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Sharp estimates for the covering numbers of the Weierstrass fractal kernel

In this paper, we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a Weierstrass fractal kernel. We investigate the properties of the reproducing kernel Hilbert space (RKHS) associated with this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.


Introduction
In 1872 K. Weierstrass presented a particular class of trigonometric series as a collection of continuous but nowhere differentiable functions (CNDF). These functions are defined in terms of the following Fourier series expansion w a,b (x) = ∞ n=0 a n cos(b n π x), x ∈ R. (1.1) For 0 < a < 1 it is clear that it defines a continuous bounded function. Under this assumption, Weierstrass proved that w a,b is nowhere differentiable provided that ab ≥ 1 + 3π 2 , with b an odd integer ( [10,11]). G.H. Hardy relaxed this condition in [9] by showing that for ab ≥ 1, the Weierstrass function is nowhere differentiable.
The Weierstrass function is an inspiration for the Weierstrass-Mandelbrot curve, a fractal curve widely explored in fractal geometry ( [5]). It is well known that the graph of w a,b is also a fractal curve. In particular, for every b ≥ 2 integer, there exists a b in the open interval (1/b, 1) such that the Hausdorff dimension of the graph of w a,b is equal to D = 2 + log(a)/ log(b), for every a ∈ (a b , 1) (see [3]). Many examples of CNDF are known, mainly in fractal theory and applications, and the Weierstrass function is definitely one of the most prominent. The function M : R → C given by where r > 1 and 0 < K < 1, originally investigated by Mandelbrot ([17,p.388]) is an object of interest in the study of fractional Brownian motion (FBM) which is intermediated by a stochastic fractal function characterized by its statistical behavior. For 0 < a < 1 and ab ≥ 1, the function w a,b,C (x) = ∞ n=0 a n e ib n x , x ∈ R, (1.3) has the Weierstass function w a,b as its real part. In [18], among other things, it is shown that w a,b,C can be modified and randomized to approximate a FBM. We suggest [5] and references quoted for more details of the Weierstrass-Mandelbrot curve and its fractal properties. The CNDF class also have important topological properties in C([a, b]). In fact, as proved independently by Banach and Mazurkiewicz ( [24]), it follows a consequence of Baire category theorem that the class of CNDF on [a, b] is of the second category in C([a, b]). Here, as usual, C([a, b]) stands for the normed real vector space of real-valued continuous functions on [a, b] with the supremum norm.
Let I = [−1, 1]. We consider here W : I × I −→ R, the 'Weierstrass Fractal kernel', defined for 0 < a < 1, b an integer such that ab ≥ 1, and given by It is expected that the functions in a RKHS inherit some of the properties of the generating kernel such as smoothness. In [13] the authors showed that functions in the RKHS associated to a kernel having a Mercer series expansion (it means a Fourier-like series given in terms of tensorial product of orthonormal functions) have the same "degree" of smoothness in terms of the fractional Laplace-Beltrami derivative of the original kernel. More general than that, in [12] similar harmonic analysis structure was enough to prove similar reproducing results on a very general context.
Reproducing kernel Hilbert spaces (RKHS) are in the formulation of many problems in Approximation Theory, Signal Analysis, Learning Theory, Geomathematics, etc. They comprise an interesting, relevant and useful class of function spaces in these areas of mathematics and many others. The metric structure they carry is an important tool in theoretical and practical aspects, prevailing the manner in the implementation of procedures in the problems where they appear. We mention references [4,6,7,19,20] for a general discussion on RKHS, including potential applications.
In this paper we present upper and lower estimates for the covering numbers of the embedding I W : H W → C(I) achieving tight bounds. The technique applied is similar to the one applied in [2] for the covering numbers of the embedding of the unit ball of the RKHS associated to isotropic kernels on compact two-point homogeneous space. In short, here, the approach is mainly based on operator norm estimate of I K and some others related finite rank operators. In [15] this approach was employed to obtain estimates for the covering numbers of the embedding operator over the RKHS associated to the Gaussian kernel over non-empty interior sets of R d . The generality of the concept of metric entropy, includes covering number, and it has several applications in many branches of Mathematics, e.g., probability ( [16]), PDEs ( [8]) and operator theory ( [14]). We highlight the important role played by covering number in statistical learning theory to estimate the probabilistic error and the number of samples required for a given confidence level and a given error bound ( [6]). We suggest [21,23] and references therein for more information on metric entropy and machine learning methods. Nevertheless, only for a few infinite-dimensional spaces there has been success in determining the precise asymptotics of covering numbers.
If A is a subset of a metric space M and ǫ > 0, the covering number is the minimal number of balls in M of radius ǫ covering the set A. Clearly, C(ǫ, A) < ∞, whenever A is a compact subset of M . For X, Y Banach spaces and T : X → Y an operator the covering numbers are defined in terms of unit balls as follows. For ǫ > 0, if B X and B Y are the unit balls in X and Y , respectively, then the covering numbers of T are and given by Our main goal is to investigate the covering numbers of the embedding I W : where H W is the reproducing kernel Hilbert space associated to W defined in equation (1.4). The statement of the main result of this paper will employ the following notations. Throughout the paper, for functions f, g : (0, ∞) → R we use the standard asymptotic notations In particular, for sequences {a k } and {b k } the asymptotic notation The main result of this paper reads as follows.
Theorem 11 Let W as in (1.4). The covering numbers of the embedding I W : H W → C(I) behave asymptotically as follows The paper is organized as follows. In Section 2 we introduce the Weierstrass fractal kernel and derive some properties. In Section 3 we give the characterization of the functions in RKHS associated to W in trigonometric series expansions. Section 4 is subdivided in three subsections, for the upper bounds, for the lower bounds and proof of the main theorem, respectively. We close the paper with some concluding remarks in Section 5.

The Weierstrass fractal kernel
In this section we bring some necessary information and properties about the Weierstrass fractal kernel defined on R × R. We consider W as in formula (1.4) but for x, y ∈ R and we present some properties of the kernel that will be useful to obtain the estimates of the covering numbers of the RKHS associate to W .
Let X be a non-empty set. A symmetric function K : For a symmetric positive-definite kernel K : X × X → R there exists a (unique) Hilbert space of real valuable function on X, such that the point evaluation linear functional, i.e., for each x ∈ R, is continuous. The Riez representation theorem implies that I x is characterized as follows In particular, H K is the closure of span{K(y, · ) : y ∈ I}. All these information and more details about the reproducing kernel Hilbert spaces (also known as Native spaces) can be found in [1,7] and others quoted there for a general discussion.

Lemma 21
The function W : R×R → R is a continuous and positive definite kernel.
Proof Since 0 < a < 1 the trigonometric series defining W is uniformly convergent and it is clear that W is continuous. In order to show that W is a positive definite kernel on R × R we observe that for any k > 1 integer, and c 1 , . . . , c k and x 1 , x 2 , · · · , x k real numbers, we have that ans this last summation is: First, we begin by showing that W is continuous and nowhere differentiable kernel.

Lemma 22
The Weierstrass fractal kernel on R×R is nowhere differentiable.
The nowhere differentiability of w as in formula (1.1), proved by Hardy in [9], implies that the partial derivatives of W do not exist and, then W is also nowhere differentiability on R × R.

Lemma 23
For all x, y ∈ R we have W (x, y) = C(x, y) + S(x, y), where C and S are the following positive definite kernels C(x, y) = ∞ n=0 a n cos(b n πx) cos(b n πy) and S(x, y) = ∞ n=0 a n sin(b n πx) sin(b n πy).
Proof The proof follows directly from the standard trigonometric identity for the cosine function. The positive definiteness of C and S follows from the same arguments in the proof Lemma 21.

The RKHS associated to W
In this section we will apply the previous results in order to prove a characterization of the RKHS associated to the kernel W and to derive some properties. Proof It is clear that (H W , · , · W ) is itself a Hilbert space. Due the unicity of the RKHS, we will show that H W is the RKHS induced by W verifying that it fits in properties R1 and R2 above.
For n ≥ 0 an integer, we consider c n (x) := a n/2 cos(b n πx) and d n (x) := a n/2 sin(b n πx), for x ∈ I. For each x ∈ I, the sequences {c n (x)}, {d n (x)} belongs to ℓ 2 and, then belongs to H W . Thus, span{W (x, · ), x ∈ I} is a subspace H W and property R1 follows. This also implies that span{W (x, ·), x ∈ I} is a dense subspace of H W . And property R2 is proved.
Property R2 above implies that W (x, ·), W (y, ·) W = W (x, y) for every x, y ∈ I. As proved above, the RKHS H W associated to W is defined as the closure of span{W (x, ·) : x ∈ I}, endowed with the inner product satisfying for all m ∈ N and x 1 , . . . , x m ∈ I.
Let L 2 (I) := L 2 (I, dx) be the usual Hilbert space of integrable functions with the usual inner product The set {cos(b n π · ), sin(b n π · ) : n ∈ N ∪ {0}}, is an orthonormal set in L 2 (I). And due the fact that 0 < a < 1 the system {a n/2 cos(b n π · ), a n/2 sin(b n π · ) : n ∈ N ∪ {0}}, is an orthonormal basis of the space H W ([22, Lemma 2.6]). We also have that H W ⊂ L 2 (I) and H W = C ⊕S with C and S the following collection of functions on I: It is important to note that Theorem 31 gives us a Fourier series-like characterization of the elements of H W . This is somehow expected, since W itself has this same type of structure (see [22]). Along with this reasoning we have a characterization for functions in H W in terms of nowhere differentiability. We will apply the next lemma. Proof First, we prove that H is dense in H W . So, for any in H W , we consider the sequence {f N } defined by u N n a n/2 cos(b n πx) + v N n a n/2 sin(b n πx), x ∈ I, It is clear that {f N } ⊂ H W . Since ab ≥ 1, we have that lim n→∞ u N n a n/2 b n = lim n→∞ v N n a n/2 b n = lim n→∞ a n b n = ∞, and we conclude that {f N } ⊂ H. Finally, the orthonormality of {a n/2 cos(b n π · ), a n/2 sin(b n π · ) : n ∈ N ∪ {0}} in H W , implies that lim N →∞ f − f N W = 0. From an application of Lemma 32 it follows that H is composed of CNDF.

Estimates for the covering numbers
We proceed by splitting this section in three subsections in order to prove the estimates for the upper (Subsection 3.1) and lower bounds (Subsection 3.2) for the covering numbers for C(ǫ, I W ). In Subsection 3.3 we present the proof of Theorem 11. The covering number was defined in equation (1.6), Section 1.
Our framework implies directly the norm estimates Hence, it is enough to prove the upper bound for the covering numbers with respect to the sup-norm and the lower bound with respect to the L2-norm.

Upper bounds
The next result presents some properties of covering numbers. The proofs are along the same lines as the proofs for the corresponding properties of entropy numbers in [15,Lemma 1] and [14, Section 2.d].

Proposition 42
The embedding I W : H W −→ C(I) satisfies Proof By the reproducing property R2, Section 3, and an application of the Cauchy-Schwarz inequality we have that If we consider f ∈ H W , f (x) = ∞ n=0 a n/2 [c n cos(b n πx)) + d n sin(b n πx)] then g(x) = f (x)/ f W , x ∈ I, is such that g ∈ H W and g W = 1. For f as above and c n = a n/2 , n ∈ N, we have that g at x = 0 attains the upper bound 1/(1 − a) and the proof follows.
Proposition above can be seen as special case of a general case of Lemma 4.23 in [21]. If H K (X) is the reproducing kernel Hilbert space generated by a bounded kernel K on X, then We will need a similar result for certain projection operators, defined as follows. Given N > 1 integer we define P N U and P N V as being the projections onto U := span a n/2 cos(b n π · ), a n/2 sin(b n π · ) : n = 0, . . . , N − 1 and V := span a n/2 cos(b n π · ), a n/2 sin(b n π · ) : n = N, . . . , ∞ respectively.
Precisely, if g ∈ U, then c n (g)a n/2 cos(b n πx) + d n (g)a n/2 sin(b n πx) for some real numbers c n (g) and d n (g), for n ≥ 0. And similarly for the space V. Thus, for any f ∈ H W , we have Lemma 43 If P N U and P N V are the projections defined above, then Proof We first observe that and the right-hand side o equality above is upper bounded by Thus, Analogously as proof above, if we consider g(x) = g(x)/ g W , x ∈ I, with g ∈ U and c n (g) = a n/2 , then and we finish the proof of The proof of the another equality is completely analogous to the one we just presented the proof and we choose to omit it here.
From now on, for each N ≥ 1, we will employ the notation Theorem 44 For any ǫ > 0, there exist a positive constant M 1 such that Now, we consider ǫ > 0 and observe that there exists a natural number N = N (ǫ) such that If ǫ → 0 + , then µ N → 0 and it follows that From Lemma 43 we have I W P N V = µ N and an application of Lemma 41, item 3, we obtain C(µ N , I W P N V ) = 1. Lemma 41, item 2, implies that By the formula (4.2), we obtain that Combining these estimates, due Lemma 41, we have Thus, ln(C(2ǫ, I W )) < N ln 1 + 2 ǫ

Lower bounds
In order to present lower bounds for C(ǫ, I K ), we first introduce the following general result.
Lemma 45 [15, Lemma 1] Let X and Y two n-dimensional real Banach spaces and T : X → Y be an operator. If ǫ > 0 then

(4.4)
A lower bound for C(ǫ, I W ) is presented in the following theorem.
Theorem 46 For any ǫ > 0, there exist a positive constant M 2 such that Proof Let E := {ψ k : k ∈ N ∪ {0}} be the orthogonal basis of H W , given by {cos(b n π( · )), sin(b n π( · )), n ∈ N ∪ {0}}, as indicated in Theorem 31. Also, consider for n ∈ N the operator, defined by the following composition where J n is the embedding of E n := span {ψ k : k = 0, .., n − 1} into H W and P n is the orthogonal projection on F n := I W (J n (E n )). The operator T n : E n → F n is uniquely determined by the following relation T n ψ k (x) = ψ k (x), k = 0, 1, . . . , n − 1.
If A n is the representing matrix of the operator T * n T n : E n → E n with respect to the ONB {ψ 0 , ψ 1 , ..., ψ n−1 } of E n , then det A n = 1, n ≥ 1.

Proof of Theorem 11
We close this section presenting the proof of our main result.

Concluding remarks
In this paper, we have introduced and investigated the Weierstrass continuous nowhere differentiable kernel W given in (1.4). The characterization of the RKHS H W induced by W , presented in Theorem 31, shows that the elements have a Fourier-like expansion nature inherited by the kernel alongside the nowhere differentiability, which is manifested for a dense subset of this space.
Even though the notion of ǫ-entropy (covering numbers, for instance) play an important role in several areas, exact asymptotic behaviour of such entities has been successfully determined only for a few infinite-dimensional spaces. Finally, the (sharp) estimates obtained for the covering numbers in this paper are independent of the parameter b, which, somehow, indicates that the periodicity of W does not affect metric entropy related to H W .