Contents

1 Approximation Theory
 1.1 Introduction
 1.2 Hardy’s Inequalities
 1.3 Best and Near-Best Approximation
 1.4 Interpolation Spaces: The Real Method
 1.5 Jackson and Bernstein Inequalities
2 Dyadic maximal-smoothness Spline Approximation
 2.1 Best and Near-Best L_p Polynomial Approximation in cubes of ^d
 2.2 Markov’s Theorem
 2.3 Divided Differences
 2.4 Univariate Splines
  2.4.1 Definition and Basic Properties
  2.4.2 Quasi-Interpolant Operators
 2.5 Tensor Product Splines: Description of the problem
 2.6 Sobolev Spaces
  2.6.1 Mollifiers and Infinitelly Differentiable Partitions of Unity
  2.6.2 Distributions and Weak Derivatives
  2.6.3 Sobolev Spaces W_p^r(G) and H_p^r(G)
  2.6.4 Properties of Sobolev Spaces in one dimension
 2.7 Modulus of smoothness and Besov Spaces
  2.7.1 Definitions and Properties
  2.7.2 Whitney’s Theorem
 2.8 Other Seminorms for Besov Spaces
 2.9 Further results: K-functional of compatible couples of Besov Spaces
 2.10 Lorentz spaces
 2.11 Further results: Interpolation of Besov Spaces
 2.12 Further Results: Embedding Theorems for Besov Spaces
3 Approximation by Ridge Functions on Dyadic maximal-smoothness splines
 3.1 General Theory of Ridge Functions
4 Appendix
 4.1 Elements of Functional Analysis
  4.1.1 Linear Transformations
  4.1.2 The Hahn-Banach Theorem
 4.2 Rearrangement-Invariant spaces
  4.2.1 Riesz Spaces
  4.2.2 Resonant measure spaces
 4.3 To Do

1 Approximation Theory

1.1 Introduction

Let (X,||^.||_X) be a (quasisemi)normed linear space. Consider a countable family of spaces in X, {X_n}_n with associated error functionals E(^.,X_n)_X = inf _gXn||^.-g||_X, satisfying the following properties:

1. Homogeneity: X_n = X_n for all n and
2. C-linearity: There exists C > 1 (independent of n) such that X_n + X_n = X_Cn for all n.
3. Local (near)best approximation: Given f X, there exists an element of (near)best approximation to f from X_n for all n.

   We say g Y is an element of best approximation to f X, if ||f - g||X = E(f,Y )X. A near best approximation element to f from Y is by definition any function g such that ||f - g||X < E(f,Y )X for some value > 0. In that case, we will often refer to such a function g as a -near best approximation element.

4. Global best approximation: lim_nE(f,X_n)_X = 0 for all f X.

We will call {X_n} a family of approximants. For any such family, and given parameter values ,q > 0, consider the following (quasi)seminorms and associated subspaces:

-

Lemma 1.1 If the sequence _n is monotone decreasing, then the (quasi)seminorms |^.|_Aq(X,Xn) are equivalent to the following:

Proof. For any k > 0 and any 2^k-1 < n < 2^k, we have the estimates

Lemma 1.2 Under the same hypothesis as before, and any value > 0, the following inclusion is verified for all 0 < q < p <:

Proof. This is just an application of the well known inclusions _{q p} for 0 < q < p <: Consider the measure space (,), where (n) = 1 for all n, and consider for each function f X the (measurable in (,)) functions : k2^kE(f;X_2^k)_X ⁺; then,

A

Remark. In the following sections we will learn to find descriptions of these spaces in terms of classical spaces. The main tools used in this sense are given in the following order:

• The (quasi)seminorms (1.1), (1.2), (1.3) and (1.4) are discretizations of p-norms over ⁺ with the Haar measure dx/x. We have included some useful results related to these measures in §1.2.
• §1.3 deals with the existence and properties of general (abstract) best and near-best approximations.
• Both §1.4 and §1.5 introduce us to Interpolation Spaces and how these are related to our problem of approximation via Jackson and Bernstein’s inequalities.

1.2 Hardy’s Inequalities

Theorem 1 (Hardy) Given > 0 and 1 < q < , the following inequalities hold for each nonnegative measurable function :

For q = , the integral is replaced by the L norms:

Proof. Let us prove the estimate (1.5). For any value > 0 we estimate first the interior integral using Hölder’s Inequality; let p be the conjugate exponent of q:

<

>

-p(1-)

q,

1-p(1-)

<

/p

q,

The remaining estimates can be obtained from this one by changes of variable or taking limits, so we skip their proofs.

Remark. It is possible to discretize integrals of the kind ₀^q when the functions (t) are nonnegative and monotone, using the same technique we used in the proof of Lemma 1.1:

Lemma 1.3 (Discrete Hardy’s Inequalities) Let a = (a_n)_n, b = (b_n)_n be two nonnegative sequences such that there exist C₀,, > 0 so that for all n, either

then, for all q > 0, and > 0 (in the first case) or 0 < < (in the second),

Proof. Let us assume that the first condition is satisfied. From the inclusions for 0 < < <, we infer that it must also be b_n < C_{0 k=n}a_k for all < . We can therefore assume that < q (if it is not, then we can certainly pick < q < ). We are now able to use Hölder’s Inequality; let 0 < < , and let r > 0 so that /q + /r = 1:

A similar proof serves to show that the second condition also gives the same estimate, but this time only for values 0 < < .

Theorem 2 Given > 0 and 0 < q < 1, there exists C > 0 that depends at most on and q such that the following inequalities hold for each nonnegative monotone function :

Proof. Given t > 0, there exists n such that 2^-(n+1) < t < 2^-n; therefore,

1.3 Best and Near-Best Approximation

Lemma 1.4 Let (X,d) be a metric space and let Y be a compact subset of X; then for each f X there exists an element of best approximation to f from Y .

Proof. Consider for each n an element g_n Y such that d(f,g_n) < inf _gY d(f,g) + 1/n. Any sequence in a compact set has at least a limit element g₀ Y ; this element is the best approximation to f from Y by definition.

Proof. If X_d = {0} then there is nothing to prove. Otherwise, consider for each f X the set Y _f = . Y _f is a closed set:

Lemma 1.5 Given (X,||^.||_X) a (quasi)normed space, and Z X a linear subspace of X such that there exists an element of best approximation from it to every element in X. If x X and z Z is a near-best approximation to x from Z with constant > 1, then for each y X there is an element z' Z of near-best approximation to y from Z such that z -z' is also near-best approximation to x-y from Z with constant ' depending at most on X and .

Proof. Given x,y X, assume E(x-y,Z)_X < E(y,Z)_X (it’s not a lost of generality, since one can switch elements). Let z Z be a near-best approximation to x from Z, and z'',z''' Z elements of best approximation from Z to y - x and y respectively. We have then for z' = z'' + z,

Definition 2 A subset Y of a (quasi)normed space X is said to be a unicity space, if for each f X there exists a unique element of best approximation from Y .

Example. The spaces L_p are unicity spaces for 1 < p < , but not for 0 < p < 1 nor p = . A characterization of spaces (at least normed) with the unicity property can be made through the use of “strict convexity”:

Definition 3 A normed space is said to be strictly convex if the following property holds:

Proof. Assume the result is not true, and there exists a function f X and two different elements g₁,g₂ Y such that ||f - g₁||_X = ||f - g₂||_X = E(f,Y )_X > 0. In that case,

Proposition 1 Given a finite dimensional unicity subspace X_d X, the operator of best approximation is continuous.

Proof. Let : X X the operator of best approximation, and let C_X > 1 be the constant in the (quasi)triangular inequality offered by the (quasi)norm ||^.||_X. Given > 0, let = (3C_X²)^-1. Notice that, if f,g X verify ||f - g||_X < , then we have

1.4 Interpolation Spaces: The Real Method

Definition 4 A compatible couple is a pair of (quasisemi)normed linear spaces, (X,||^.||_X), and (Y,||^.||_Y ) continuously embedded in a Hausdorff topological linear space H.
We define the sum and intersection of such a couple as

Definition 5 Given a compatible couple (X,Y ), we say a linear operator T : X + Y X + Y is admisible for the couple if (i) T(X) X, and _X is bounded. (ii) T(Y ) Y , and _Y is bounded.

Definition 6 A (quasisemi)normed linear space (Z,||^.||_Z) is intermediate between X and Y (or for the couple given by those spaces), if (i) X Y Z X + Y (ii) X Y is continuously embedded in Z: there exists C_XY > 0 such that ||f||_Z < C_XY ||f||_XY for all f X Y . (iii) Z is continuously embedded in X + Y : there exists C_X+Y > 0 such that ||f||_X+Y < C_X+Y ||f||_Z for all f Z.

An intermediate space Z of a compatible couple (X,Y ) is an interpolation space for the couple if T(Z) Z for all admisible operator T.

Remark. In the 70’s there were primaryly two methods for constructing interpolation spaces of a compatible couple: the complex method of Calderón [Cal1], and the real method of Lions and Peetre [Peet]. We are mainly interested in the latter, since it uses as building blocks similar quasi-seminorms to the ones in the description of the Approximation Spaces above.

Definition 7 We define the K-functional of a compatible couple (X,Y ) as follows:

for each f X + Y and t > 0.

Lemma 1.7 (Properties of the K-functional) Let (X,Y ) be a compatible couple: (i) K(f,^.;X,Y ) is a continuous subadditive convex-down monotone nondecreasing function of t that verifies K(f,t;X,Y ) = tK(f,q/t;Y,X), and K(f,nt;X,Y ) < nK(f,t;X,Y ) for all n and t > 0. (ii) For each t > 0, K(^.,t;X,Y ) is a (quasi)seminorm equivalent to ||^.||_X+Y . (iii) Let (X,Y ) be a compatible couple and let T be an admisible operator; then for each f X + Y and t > 0, where M = max.

Proof. Notice that, given f X + Y , the function K(f,^.;X,Y ) is trivially nonnegative and monotone nondecreasing. Its concavity is proved in the following way: Given t₁,t₂ > 0 and 0 < < 1, and any decomposition f = f_X + f_Y , we have, for t = t₁ + (1 - )t₂,

Lemma 1.8 Given a compatible couple (X,Y ), and parameters 0 < < 1, 0 < q <, the following functionals are (quasi)seminorms in X + Y :

Proof. |^.|_{(X,Y ),q} is trivially linear and nonnegative. The (quasi)triangular inequality is directly inferred from part (ii) in Lemma 1.7.

Proposition 2 Given a compatible couple (X,||^.||_X), (Y,||^.||_Y ), and parameters 0 < < 1, 0 < q <, the (,q) spaces

are interpolation spaces.

Proof. This is inmediate from (iii) in Lemma 1.7.

Lemma 1.9 Given a (quasisemi)normed space (X,||^.||_X), and a continuously embedded subspace Y X with ||f||_Y < C_Y ||f||_X for some C_Y > 0, and all f Y : (i) The K-functional of the compatible couple (X,Y ) can also be written as

(ii) The (quasi)seminorms |^.|_{(X,Y ),q} are equivalent, for each r > 0, to the following discretizations:

Proof. Part (i) is trivial. As for part (ii), we start noticing that

,q

Theorem 4 (Holmsted (1970)) Let (X₁,||^.||_X1), (X₂,||^.||_X2) be a compatible couple of (quasi)normed spaces, let 0 < ₁ < ₂ < 1 and 0 < q₁,q₂ < , and consider the interpolation spaces Y ₁ = (X₁,X₂)_1,q1, Y ₂ = (X₁,X₂)_2,q2; then, for f Y ₁ + Y ₂ and = ₂ - ₁, we have the following equivalence:

with constants of equivalency independent of f and t. The usual change from integral to a supremum applies when either q₁ or q₂ are infinite.

Theorem 5 (Peetre’s Reiteration Theorem (1963)) Under the same hypothesis as the previous Theorem, and for any 0 < < 1, 0 < q <, we have (Y ₁,Y ₂)_,q = (X₁,X₂)_',q, where ' = (1-)₁+₂.

1.5 Jackson and Bernstein Inequalities

Definition 8 Given a (quasisemi)normed space (X,||^.||_X), and (quasisemi)normed continuously embedded subspace (Y,|^.|_Y ) X, Jackson Inequality: We say the family of approximants {X_n}_n verifies a Jackson Inequality with respect to Y if there exist r,C > 0 such that

Bernstein Inequality: We say the family of approximants {X_n}_n verifies a Bernstein inequality with respect to Y if there exist r,C > 0 such that

Remark. In the literature of Approximation Theory, results that state Jackson’s Inequalities are refered as “direct theorems”, whereas Bernstein’s inequalities are also identified as “inverse theorems”.

Proposition 3 Given a (quasisemi)normed space (X,||^.||_X), a family of approximants {X_n}_n satisfying both Jackson and Bernstein inequalities with respect to a (quasisemi)normed continuously embedded linear subspace (Y,|^.|_Y ) X, the following estimates hold:

Proof. To prove (1.11), given f X, consider any g Y and a best approximation g_n to g in X from each X_n; then,

1. An appropriate (quasisemi)normed linear subspace (Y,|^.|_Y ) X for which the family of approximants {X_n}_n verifies the Jackson and Bernstein inequalities.
2. A characterization of the interpolation spaces (X,Y )_,q.

The second step is often provided by classical results in the Theory of Interpolation. The first step is the difficult one from the viewpoint of approximation. Theorem 6 provides a good start. Finally, Theorem 7 (proof not offered here, read it in [CDeH]) provides somehow an inverse result to Corollary 3.1.

Proof. Estimate (1.11) gives us A_q(X,X_n) (X,Y )_/r,q trivially; for example, for 0 < p < , and r that makes good both Jackson’s and Bernstein’s Inequalities, if f (X,Y )_/r,q, then

< < r

Theorem 6 (DeVore, Popov) For any (quasisemi)normed space (X,||^.||_X) and family of approximants {X_n}_X such that X_n X_n+1 for all n, as well as for any r > 0 and 0 < p <, the spaces X_n verify the Jackson and Bernstein inequalities for the exponent r > 0, with respect to Y = A_p^r(X,X_n). Therefore, for any 0 < < r and 0 < q <, we have

Proof. It is enough to show that {X_n}_n verifies both Jackson and Bernstein’s Inequalities for the exponent r > 0 with respect to A_p^r(X,X_n):

• Let f A_p^r(X,X_n); we know that there exists C > 0 such that Given n, find k such that 2^k < n < 2^k+1; then,
• Given g_n X_n, we have
  

The rest of the statement follows inmediatelly.

Theorem 7 (Cohen, DeVore, Hochmuth) Let X,Y,{X_n}_n be as before, and suppose that {X_n}_n satisfies the Jackson and Bernstein inequalities for r > 0. Suppose further that the sequence of operators {T_n}_n verifies: (i) T_n : X X_n (not necessarily linear). (ii) There exists C > 0 such that ||f - T_nf||_X < C E(f,X_n)_X for all f X. (iii) |T_nf|_Y < C|f|_Y for all n and f X.

Then, {T_n}_n realizes the K-functional; that is,

2 Dyadic maximal-smoothness Spline Approximation

In this section we want to exemplify how to obtain the approximation spaces in the following case: Given the unit cube ^d, X = L_p() for any choice of 0 < p <, and X_n is a linear space of box-splines with coordinate order r, maximal smoothness, and associated to the dyadic n-th partition of the cube .
In the search for the spaces A_q(X,X_n), we will go through different levels of abstraction: from the low-level construction of best and near-best polynomial approximation to functions in L_p() on cubes, to the high-level description of the K-functionals that will lead us into further results involving interpolation spaces for compatible couples of Besov spaces. The logic step-by-step exposition is summarized in the following table:

1. Basic Pre-requisites:
   • Construction and results on different norm estimations related to best and near best L_p() polynomial approximation in subcubes (§2.1).
   • Markov’s Theorem on norm extimation of derivatives of polynomials over compact intervals of the real line (§2.2).

2. Construction of the approximants and projectors:
   • Divided differences: definition and basic results (§2.3).
   • Univariate splines: definition, properties and description of the quasi-interpolants as projectors associated to the construction of the puB-spline basis (§2.4).
   • Multivariate tensor-product puB-splines associated to dyadic partitions of the unit cube , the linear spaces spanned by them, and the generalized quasi-interpolant for these spaces (§2.5).

3. Search for good candidates for approximation spaces:
   • Sobolev Spaces (§2.6).
   • Besov Spaces (§2.7).

4. Solution of the problem of approximation and related results:
   • Construction of equivalent seminorm functionals for Besov Spaces that link them with the approximation spaces we are looking for (§2.8).
   • Interpolation of compatible couples of Besov Spaces (sections 2.9, 2.10 and 2.11).
   • Embedding Theorems for Besov Spaces (§2.12).

2.1 Best and Near-Best L_p Polynomial Approximation in cubes of ^d

Lemma 2.1 Given r > 0, a cube ^d and 0 < q < p <, there is a constant C > 0 depending at most on p, q and d such that

for all g (r).

Proof. Consider for all p > 0 the (quasi)norms |||^.|||_Lp() = ||^-1/p||^.||_Lp() in (r), and apply Theorem 27 (page 101).

Lemma 2.2 Given r > 0, cubes I J ^d such that |J|< |I| for some > 0, and 0 < q <, there are constants C₁,C₂ > 0 depending at most on q and d such that

for all g (r). In particular,

Proof. Consider the (quasi)norms ||^.||_I,Lq() = |I|^-1/q||^.||_Lq() in (r) and apply Theorem 27 again.

Proof. Let P be the best L_p() approximation element to f from (r); then we have:

Proof. Let P be the best L_p(J) approximation element to f from (r). First, notice that for any cube I J,

2.2 Markov’s Theorem

Theorem 8 (Szegö (1928)) For each trigonometric polynomial T_r of order r,

Corollary 8.1 (Bernstein’s Inequalities) ||T'_r||_L() < r||T_r||_L(). ||T_r^(k)||_L() < r^k||T_r||_L().

Corollary 8.2 For an algebraic polynomial P_r (r), and all x (-1,1),

Corollary 8.3 For an algebraic polynomial P_r (r) of order r with complex coefficients on the disk D = {z : |z|< 1},

Theorem 9 (Markov) For an algebraic polynomial P_r (r),

2.3 Divided Differences

Definition 9 Given a function f : and a finite collection of real numbers {t₀,t₁,...,t_n}, we denote with | (f;t₀,t₁,...,t_n) the leading coefficient of the polynomial of degree n that interpolates f at t₀,...,t_n. We call it the n-th divided difference of f. Divided differences are computed recursively as follows:

Lemma 2.5 (Newton’s Interpolation polynomial) Given a function f : , and knots t₀ < ... < t_n, the interpolation polynomial of f at those knots, P_f(x;t₀,...,t_n) can be written in terms of the divided differences as follows:

Proof. Given f : , consider the following interpolation polynomials for each k = 0,...,n - 1: Q_k(x) = P_f(x;t₀,...,t_k) (k), Q_k+1(x) = P_f(x;t₀,...,t_k+1) (k + 1). Notice that g = Q_k+1 - Q_k (k + 1) vanishes at the knots t₀,...,t_k, and by definition its leading coefficient is the divided difference | (f;t₀,...,t_k+1); hence,

Lemma 2.6 If f Cⁿ[a,b] and a < t_k < b for all k = 0,...,n, then there exists (a,b) such that | (f;t₀,...,t_n) = f⁽ⁿ⁾().

Proof. This is a direct consequence of Rolle’s Theorem.

Lemma 2.7 (Leibnitz Formula for Divided Differences) Given functions f,g and knots t₀, ..., t_n, the n-th divided difference of their product is given by the following formula:

(2.1)

Proof. Assume all the knots are different; consider the polynomial of interpolation of h = fg at those knots (using Newton’s expression with the divided differences as coefficients).

|

2.4 Univariate Splines

2.4.1 Definition and Basic Properties

Definition 10 (Schoenberg spaces: knots-multiplicity form) Given a interval A = [a,b] in , we define initially spaces of splines in the following way: Fix r > 0 and let {a < t₁ < t₂ < < t_n < b} be a partition of the interval, and associated to these knots, multiplicities 0 < m_k < r. We denote t = (t₁,...,t_n), m = (m₁,...,m_n), and

The multiplicity m_k indicates the degrees of freedom (associated to polynomials with order r) on each knot t_k; hence, r-m_k-1 gives the smoothness of the spline function at those points (-1 meaning discontinuity).

Classically, these are called Schoenberg spaces on A.

Remarks. For instance, m = 1 gives one degree of freedom: the location of the image of f(t). In that case, the smoothness of f at t is r - 1, which is the maximum possible degree. In particular, this shows that (r) = S_r(t,1;A), where 1 = (1,...,1).
On the other hand, if m = r, then we have all possible degrees of freedom: we can choose location, and all derivatives (both sides); this leaves us piecewise polynomials with possible discontinuities on each knot.
With a slight abuse of notation, we can write S_r(t,r;A) = _k=1ⁿ(r)|_(tk,tk+1), where r = (r,...,r).

Proposition 4 The space S_r(t,m;[a,b]) has the basis

where x₊ = x_{x>0} denotes truncated powers. The associated dual functionals are as follows:

In particular, dimS_r(t,m;[a,b]) = r + |m| = r + _k=1ⁿm_k.

Definition 11 (Schoenberg spaces: basic knots form) Given an interval A = [a,b] in the real line, r > 0 and an increasing sequence of knots t = {a < t₁ << t_n < b}, where t_k < t_k+r for all k, we define the Schoenberg space S_r(t;A) to be the space of splines of order r with knots given by the partition generated by t, and multiplicities given by the number of repetitions of each knot in the sequence.

Example. Consider A = [0,1] and the basic knot sequence t = {0,0,0,1/2,1/2,/12,1,1,1}. In this case, we have

Definition 12 (puB-splines) If t_k << t_k+r is a sequence of r + 1 knots with t_kt_k+r, we define the puB-spline N_k,r as follows:

Lemma 2.8 (Properties of puB-splines) (i) N is a spline function. (ii) supp(N_k,r) = [t_k,t_k+r] (iii) Recurrence formula:

Proof. Notice that N is by definition a linear combination of truncated powers (t_j - x)₊^r-k_j, where k_j is the number of repetitions t_i = t_j for i < j; therefore, it is a spline function. Furthermore, since any r-th order divided difference of a polynomial of degree r - 1 is zero, N_k,r vanishes identically when x < t_k and x > t_k+r (think leading coefficients).

The recurrence formula is a direct consecuence of the recurrence formula for divided differences and Leibnitz formula for the divided difference of a product of two functions:

Remarks. On his classnotes [deBo], Carl de Boor expresses the previous recurrence formula in the following way:

Proof. Although the result was first proved by Curry and Schoenberg [CuSc] in 1966, we will offer here a different proof by de Boor and Fix [dBFi], based on the Marsden identities:

Step 1. The restriction to the interval [a,b] of the previously defined puB-splines N_k,r(x) gives a partition of unity: This is a direct consequence of the recurrence formula for puB-splines.

We can therefore use induction on r, since for r = 1 we have trivially:

Step 2. Marsden identities: For any , where for all k = 1 - r,...,n,

We will prove this statement by induction on r: It has been proved true for r = 1 above. Assume the property holds up to r - 1; then,

Let us rewrite the coefficients of each puB-spline in the previous expression:

therefore,

Step 3. puB-spline series of polynomials in (r): For any , and any polynomial P (r), we can write The functionals _k,r are called de Boor-Fix functionals.

Notice first that the previous Marsden’s identities can be completed with the expression of any power ( -x) for < r - 1 by differentiation:

Given now any polynomial P (r), consider any value and the Taylor expansion of P around : which, after rearrangement, gives the desired coefficients.

It only remains to prove that a different choice of leads to the same coefficients, and therefore the functionals _k,r : (r) do not depend on this choice; let ', and write each derivative P() in Taylor expansion around ':

Notice that then,

Step 4. puB-spline series of truncated powers: For any basic knot t_j and a power for 0 < < r -m_j (being m_j the multiplicity of the knot t_j), This is a direct consequence of the previous step; for any basic knot t_j, it must be 1 < j < n, and we can write:

as we wanted to prove.

Conclusion: We have expressions of every basic element of S_r(t;[a,b]) in terms of the constructed puB-splines. This means that they span the Schoenberg space, and because of their cardinality, they must form a basis of the space.

Remark. Consider the dual functionals associated to the basis of S_r(t;[a,b]) given by the puB-splines; let us denote them _k,r. There are different ways of expressing these functionals; de Boor and Fix offer the most useful for our purposes: For each k = 1 - r,...,n, choose _k,r supp(N_k,r) = (t_k,t_k+r) [a,b], and write

Consider the functional Q_t : S_r(t,[a,b]) given by

2.4.2 Quasi-Interpolant Operators

It is not hard to show that the projector Q_t is a bounded operator on the Schoenberg spaces; given S S_r(t;[a,b]), we have

Lemma 2.9 There exists C > 0 (depending at most on r), such that for all 0 < p <, k = 1 - r,...,n and S S_r(t;[a,b]),

Proof. We will use Lemma 2.1 (page 30), Markov’s Theorem (page 33), and the fact that the functions g_k,r and their derivatives are polynomials, hence bounded in any compact. Consider any point _k,r J_k,r:

Proposition 5 For all 1 < p <, and all f L_p[a,b], there exists a constant C > 0 that depends at most on p and the order r, such that the following local and global estimates hold:

Proof. Estimate (2.3) is a direct consecuence of the remark after Lemma 2.9, the “partition of unity” property of the puB-splines, and the fact that |J_k,r|> (t_k+r - t_k)/r and J_k,r I_j,r for all suitable index j:

Remark. Unfortunatelly, one cannot use Hahn-Banach to find the same kind of results for 0 < p < 1, although similar approximation results will remain valid. In §2.5 we will show how in a more general setup.

2.5 Tensor Product Splines: Description of the problem

We have all the necessary ingredients to pose the problem of approximation on cubes of ^d by dyadic splines. Let = [0,1]^d be the unit cube in ^d, and let X = L_p() with the corresponding (quasi)norm for all 0 < p <. The family of approximants will be constructed as spaces spanned by tensor product puB-splines. The construction of those spaces starts in the real line:

Consider for each n the basic knots in [0,1] given by t_n = {k2^-n : 0 < k < 2ⁿ}, and t₀ = Ø by definition. Associated to these basic knots we will use the following Schoenberg spaces:

n = 0 :

Consider auxiliary knots 1 - r,...,0 and 1,...,r, and the corresponding puB-splines N_k,r = N(^. | k,...,k + r) for all k = 1 - r,...,0. Notice that all of those functions may be obtained as (restrictions on [0,1] of) horizontal shifts of N_0,r. We can then write N_k,r(x) = N_0,r(x - k).

n > 1 :

Consider auxiliary knots k2^-n for k = 1 - r,...,0 and k = 2ⁿ,...,2ⁿ + r - 1, and the corresponding puB-splines. Notice that, as in the previous case, we can obtain each of them as (restrictions on [0,1] of) horizontal shifts of dyadic dilations of N_0,r. We need to update our notation in order to handle the new situation:

For each n > 0, denote N_0,r^[n](x) = N_0,r(2ⁿx) (the dyadic dilation of order 2^-n), and then for each k = 1 - r,...,2ⁿ - r - 1, N_k,r^[n](x) = N_0,r(2ⁿx - k) (horizontal left-shifts of length k2^-n).

Notice that {N_k,r^[n]}_k=1-r^2n-1 is a basis for S_r(t_n,[0,1]); let us denote _k,r^[n] the corresponding dual functionals.

Let us move into d dimensions, where we will write x = (x₁,...,x_d) ^d. Consider for each multi-index k = (k₁,...,k_d), the tensor product puB-spline N_k,r^[n](x) = N_k1,r^[n](x₁)N_kd,r^[n](x_d), and the functionals _k,r^[n] = _k1,r^[n] oo _kd,r^[n].

Lemma 2.10 For each S X_n, any tensor product puB-spline N_k,r^[n] = N_k1,rN_kd,r, and any point _k,r = (_k1,r,...,_kd,r) supp(N_k,r^[n]) , (_kj,r supp(N_kj,r) proj_j()),

Proof. Given a multi-index k = (k₁,...,k_d), and a tensor product spline S X_n, write

It follows that these are the dual functionals of the constructed tensor product puB-splines, and therefore the functions N_k,r^[n] are linearly independent in L_p(). Let us denote X_n = span{N_k,r^[n]}, where k has all its indices between 1 - r and 2ⁿ - 1. This is the space of all piecewise polynomials with coordinate order r and maximal smoothness on dyadic subcubes of size 2^-n in the cube (since DN_k,r^[n] L() for all multi-index 0 < < r - 1, and DN_k,r^[n] is continuous for multi-indices 0 < < r - 2). As in the univariate case, each tensor product puB-spline N_k,r^[n] can be obtained from N_0,r^[0] by shifts and dilations:

The family {X_n}_n is our family of approximants:

1. Each X_n is a linear space: X_n = X_n, and X_n + X_n = X_n trivially.
2. By Theorem 3 (page 14), there exist elements of best approximation, although so far we have no means of computing them for a given f L_p(). By using the construction below and the results in sections 2.1, 2.4.2, we will be nevertheless able to construct suitable elements of near-best approximation.
3. Notice that X_n+1 X_n for all n. It can be proved that _nX_n is dense in X, and therefore lim_nE(^.,X_n) = 0 monotonically decreasing. We can therefore use Lemma 1.1 (page 5) if needed.

We would like now to have a projector, but this is not an easy task. The quasi-interpolant of §2.4.2 does not work for 0 < p < 1, but is still useful. The trick is to find first intermediate spaces X_n Y _n X for each n, where the quasi-interpolants can be easily extended, and such that we can effectively compute (near)best approximations from Y _n to elements of X. In the case we are studying here, the obvious choice works just fine:

For each multi-index j = (j₁,...,j_d) with 1 < j_i < 2ⁿ, consider the dyadic cubes of , _j,n = [(j₁ - 1)2^-n,j₁2^-n] ×× [(j_d - 1)2^-n,j_d2^-n], and D_n = {_j,r : 1 < j < 2ⁿ} the family of those cubes. Let us denote Y _n = _j=1²ⁿ (r;_j,n) the space of piecewise polynomials of coordinate order r associated to the dyadic partition D_n.

As we did before for univariate puB-splines, we need to consider for each tensor product puB-spline N_k,r^[n], two especial cubes:

J_k,r:

J_k,r = J_k1,r ×× J_kd,r D_n (read §2.4.2 for the definition of the intervals J_j).

I_j,r:

For each 1 < j < 2ⁿ, denote I_j,r the smallest cube that contains each of the J_k,r for which supp(N_k,r^[n]) _j,nØ. Notice that I_j,r D_m for some m < n; therefore, = 2^(n-m)d. Also, and as before, the number of subcubes J_j,r contained on each I_k,r depends solely on r and d.

In order to obtain the de Boor-Fix expression of the dual functionals of the tensor product puB-splines N_k,r^[n], we will choose canonically _k,r to be the center of the cubes J_k,r. In that case, one realizes that these functionals can also be applied to any function f L_p() which is differentiable enough on each of the points _k,r; in particular, any piecewise polynomial P Y _n:

Lemma 2.11 For any 0 < p < and any piecewise polynomial P Y _n, there exists a constant C > 0 which depends at most on the order r, such that

The proof of this lemma follows the same steps that the one for Lemma 2.9 (page 47) and its posterior remark. We can also construct quasi-interpolant operators Q_n : Y _n X_n, which act naturally as projectors.

Proposition 6 For 0 < p < and any piecewise polynomial P Y _n, there is a constant c > 0 depending at most on r, d and p, such that the following estimates hold:

The proof is trivial; it uses the previous lemma, and follows the same steps that the proof of Proposition 5 and its posterior remark (page 48).

We are ready to construct the method of approximation: Given > 0, consider any operator of near-best L_p approximation by elements of Y _n, say _n : X Y _n such that _Lp() < E(f,Y _n)_p. Notice that such operators may be constructed by collecting the restriction on cubes of the near-best L_p approximations to f by polynomials in (r) on each cube _j,n D_n, and patching them together. Let then

Proof. Notice first that, for all subcube _j,n D_n, we have

Proof. Given f L_p(), let S_n X_n be its best L_p approximation from X_n; then, we have the estimate:

Remark. Notice that now we may use indisctinctly for each f L_p() either E(f,X_n)_p or ||f -T_n(f)||_LP(), since they are equivalent. Moreover, when searching for the approximation spaces, we may use the (quasi)seminorm functions

L_p()

<

2.6 Sobolev Spaces

2.6.1 Mollifiers and Infinitelly Differentiable Partitions of Unity

In this section we introduce two important tools in the Theory of Sobolev Spaces: mollifiers and infinite differentiable partitions of unity. We also illustrate how to use the former to construct plateau functions and prove the density of C₀(G) on L_p(G) for 1 < p < on domains G ^d.

Example. An example of mollifying kernels are the “bump” functions _d : ^d given by

Figure 2.1: 2(x) = expB(0,1)

Proof. Assume f M(^d,|^.|). Let (s_n)_n be a monotonically increasing sequence of nonnegative simple functions converging pointwise to f. As p > 1, we have 0 < s_n(x)^p < f(x)^p a.e., and therefore, it must be s_n L_p(G), and furthermore, by the Dominated convergence Theorem, lim_n||f - s_n||_Lp(G) = 0 (since |f(x) - s_n(x)|^p < f(x)^p for all x). Given > 0, find s_n such that ||f - s_n||_Lp(G) < /2. Use now Lusin’s Theorem to find a continuous function g C(G) such that |g(x)|<||s_n||_L(G) and more importantly,

Lemma 2.12 Given a domain G ^d, a mollification kernel and a function f M(^d,|^.|) such that f(x) = 0 for x / G, the following holds: (i) If f L₁^loc(G), then * f C(^d) for all > 0. (ii) If also supp(f) is compact, then * f C₀(G) for all 0 < < dist(supp(f),G). (iii) If f L_p(G) for any 1 < p < , then * f L_p(G); moreover,

(iv) If f C(G) and K G is compact, then lim_0⁺ * f = f uniformly on K. (v) If f C(G), then lim_0⁺ * f = f uniformly on G.

Theorem 12 Given a domain G ^d, a compact subset K G and 0 < < dist(K,G), there is a plateau function C₀(G) such that 0 < (x) < 1 for all x G, and (x) = 1 for all x K = _yKB(y,).

Proof. Let : ^d be a mollifying kernel, and consider = *_{K 3/2}, the mollification of _K3/2 with . This is the function we are looking for.

Proof. This is a direct consequence of Theorem 11 and parts (ii) and (v) of the previous Lemma.

2.6.2 Distributions and Weak Derivatives

Remark. The space L₁^loc(G) can be identified with a subspace of D'(G) as follows: given f L₁^loc(G), let T_f : C₀(G) g _Gf(x)g(x)dx . These functionals are trivially linear. Notice that it is also continuous: Given a sequence (g_n)_n in C₀(G) such that there exists a compact K G so that supp(g -g_n) K for all n, and lim_ng_n(x) = g(x) uniformly on K; we have

D'

2.6.3 Sobolev Spaces W_p^r(G) and H_p^r(G)

Definition 17 Given a domain G ^d, and r {0}, we define the following functionals:

Functionals (2.8) and (2.9) are trivially seminorms, and (2.10), (2.11) are norms. Associated to these functionals, we define the following spaces:

• H_p^r(G), the completion of {f C^r(G) : ||f||_Wp^r(G) < } with respect to the norm ||^.||_Wp^r(G).
• W_p^r(G) = .
• W_p^r(G), the closure of C₀ in the space W_p^r(G).

Remark. We have trivially W_p⁰(G) = L_p(G) for 1 < p <, and W_p⁰(G) = L_p(G) for 1 < p < (by Theorem 13). Notice also the chain of (continuous) embeddings for all r :

Proof. Let (f_n)_n be a Cauchy sequence in W_p^r(G) L_p; then trivially (D^kf_n)_n are Cauchy sequences in L_p(G) for all multi-index k with 0 <|k|< r. Let f, ^(k) L_p(G) be such that lim_nf_n = f and lim_nD^kf = ^(k) both in L_p(G). As L_p(G) L₁^loc(G), each of those functions determines distributions T_f,T ^_(k) D'(G). For any g D(G), we have then (let q be the conjugate exponent of p):

^(k)

D

Lemma 2.13 Given domains G' G ^d, such that the closure of G' in G is a compact, 1 < p < , r , a mollifying kernel C₀(^d), and f W_p^r(G), we have lim_0⁺ * f = f in W_p^r(G').

Theorem 16 (Meyers, Serrin) Given a domain G ^d, 1 < p < , and r {0}, we have H_p^r(G) = W_p^r(G).

2.6.4 Properties of Sobolev Spaces in one dimension

For functions of one variable, both ordinary and generalized derivatives produce the dame space. This is proved in the following two results:

Proposition 9 If f L₁^loc(A) has a weak r-th derivative ^(r) L₁^loc(A), then it can be redefined on a set of measure zero so that f^(r-1) is absolutelly continuous, and f^(r) = ga.e. on A.