of fluid mechanics.
Let us briefly explain this procedure. Consider a dynamical system given by a function f : E → E, f (x k ) = x k+1 . The Koopman operator is a linear operator U that acts on the space of all scalar functions F : X → R as U(ϕ)(x) = ϕ( f (x)). Since U is linear, we can consider its diagonalization in order to obtain a set of eigenfunctions ϕ j associated with eigenvalues λ j ∈ C belonging to a certain function space. Consider an observable g, that is, a function g : E → R p for a given natural number p. If a countable number of eigenfunctions is sufficient to act as a basis of the subspace to which the p coordinate functions of g belong, we can expand g as where the p coordinates of the vectors v j are the corresponding coefficients of the representations on the system {ϕ j } ∞ j=1 of each coordinate function of g. The functions ϕ j are called Koopman eigenfunctions, and the vectors v j the Koopman modes of the map f .
Our setting is similar in various aspects to the ones of the Koopman method, but we use measures and integral representations to facilitate the further use of integral averages to get approximation formulas. Let us see what happens in the case where the function f is a symmetric linear map f : E → E on the Euclidean space E. Take an orthonormal basis of eigenvectors v 1 , . . . , v n for f with eigenvalues λ 1 , . . . , λ n , and the dual basis w 1 , . . . , w n of functions of the dual space E * . Note that in this case the Koopman operator U : Take now the identity map I : E → E as g : the expansion formula is x, w j v j , that is, the Koopman nodes are in this case the eigenvectors of f , and we get a representation of f as Using our finite measure integration formalism, we can consider the measure ν = 1 n ∑ n j=1 δ w j and the eigenmeasure µ = 1 n ∑ n j=1 δ v j to get an integral representation of f as the Bochner integral where λ(x) = λ j for x = v j and 0 otherwise. This gives a result equivalent to that obtained by the Koopman operator method. This procedure, which can also be applied in the nonlinear case by following the same steps, shows another example of a linear rule; other examples of application of Koopman nodes to nonlinear problems (and which could be rewritten in our terms) can be found in [29] (see Section 2.2 for application to periodic solutions of dynamical systems).

Example 3 (Quasi-linear rules).
More sophisticated adaptations of the linear rule could be necessary to cover more complex applications to non-linear operators of interest in physics and other sciences. For example, the moment of inertia is a fundamental magnitude that is used in mechanical systems, and sometimes its time dependence has to be modelled, for example, in the description of the vibratory motion [30]. Let us show a related operator here. Let us consider a 2-dimensional system and suppose that the time dependence of an moment of inertia is given by the operator T : R 3 → R 3 (the third coordinate for the variable time t), given by the formula where x and y are the usual Cartesian coordinates of the points in R 2 , and the third coordinate represents a weighted counter of the time t. After a diagonalization process of T, the analyst can get (time-dependent) information about the principal axis of inertia of the physical systems that are represented by this operator. As in the linear case, a direct computation involving the characteristic polynomial of the matrix above provides eigenvalues of T. Indeed, the (time-dependent) solutions of the equation are λ 1 (t) = 2t − 1, λ 2 (t) = 2t + 1, and λ 3 (t) = 3. Note that the system is not linear, so no information about the general structure of the sets of eigenvectors is available. However, the vectors (1, −1, 0), (1, 1, 0), and (0, 0, 1) provide a basis for R 3 of eigenvectors of T associated with λ 1 , λ 2 , and λ 3 , respectively. If we write x , y , and t for the coordinates of the vectors in R 3 with respect to this new basis, we get the change of variables are given by the equations x = (x − y)/2, y = (x + y)/2 and t = t . A straightforward calculation shows that it provides an "integral representation" of T as Thus, a quasi-linear "integral representation" as has been obtained, where λ i (v) is the eigenvalue associated with e i depending on v, and where P e i (v) gives the coordinate of v in the new basis.

Topological Rules
Although the approximation of a linear operator is essentially a topological question, the usual methods and techniques are strongly influenced by linearity. Point approximation can also be performed using sequences of locally constant functions, and, in fact, these functions are primarily used for approximation in the context of vector-valued integrable functions. These functions can usually be written as pointwise limits of simple functions, provided they belong to any Köthe-Bochner function space under some weak requirements of density or order continuity.
Let us recall the definition of (strong) eigenmeasure µ for an operator, applied to the context of functions f : E → E, that will be useful in the present section. Although the definition provided in [26] is rather abstract, for the aim of this paper we can use the following version: let B(E 0 ) be the sigma-algebra of Borel sets of a multiple E 0 of the unit ball of the Banach space E associated with the norm topology. A measure µ : B(E 0 ) → R + is an eigenmeasure for a Bochner µ-integrable map f : E 0 → E (that is, the restriction of f : E → E to E 0 ) if it is a regular Borel probability measure such that Note that, in this case, the Bochner integral of f (x) can be used to reconstruct the diagonal part of the function f by the integral equations In the case that the set ev( f ) is finite, µ can always be defined as a proper convex combination of Dirac's deltas δ x i , x i ∈ ev( f ). Let us use these integral equations for the next construction.
Consider a function f : R n → R n , n ∈ N, which has a finite set of eigenvectors ev( f ) = {x 1 , . . . , x m }. Define a probability measure µ : B(E 0 ) → R + by Let us define the metric rule for the extension of such a function f 0 : ev( f ) → R + as follows: for every point x ∈ E, we compute the (measurable) set Then, we define the extension f by Therefore, we say that a function f : E → E with a finite number of eigenvectors is diagonalizable under a metric rule if there is an eigenmeasure µ such that Example 4. Consider the function φ : (R 2 , · 2 ) → (R 2 , · 2 ) given by and having in the diagonal lines the averages of the values of the adjacent sectors. Thus, it is a constant function inside any of the regions defined by the diagonal lines in Figure 2, for which the minimal distance to any of the reference points (1, 0), (−1, 0), (0, 1), and (0, −1) provides the value of the constant image vector. The description of the function φ is easy: if we divide R 2 into four sectors by the diagonal lines that intersect at (0, 0) that can be seen in Figure 2, we have that the function is constant in each of them, and it is also constant in the diagonal lines themselves.
It can also be equivalently defined, as explained in the explanation of the metric rule above. Consider the points e 1 = (1, 0), e 2 = (0, 1), e 3 = (−1, 0) and e 4 = (0, −1). It is clear that they are eigenvectors for φ with eigenvalues equal to 1. Consider the eigenmeasure µ(·) = 1 4 ∑ 4 k=1 δ e k (·). For each point x that is not in the diagonal lines, there is a point e k of these ones such that the set M x is given by M x = {e k } since the distance to these points actually defines in which sector the point x is (see Figure 2). Therefore, for every element x inside of a sector, we apply the linear rule Case x belongs to the diagonal lines, but it is not (0, 0); we have that there are two eigenvectors e k and e j that are at the same distance to x, getting in this case M x = {e k , e j }, and so Finally, M (0,0) = {e 1 , e 2 , e 3 , e 4 } and then

Example 5.
This easy procedure can be adapted to other situations to approximate non-linear operators using the minimal distance criterion. The idea is to use the suitable inequalities appearing in the description of the operator to control the error committed. For instance, in [31] the authors analyze a nonlinear multiparameter eigenvalue problem arising in the theory of nonlinear multifrequency electromagnetic wave propagation. Under certain conditions, the problem can be written as a system of n nonlinear single parameter eigenvalue problems, with linear and non-linear components. It is proved that the corresponding linear problems have a finite number of positive eigenvalues; however, the nonlinear ones have infinite positive eigenvalues, which provide eigentuples of the original multiparemeter problem not related to the solutions of the linear one-parameter problems. After some transformation, an eigenvector equation appearing in the formalism is defined as with f belonging to the space of functions with continuous second derivatives (see Equation (6) in [31]; see also Equation (15) in [32]). We can then search for the eigenvectors of the operator T( f ) = f + α f 3 + f . Once we get a set ev(T) 0 of these eigenfunctions, we can find a naive approximation to T( f ) the value T(u) for the function u ∈ ev(T) 0 that minimizes a certain "error function", using the inequality where we consider for example the norm · 2 of function space L 2 . Suppose that we can assume that map g → g is a Lipschitz map when considered from ev(T) 0 ∪ { f } to L 2 and that all these functions are pointwise uniformly bounded in their domains by a constant Q. Then, we can get a Lipschitz type estimate for T( f ) − T(u) 2 as for R = K + |α|3Q 2 + . The best approximation by the metric rule would be given in this case by f ≈ λ u u, where u is the function in ev(T) 0 for which the minimal L 2 distance to f is attained. A more precise calculation would lead to a domination by the norm of the Sobolev space W 2,2 .
There are a lot of nonlinear problems directly connected to differential operators arising in physics, biology, economics, and other sciences. Just to mention some recent developments, the papers [33,34] study the principal eigenvalue of p-biharmonic and (p, q)-biharmonic operators, which is applied to various differential equation problems appearing in physics. More specific results could be obtained by setting some requirements on the functions involved beforehand. Deeper assumptions of the functional setting would provide more possibilities to find more concrete applications, both in terms of how to obtain the eigenvectors of the operators and the general metric structure of the function spaces under consideration. For example, fixed-point results are often used in the search for eigenvalues of nonlinear functions, and more general notions of distance and function spaces can be used, including order-based tools [35][36][37][38].

The Lipschitz Case
Let us show now how a diagonalization procedure can be implemented in the context of Lipschitz maps. Obviously, the essential requirement in this case is that the operators to be diagonalized have to be Lipschitz; let usnote the example of application of a linear rule proposed in Section 3.1 (Example 1 is not Lipschitz, so the method presented here is not an extension of a linear-type rule). In fact, metric topology is used instead as it corresponds to the use of Lipschitz operators, but an additional tool is available in this case, so this situation is not an extension of the metric rule shown in Section 3.2: note that Example 4 is not given by a Lipschitz map.
Let us show a non-trivial case of diagonalizable Lipschitz map.
Example 6. Let E be a Banach space, and consider the function T : E → E given by Let us show that T is a Lipschitz map. First, suppose that y ≤ 1. Then, we have that Of course, the same computations give the same result in case x ≤ 1. Finally, if x > 1 and y > 1, we obtain which gives that T is Lipschitz with constant K ≤ 2.
The main property of the function T is that it is "absolutely" diagonalizable: indeed, all the vectors in the space are eigenvectors, and for each of them the eigenvalue is given by min{1, x }. Similar examples could be obtained for special operators defined as T(x) = f (x) · x, with f being a real bounded Lipschitz function.

Approximation of Lipschitz Maps by the Proximity Rule
In case we have no more information than that provided by the fact that T is Lipschitz, we have to apply direct algorithms for obtaining an approximation to T. Let us show two methods to do it.

The Minimal Distance Extension of T| ev(T)
Take a strictly decreasing sequence of positive numbers ( i ) +∞ i=1 such that lim i i = 0, and consider an eigenmeasure µ for T. We can get an approximation to T as follows. We have to assume that norm-bounded subsets of ev(T) are relatively norm-compact, which is obviously true if E is finite dimensional. • Fix an element x ∈ E and define d 0 = inf{ x − y : y ∈ ev(T)}. Consider the set We consider the value of the first extension of T as λ y 1 · y 1 , where λ y 1 is the eigenvalue associated with y 1 . • Repeat the procedure to get a sequence (y i ) n i=1 in ev(T) with associated eigenvalues (λ y i ) +∞ i=1 . By the assumption on compactness, and taking into account that the sequence of vectors is bounded, there is a subsequence that converges to an element y. Let us write again (y i ) +∞ i=1 for the subsequence. Note that for each i ∈ N, we have and lim i y i − y = 0. This allows one to define the extension T as T(x) = T(y) = lim i λ y i · y i . • Note that, due to the Lipschitz inequality for T, we can write the following bound for the error committed.
This gives a basic extension procedure, for which the only requirements are the existence of a convenient set of eigenvectors and the Lipschitz inequality for T. Indeed, in case we have no information about the map T other than it is Lipschitz; we have to choose the point y in ev(T) that makes the bound inf x i ∈ev(T) x i − x to be (at least almost) attained. That is, if a point x i 0 ∈ ev(T) attains this infimum, we approximate T(x) by T(x i 0 ). However, this is not the best way of obtaining a good approximation for T(x) if we can assume certain convexity properties for the map T. Essentially, we have to assume that the map T is nearly convex, that is, we can approximate it by a convex combination of eigenvectors as In case if we assume such property, we can find better error bounds, which suggests other approximation methods instead of the one provided by the nearest point in ev(T). Recall that for this point to exist and to be unique for the next definition makes sense, we need some strict convexity conditions on X; we are assuming that the space X is uniformly convex. This is explained in Section 5. Before doing it, let us present a most elaborated version of the minimal distance extension just explained.

The Average Value Extension of T| ev(T)
As in the previous case, take a strictly decreasing sequence of positive numbers ( i ) +∞ i=1 such that lim i i = +∞ and consider an eigenmeasure µ for T. We can get an approximation to T as follows.
Fix an element x ∈ E, define d 0 = inf{ x − y : y ∈ ev(T)}, and consider the sets Let us assume first that lim i µ(M i (x)) > 0; we will see after about what to do in the case lim i µ(M i (x)) = 0. Write M 0 (x) for the measurable set ∩ +∞ i=1 M i (x). Note that, by the monotone convergence theorem, , and define T ε i (x) in the same way, as The set {λ y · y : y ∈ M 1 (x)} is norm bounded: indeed, we have that for every λ y · y · χ {y} dµ(y) that converges with i, j to 0. Consequently, T i (x) − T j (x) → i,j 0, and so we can define the approximation at x by Finally, in case µ(M 0 (x)) = 0, we fix a small δ > 0 and change d 0 in the definition of the sets M i (x) by d 0 = inf{ x − y : y ∈ ev(T)} + δ to get a non-µ-null set M 0 (x).

Approximation of Lipschitz Maps by the Convexity Rule
In this section, we show how a Lipschitz map T can be approximated using the evaluation of the function just in the set of eigenvectors, under some convexity assumptions on T. Although the explanation we develop here is of theoretical nature, we are thinking in effective algorithms to approximate functions by this method, so minimization of the errors explained in Section 2 of the paper for getting eigenvectors or "almost" eigenvectors to provide a general method for extending the idea of average reconstruction of a Lipschitz map by means of its eigenvectors. We assume here that the involved maps are "approximately convex", that is, the values of the maps at all the points are near to a convex combination of the eigenvectors that are near them. This is the main difference with respect to the methods explained in the previous section.
In order to obtain reasonable error bounds, we need the functions to satisfy some requirements. The main idea is that we can perform approximations of the value of the function at a given point using its metric properties, which are essentially determined by the Lipschitz condition of the function if nothing else is assumed for the function. So, we have to argue in topological terms, in the line of what explained in Section 3.2. We propose to define the pointwise approximation to the Lipschitz map by convex combination of eigenvector evaluations. Consequently, the formalism of eigenmeasure, as well as the computation of nearest points to a given set in a Banach space endowed with a uniformly convex norm, and the inequalities associated with the Lipschitz character of the function, will be required.
Given a Lipschitz function T : X → X, we will reconstruct the function T by convex combinations of eigenvectors. We have to assume that ev(T) is non-empty. In order to do it, we need to find what is the convex hull of finite sets of eigenvectors that is as near as possible to the point x. In case a given point x ∈ X belongs to any convex combination of elements x 1 , . . . , x n of ev(T) with eigenvalues λ 1 , . . . , λ n , we can approximate T(x) as the convex combination Since several convex combinations involving different elements of ev(T) can occur, we have to choose one of them: the Lipschitz inequality for T gives the key of how to do it. The idea is to divide the error in two terms, with the first one corresponding to the distance of x to the convex hull of x 1 , . . . , x n and the other one measuring the error committed by the approximation of T as a convex function.
If we write co(S) for the convex hull of any subset S of the Banach space X and co(S) for its norm closure, we should write d(x, S) = inf z∈S x − z for the distance from S to x. Recall the following fundamental result on optimal approximations in Banach spaces: if X is a uniformly convex Banach space, and x ∈ X and C ⊂ X are closed and convex, then there is a unique y ∈ C such that x − y = inf z∈C x − z = d(x, C). We say as usual that y is the nearest point to x in C. A direct consequence of this general fact is the following.
is attained at z, and (ii) there is a bound for this minimal distance as Proof.
(i) Consider the convex hull of the points λ 1 x 1 , . . . , λ n x n . By the quoted result, there is an optimal point y in the convex hull in which the minimal distance to T(x) is attained that is also an element of X; let us write it as the convex addition ∑ n i=1 α 0 i λ i x i . (ii) Then, since x i are eigenvectors, if we consider any other convex combination ∑ n i=1 α i λ i x i we obtain Thus, computing the infimum on the right hand side, we get as we wanted to prove.

Remark 3.
Proposition 4 provides an approximation tool for Lipschitz maps, which works best whenever T behaves more or less like a convex function. Note that the nearest point provides another way to produce an approximation of a Lipschitz map in terms of eigenmeasures, again as an integral average. Let us show this in the simple case of a finite set of eigenvalues; the general procedure is explained below, although we do not use the integral formalism in the subsequent explanation for the sake of clarity. Suppose that we have a set S = {x 1 , . . . , x n } of eigenvectors of a Lipschitz map T. Write N S (x) for the nearest point of co(S) to x, and P {y} N S (x) (where y ∈ S), for the coefficient α 0 i associated with y = x i of the convex combination that gives N S (x). Note that, by the requirements on E, these coefficients are unique so the expression is well-defined. Take an eigenmeasure µ associated with S, µ(·) = ∑ n i=1 µ({x i }) · δ x i (·). Then, we have that the best approximation T is given by the spectral integral Fixing T, and assuming that convexity is a good method for approximating it, we can define two different errors that can be controlled using complementary arguments. We have to point out that getting the exact value of T(x) just knowing the eigenvectors of T is not possible in general: stronger requirements (such as linearity, for example) are needed. However, we provide an approximate value of T(x) and fix small bounds for the errors committed.
Definition 3. Let X be uniformly convex and consider a Lipschitz map T : X → X. For each set, S = {x 1 , . . . , x n } ⊂ ev(T) and x ∈ X; if ∑ n i=1 α 0 i x i is the nearest point of S to x, we define two approximation errors as follows .
(1) The convexity error ε c , which measures how far the map T is from being convex, is given by (2) The distance error ε d (T, S, x), which gives the distance of T(x) to the value of the map to the nearest point in S, is given by Next result gives the main tool for the approximation of T; the way the error is decomposed allows one to evaluate how far the image of the nearest point of S to x by T could be a good approximation for T(x). On one hand, we have that the possibility of writing x as a convex combination of eigenvectors makes the distance error equal to 0, and so we only have to consider the convexity error, which measures how far the function T fails to be convex. Indeed, if it is convex, we have that On the other hand, in case T is convex, for getting an exact formula for T in terms of its eigenvectors, we only need the set ev(T) of these eigenvectors to be big enough to cover all the space as convex combinations of the elements of ev(T).
Let us write this explicitly in the next result, which can be considered as a basic form of an approximated Spectral Theorem for Lipschitz operators.
Theorem 2. Let X be a uniformly convex Banach space. Let T : X → X be a Lipschitz map. Fix a set S = {x 1 , . . . , x n } ⊂ ev(T) and x ∈ X. Then, there is a convex combination ∑ n i=1 α 0 i x i ∈ co(S) such that Note that the bound obtained above for the convexity error ε c (T, S, x) does not allow one to prove that it equals 0 in case T is convex: this has to be done directly by the assumption of certain convexity properties for T, which have to be applied in a second step. If we do not have such information, it could be better to approximate Consequences of this result under reasonable assumptions can be directly obtained. We can control both errors separately since each of them has different meaning.
To obatin the definition of the approximation to T, we try to find a compromise between the approximate convexity expected for T and the error control by using the distance between points, taking into account that T is Lipschitz. We describe the construction of through several steps; as we explained before, we are searching for an extension T : E → E for the function T restricted to a subset S ⊆ ev(T), T S . For helping in the construction process, we first fix an eigenmeasure µ for the set S.
Let us define our general technique under certain restrictions. For the aim of simplicity, we choose a countable set S ⊆ ev(T) to start with, which has the property that for every R > 0, S ∩ R · B E is finite. We will use metric tools through the entire process to obtain a pointwise definition of the function T. Take also a positive strictly increasing sequence ( i ) +∞ i=1 such that lim i i = +∞. The approximation obtained will depend both on the elections of S and ( i ) +∞ i=1 . Fix also r > 1. • Fix a point x ∈ X and define d 0 = inf y∈S y − x . Our aim is to work with a finite subset of S that is "around" the point x. To take it, consider the ratio r x to get the set S ∩ (d 0 + r x ) · B 1 (x). As we said, this ratio r is chosen at the beginning and has to be fixed in order the function T to be univocally defined. • Take 0 < ε 1 < r x , be sure that r is big enough for this to hold, and define the set M 1 (x) = {y ∈ S : x − y ≤ d 0 + 1 }. Let us write n 1 for the finite cardinal of this set and write x 1 , . . . , x n 1 for the elements of M 1 (x). Compute the best approximation in co(M 1 (x)) to x, which is written as ∑ We define the approximation T 1 at the point x as T 1 (x) := ∑ n 1 i=1 α 1 i λ i x i . We also compute the associated bounds for the errors ε c (T, M 1 , x) and ε d (T, M 1 , x), given by the product of Lip(T) and We have to balance these errors to choose the best value for the approximation. In general, both bounds are independent, and since we want to obtain a decision rule, we choose a parameter 0 ≤ τ ≤ 1 that will depend on the election of the decision maker. So we will use the value of for further comparison. • Consider now 2 and repeat the construction of the approximation obtained above for 2 , which gives We compute also the associated bounds and the values of V( 1 ) and V( 2 ). • After comparing V( 1 ) and V( 2 ), we consider the lower value as the better approximation. • We follow the same procedure until the last approximation is computed; note that by the construction, the number of steps needed to finish it is finite.

Some Hints for Practical Application: An Algorithm for Convex Approximation of Lipschitz Maps in Euclidean Space
We have already fixed some general procedures for approximating a given map by means of the properties that it satisfies. We have focused our attention firstly on Lipschitz maps and secondly on those operators for which some convexity assumption can be assumed.
We assume that we have already fixed a method for computing a significant subset of eigenvectors; we have not developed this question in this paper, but we have suggested that Monte Carlo integration together with our error formulas would help for the determination of such a subset. The scientific literature provides a large source of procedures for doing this (see, for example, [10,14]; see also [17,18,20] for concrete applications in quantum physics).

•
For the aim of simplicity, fix E to be a (finite dimensional) Euclidean space; we center the attention in this case because it is the standard case for applications. Once a Lipschitz map T : E → E is fixed, the first step consists on computing an approximation to the set ev(T). In order to do it, we use any optimization method. • The most basic one is a Monte Carlo calculation: just fix a certain > 0 and define a set ev (T) by taking vectors around the region we want, and satisfying that cos(x, T(x)) = | x x) ,

T(x)
T(x) | > 1 − . The idea is to minimize the diagonalization error provided in the first section using the norm, T(x) − λ · x . • The corresponding eigenvalues can be fixed by using the approximate equation T(x) = |λ| x and taking the sign of the scalar product for these points. • Use the set ev (T) and any of the methods explained in Sections 4 and 5 to get an approximation of the errors. The error bounds explained there can be adapted for this case by including the error associated with the previously explained approximation to the set ev(T), which is given by the numbers T(x) − λ · x , x ∈ ev (x).
Author Contributions: Conceptualization, E.E. and E.A.S.P.; methodology, E.E. and E.A.S.P.; formal analysis, E.E. and E.A.S.P.; investigation, E.E.; writing-original draft preparation, E.A.S.P.; writing-review and editing, E.E. and E.A.S.P. All authors have read and agreed to the published version of the manuscript.
Funding: This research was partially supported by the Grant PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".

Conflicts of Interest:
The authors declare no conflict of interest.