Fractional integrals, derivatives and integral equations with weighted Takagi-Landsberg functions

In this paper we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg functions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg functions which have arbitrary bounded coefficients in the expansion under Schauder basis. The class of the weighted Takagi- Landsberg functions of order $H>0$ on $[0,1]$ coincides with the $H$-H\"{o}lder continuous functions on $[0,1]$. Based on computed fractional integrals and derivatives of the Haar and Schauder functions, we get a new series representation of the fractional derivatives of a H\"{o}lder continuous function. This result allows to get the new formula of a Riemann-Stieltjes integral. The application of such series representation is the new method of numerical solution of the Volterra and linear integral equations driven by a H\"{o}lder continuous function.


Introduction
The aim of this paper is to get a broad class of continuous functions on [0,1], which are nowhere differentiable but have fractional derivatives. The prominent example is the Takagi-Landsberg function with Hurst parameter H > 0 introduced in [ where {e m,k , m ∈ N 0 , k = 0, . . . , 2 m − 1} are the Faber-Schauder functions on [0,1]. In the present paper, we find the fractional derivatives of the Takagi-Landsberg functions, and for other properties we refer to the surveys [2] and [9]. In the case H = 1/2, the function x H is known as the Takagi function.
There are several generalizations of the function x H . In the paper of Mishura and Schied [13], the signed Takagi-Landsberg functions of the form are considered. Their results concern the maximum, the maximizers, and the modulus of continuity. Particularly, it was shown that max t∈[0, 1] x H (t) = 1 3(1−2 −H ) . The case of H = 1/2 is considered in [16], where the connections to the Fölmer's pathwise Itô calculus (e.g. [5]) is also described.
In the present paper we go further and introduce so-called weighted Takagi-Landsberg functions, for which we let θ m,k be arbitrary bounded coefficients. We show that such weighted Takagi-Landsberg functions coincides with the Hölder continuous functions which immediately gives the new series representation for them, which we call a Takagi-Landsberg representation. Then we compute the fractional Riemann-Liouville derivatives and integrals of the Faber-Schauder functions, and therefore we obtain the fractional derivatives of the (weighted) Takagi-Landsberg functions. Such a new series representation of the fractional derivative for Hölder continuous functions is very promising for further development of the continuous functions without derivatives. Particularly, the Takagi-Landsberg representation gives the new method for numerical solution of the integral equations involving Hölder continuous functions.
As an example, we consider the Volterra integral equation with fractional noise, called also fractional Langevin equation, e.g. [12,4]. This equation is of interest for modeling of anomalous diffusion in physics (e.g. [11], [8]) and financial markets (e.g. [17]). Our method of its numerical solution allows to reduce it to the system of linear algebraic equations, which is computationally effective. We prove that the numerical solution of the fractional Langevin equation, due to our method, approaches the theoretical solution, which is illustrated by numerical examples.
We obtain also the series expansion of the Riemann-Stieltjes integral applying methodology based on fractional Rieman-Liuville integrals introduced in [18] and developed in [14]. As an illustration we consider the linear differential equation driven by Hölder continuous function and prove that its numerical solution due to our method tends to the exact solution in the specific norm. This result are supported also by numerical examples.
The paper is organized as follows. In Section 2, we recal some basic definitions from fractional calculus and Schauder basis. In Section 3, we compute fractional Riemann-Liouville integrals and derivatives of the Haar (Section 3.1) and the Faber-Schauder (Section 3.2) functions. In Section 4, we introduce the weighted Takagi-Landsberg functions and obtain the series representations of their Riemann-Liouville derivatives. The series expansion of the Riemann-Stieltjes integral is given in Section 5. In Section 6, we consider the application of the Takagi-Landsberg representation for the solution of the Volterra integral (Section 6.1) and linear differential (Section 6.2) equations. The numerical results are presented in Sections 6.3 and 6.4.

The Faber-Schauder functions
Here, we find the fractional integrals and derivatives of the Faber-Schauder functions.
Lemma 3.2. Let α ∈ (0, 1), T > 0, k, m ∈ N 0 and 0 ≤ k < 2 m . Then for t ∈ (0, 1) we have Proof. It follows from [15, formula (2.65)] that I α 0+ e m,k = I α Finally, we note that and Proof. Formula (17) Remark 3.4. We can write the fractional derivatives D α 0+ e m,k and D α T − e m,k as 2) Let ∞ n=0 a n (t), t ∈ [0, T ] be a convergent in L 1 [0, T ] series with a n ∈ I α Proof. 1) The first statement follows from the Lebesgue dominated convergence theorem, that is 2) Note that D α 0+ a n (t) = d dt I 1−α 0+ a n (t). Since ∞ n=0 a n ∈ L 1 [0, T ], we have from the first part that Consider the partial sums of the fractional derivatives of the Faber-Schauder functions D α (18) and (19), we have where Proof. From Remarks 3.1 and 3.3, it follows that Consider the case of D α T − . Let 2 m t + 1 ≤ 2 m T − 1, then it follows from Remarks 3.1 and 3.3 that Then there are at most two non-zero τ 1−α 2,2 m +k (t, T ). Thus, we get the upper bound

It follows from Proposition 3.2 that the series
Now consider the special case α = H and the values of D α 0+ x H (t) at points of the m 0 th dyadic partition of [0, 1], that is the set T m0 := {k2 −m0 |k = 0, . . . , 2 m0 }.
Proof. In the case α = H, m ≥ m 0 , it follows from Remark 3.2 that which gives that the right hand side of (24) is negative.
We get from the last relation and (25)

A weighted Takagi-Landsberg function
In this section we consider the extension of the class of the Takagi-Landsberg functions. Namely, for constants c m,k ∈ [−L, L], k, m ∈ N 0 , we define a weighted Takagi-Landsberg function as y c,H : Since We call formula (27) the Takagi-Landsberg representation of function f.

Proof.
To show this, we first provide the relation between coefficients a m,k in expansion (7) and c m,k in (26) that is Theorem 3 on p. 191 in [7] states that f is H-Hölder continuous if and only if coefficients a m,k in expansion (7) Proof. Due to (17), the fractional derivatives of summands in (26) equal From (22) we have the following uniform bound Analogously, for any α ∈ (0, 1) such that α < H 1 , 1 − α < H 2 , see, e.g. [18]. We use the Takagi-Landsberg representation of functions f and g (26) to give the series expansion of integral n,l ∆ 2 2 m +k,2 n +l (t).
Proof. Due to Theorem 4.1, we have that D α 0+ f and D 1−α t− (g(·) − g(t)) exist and converge uniformly as series (29) and (30). Therefore, D α 0+ f (s)D 1−α t− [g(·) − g(t)](s) converges uniformly on s ∈ (0, t) as well with the following bound for all s ∈ (0, t). So, we apply the Lebesgue dominated convergence theorem to the integral Obviously, if t < k 2 m ∨ l 2 m the last integral equals zero. Let t ∈ J n,l , then If t ∈ J n,l+0.5 , then The case t > l+1 2 n is similar. Thus, we have Then the statement follows from Lemma 3.1, relations (16) and (31).
Remark 5.1. The Riemann-Stieltjes integral in Theorem 5.1 can be written as .
n2,l2 ∆ 2 2 n 1 +l1,2 n 2 +l2 (1), Proof. The value of x R 1 follows from (32). Denote by R(t) the value of the integral t 0 f dg. The function R ∈ H H2 [0, 1] possesses the representation as a weighted Takagi-Landsberg function with coefficients c R given by c R m,k = 2 mH 2R k+0.5 . Then for m ∈ N 0 and k = 0, . . . , 2 m − 1 we have from (35) that We can rewrite the last integral as From the properties of the Schauder system we get that g k In this section it is also convenient to make the new indexation of c g . We write c g n for c g m,k if n = 2 m + k, m ≥ 0, k = 0, . . . , 2 m − 1. and Thus, coefficients c f m are determined by coefficients {c x n , n ≤ m}.

Volterra integral equation
Let H < α ∈ (0, 1), θ = 0 and g ∈ H H [0, 1], that is g has the Takagi-Landsberg representation with bounded coefficients c g = {c g m,k }. Consider the Volterra integral equation given by Equation (40) is called also as the fractional Langevin equation, e.g. [4]. It follows from the general theory of integral equations that (40) has a unique solution in C[0, 1], e.g. [6, Section XII.6.2]. Indeed, the operator I α 0+ has the norm I α Thus, X posses the Takagi-Landsberg representation (27) with x 1 ∈ R and bounded coefficients c x = {c x m , m ≥ 0} Then we apply Lemma 3.3 and formula (29) to get that [I α X](t) has the following series representation We introduce a truncated fractional integral I α Denote by X p the solution of the following truncated equation Obviously, I α 0+ S p ∞ ≤ I α 0+ ∞ , thus (41) has a unique solution in C[0, 1]. By construction I α 0+ S p X p ∈ H α [0, 1], so X p is H−Hölder continuous on [0, 1] as well.
Here we give the solution of (41) by finding the coefficients c p and x p 1 in the Takagi-Landsberg expansion (27) of X p .
Lemma 6.2. Let X p be the solution of equation (41), then X p tends to the solution of (40) in the supremum norm on [0,1].
Proof. Let X be the solution of (40). Denote by err p = X p − X. Note that Due to the power series expansion of err p as a solution of equation (48) we have , where x H is a Takagi-Landsberg function. The second term in the right-hand side of (48) is bounded by Thus, I α 0+ [S p X p − X p ](t) ∞ → 0 as p → ∞. This yields that X − X p ∞ → 0, p → ∞.

A linear differential equation
Let β, γ ∈ R and β = 0, γ = 0. Let g : [0, 1] → R be a Hölder continuous of order H > 1 2 , with g(0) = g(1) = 0, that is g be a weighted Takagi Let us apply the Takagi-Landsberg expansion to solve (50). Using notation (16), we get that the first integral in the right hand side of (50) has the following representation Denote by X p the solution of the following truncated equation the exact and truncated solution. Moreover, if we increase the value of p = 7, then the graphs of X and X p for H = 0.501 become visually indistinguishable and the computed norm of the error X − X p ∞ is 0.01888 for this example. From the other hand, the wrong value of H, which is greater than the Hölder exponent of g, affects on solution X p and the error between X p and X increases. We illustrate such mis-specification of H on Figure 2, where one clearly see the difference between the exact solution X and numerical solution X p when H is significantly larger than true value 0.5. Figure 1: Left: function g. Right: solutions X (black) and X p (red) for H = 0.51.