Sequential generalized Riemann–Liouville derivatives based on distributional convolution

Sequential generalized fractional Riemann–Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.


Introduction
An important issue in applications of fractional differential equations are initial conditions, as emphasized already in [8], [9, p.115] or [10] for fractional relaxation and fractional diffusion. Derivatives D α,β 0+ of fractional order α and type β were introduced precisely because their type β parametrizes different types of initial conditions.
Most investigations of fractional initial value problems were until recently concerned with the simplest fractional derivatives D α,0 0+ of type 0 (Riemann-Liouville) or D α,1 0+ of type 1 (Liouville-Caputo) [7,12,16,22,31] considered initial value problems for sequential fractional derivatives [3,24,25,30] thereby continuing the classical investigations in [5] and [31,Sec. 42.2]. In [3,24,25] generalized fractional derivatives of various types were studied all of which were based on the standard kernel t α−1 / (α). Other recent works studied generalized fractional derivatives where the kernel t α−1 / (α) is replaced by a Sonine kernel [19,20,26,27,29]. Riemann-Liouville derivatives and their generalizations D α,β 0+ have β-dependent domains and null spaces, but they all coincide on a specific complement of the null space of D α,1 0+ [10,Sec.7]. Exactly on this complement they also coincide with the causal distributional fractional derivative D α + of order α defined as [32] D α where D 0+ is the space of distributions supported on the right half axis and Y α ∈ D 0+ are causal power distributions that satisfy Y α (t) = t α−1 / (α) for t > 0 (see Equation (2.7)), provided the function spaces for D α,β 0+ are canonically identified via restriction and zero extrapolation. Motivated by this observation and the recent research activity in sequential fractional derivatives, the main purpose of this work is to generalize and unify the theory of fractional initial value problems that involve several sequential derivatives.
Distributional convolution operators on the full axis modified by certain partially defined operators will be used in this work to reinterpret sequential fractional derivatives with D α + from (1.1) are investigated, where each first-order derivative in Eq. (1.2) gives rise to an operator R in Eq. (1.3). In this representation the operator R is a partially defined operator on D 0+ that is called δ-eliminator, because it removes the δ-part of a distribution at the origin.
Given a generalized sequential fractional derivative of the form (1.3), it can be rewritten in the normal form D α|γ 1 ,...,γ m 0+ where α = α 0 + α 1 + . . . + α n ∈ R is its fractional order, (γ 1 , . . . , γ m ) is called its sequential type, and R γ := D −γ Linear combinations of generalized sequential fractional derivatives can then be written in the form D = m l=0 D l • R γ m • · · · • R γ 1 with γ 1 < · · · < γ m , m ∈ N 0 , (1.5) where the operators D l are linear combinations of distributional fractional derivatives D α + , α ∈ R (Theorem 2, p. 21). Our Theorem 3 on p. 23 derives a set of basis functions K 1 , . . . , K m such that every distribution from the null space of D is a linear combination of the K 1 , . . . , K m . Ranges and maximal domains such that D becomes a bijective operator are calculated in Theorem 4, p. 24. If a distribution g belongs to the range of D, then it is an admissible inhomogeneity for the equation D f = g. And a distribution f with D f = 0 is a linear combination of the basis functions K 1 , . . . , K m that solves an initial value problem with initial values obtained by applying the operators V • D γ l + • R γ l−1 • · · · • R γ 1 to K k for k, l = 1, . . . , m, where the δ-value operator V extracts the coefficient multiplying a δ-distribution.
While Theorem 3 provides a useful result, a complete characterization of the null space of D requires an analysis of the value and well defined-ness of V • D γ l + • R γ l−1 • · · · • R γ 1 (K ) with K a linear combination of the K 1 , . . . , K m . In the case where D is a linear combination of sequential derivatives with distinct orders Theorem 5, p. 25, gives a helpful simplification for this characterization. This leads to a generalization of the existence and uniqueness results from [3] and [5,Theorem 4]. In the general case however, a complete characterization is quite complicated. For the case with only two sequential types, γ 1 and γ 2 , the null space of D is fully characterized here in Section 5.4 below.
Analogous to the generalized fractional derivatives from [19,20,29] the operators (1.4) can be generalized to C K • R γ n • · · · • R γ 1 where C K denotes the convolution operator f → K * f on D 0+ with convolution kernel K ∈ D 0+ . Theorems 2, 3 and 4 will be proved for this more general class of operators. Section 2 summarizes some basic mathematical notations on causal distributional convolution operators, the convolution field generated by causal power distributions [17] and partially defined linear operators. Section 3 studies coefficient operators, projectors and eliminators and their application to certain series of distributions. The generalized sequential fractional derivatives are introduced in Section 4. Their fundamental properties are established and their relation to classical Fractional Calculus operators is elucidated. In the final Section 5, the kernels and maximal injective domains of linear combinations of sequential derivatives are studied.

Preliminaries and notations
Subsection 2.1 recalls some basic properties of convolution of distributions with support on the right half axis. Subsection 2.2 summarizes some results from [17] about the field of convolution operators that arises naturally from L. Schwartz' approach to Fractional Calculus. Subsection 2.3 summarizes definitions for partially defined linear operators and forms.

Convolution of distributions with support on the right half axis
The following describes some important properties of convolution of distributions with support contained in the right half axis. The details can be found in [32,Chap. IV,§5] or [15]. Convolution of distributions is a bilinear continuous operation on the space of causal distributions. In other words, D 0+ is a convolution algebra with continuous convolution operation. It follows that any distribution U ∈ D 0+ gives rise to a continuous linear translation-invariant endomorphism by means of the definition Conversely, any continuous linear translation-invariant endomorphism of D 0+ is a convolution operator C U for some kernel distribution U ∈ D 0+ . The associative law for (D 0+ , * ) entails the composition rule In particular, the convolution operator C δ coincides with the identity operator E. A convolution operator C U with U ∈ D 0+ is continuously invertible if and only if there exists a distribution V ∈ D 0+ such that U * V = δ, where δ denotes the "Diracfunction". Note, that V is unique because the convolution algebra D 0+ has no zero divisors. A Summation of sequences is compatible with convolution: is well defined as an absolutely convergent series in D 0+ , then f * h = δ.

Causal distributional fractional calculus
In his book [32] L. Schwartz considered fractional integrals and derivatives, I α + and D α + , with orders α ∈ C, as convolution operators with distributional kernels operating on the space D + of distributions with support bounded on the left. In this work their restrictions to D 0+ are used. That is, one defines I α + := C Y α and D α Due to Equation (2.3) the semigroup property Y α * Y β = Y α+β automatically translates to the well known index laws The operators I α + and D α + are continuous, linear and inverse to each other.

Remark 1
The presented approach to Fractional Calculus resembles operational calculus approaches [13,23,28] in the sense that a quotient field construction is used in both cases. However, the restriction to convolution quotients of a certain class of special distributions, whose inverses are well understood distributions (see [17]), has the advantage that the concrete interpretation of the abstract quotients is known very early on.

Partially defined linear operators and forms
A partially defined linear operator A : dom A ⊆ X → X is a linear operator A : dom A → X with dom A ⊆ X a linear subspace. Similarly, a partially defined linear form L : dom L ⊆ X → K is a linear form L : dom L → X with dom L ⊆ X a linear subspace. The following recalls basic operations on partially defined linear operators and forms. Let A, B partially defined linear operators on X . One denotes The sum A + B of A and B is defined as The scalar multiple λ · A with λ ∈ K is defined as The composition A • B of A and B is defined as Addition (2.14) respectively composition (2.16) defines a semigroup with neutral element given by the zero operator 0 with dom 0 = X respectively the identity operator E with dom E = X . The distributive laws read Equality holds in Equation (2.17b) as well if dom C = X . Addition of partially defined linear forms is defined analogous to (2.14). The composition L • A of a partially defined linear form L and a partially defined linear operator A is defined analogous to (2.16). This composition defines a semigroup operation of the partially defined linear operators on the partially defined linear forms on X .
The operations for partially defined operators and forms exhibit strange behaviour. For instance, A − A = 0 dom A ⊆ 0 and 0 · A = 0 dom A ⊆ 0, but equality holds if and only if dom A = X . For every linear subspace Y ⊆ X one obtains a linear space of partially defined linear operators A : dom A ⊆ X → X with dom A = Y , but the set of all partially defined linear operators or linear forms does not define a linear space.

Eliminators, initial values and series expansions
This section introduces and studies Y α -coefficient operators, -projectors and -eliminators. These are defined for α = 0 in Subsection 3.1. Via fractional integration and differentiation the definition is transported to general α in Subsection 3.2. Then, in Subsection 3.3, the operators are applied to represent the coefficients of a series of distributions Y α .

The δ -limit operator, the δ -projector and the δ -eliminator
Roughly speaking, the δ-elminator, which is denoted by R, removes the "δ-component" of a distribution. This "surgery" does not go well without careful preparations. Therefore, the δ-eliminator needs to be introduced as a partially defined operator on D 0+ .
Let I ⊆ R be open. The space of distributions of order zero on I , denoted by D 0 (I ), is the set of distributions μ ∈ D (I ) satisfying The δ-value operator is the partially defined linear form V given by where the right hand side of (3.3b) is defined for ε > 0 small enough by for any sequence (ϕ n ) ⊆ D(R) such that ϕ n 1 ]−ε,ε[ for n → ∞. The δ-projector is the partially defined operator The δ-eliminator is the complementary projection of P, that is The δ-residual space of causal distributions is the range of the δ-remover It is immediate from the definition of the space D 0+,δ that, for any ε > 0 and any f ∈ D 0+,δ , the distribution f can be written as The δ-coefficient can be defined equivalently as V f = μ({0}) for f = μ + g with μ, g as in (3.6), due to the continuity of Radon measures [1,6]. Here μ({0}) denotes the μ-measure of the Borel set {0} ⊆ R.
Proof The proposition is immediate from Equation (3.6) and the following three facts: It holds inf supp( f * g) ≥ inf supp f +inf supp g for all f , g ∈ D 0+ , the set D 0+ ∩D 0 is closed with respect to convolution, and δ ∈ D 0+,δ .

Proposition 2 The operators P and R are complementary linear projection operators on
where a and g are given by a = V f and g = R f . In addition, Proof It is immediate from the Definitions (3.3) that P δ = δ. Thus, P • P = P and the first part of the proposition follows from basic linear algebra. For Equation (3.8), recall the remarks below Equations (3.1) and (3.6). Then use that for all μ, ν ∈ D 0 ∩ D 0+ . The latter follows from Equation (1)  Equation (3.8) means, that the operator V defines an "augmentation" of the convolution algebra D 0+,δ . That is, the operator V defines a linear homomorphism (3.10) In particular, D 0+,δ / is a non-unitary convolution subalgebra of D 0+,δ .

Generalized initial value operators, eliminators and projectors
Composing the operators V, R and P with the distributional fractional integrals and derivatives from Subsection 2.2 yields operators that act on the Y γ -part of a distribution in an analogous way. This section studies these operators and, for the case of real orders, the generated algebra of partially defined linear operators.
It is immediate from the definitions, that the operators I γ + and D γ + induce bijections between spaces D Y,γ and D / Y,γ of different orders. For instance, one has a bijection The domains of the operators from Definition 2 are (3.14) Ranges and kernels of these operators are given by The brackets − denote the (complex) linear span. There hold the composition and commutation rules Analogous to the operators V, P and R, one has the relations The operators P γ and R γ are complementary projections, just as P and R, and a statement analogous to (3.7) holds. However, the operator V γ does not define an augmentation. From Equation (2.8) and Equation (3.10), one obtains the convolution inclusions

Series expansions in causal power distributions
Absolutely convergent series over expressions c γ Y γ , with c γ ∈ C and γ ∈ R, are now considered as a kind of fractional Taylor series expansion. Theorem 1 shows how to extract the coefficients c γ using coefficient operators V γ and composite eliminators R γ 1 ,...,γ n .
The statement O (U ) = γ will be used as equivalent notation for conditions (3.29).
The notation O (U ) ∈ R means that there exists γ ∈ R, such that conditions (3.29) hold true.
where U k is recursively defined by the initial condition U 0 := U and Lemma 6 Let (γ n ) be a sequence of strictly increasing positive numbers and (c n ) a sequence of complex numbers. If the series ∞ n=0 c n Y γ n converges absolutely in D , then it converges absolutely in L 1 Corollary 2 If the series from Lemma 6 converges absolutely in D , then its limit belongs to D 0+,δ / .
for all γ, δ, c, d ∈ C, which proves the proposition.

Generalized sequential fractional derivatives
The

Definition and fundamental properties
Generalized sequential fractional derivatives are introduced as catenations of a distributional fractional derivative and a composite of eliminators.

Definition 5
Let α ∈ R, γ 1 < · · · < γ n and n ∈ N. The sequential fractional derivative of order α and sequential type (γ 1 , . . . , γ n ) is defined as where D α| 0+ := D α + is used to link the notation for sequential fractional derivatives with that for simple fractional derivatives.
The operators defined in Equation (4.1) coincide with alternating compositions of eliminators and fractional derivatives with real orders. This means operators Note, that R • D α + • R = D α + • R holds for all α < 0, so that α 1 , . . . , α n−1 > 0 may be assumed without loss of generality. The expression on the right hand side of Equation because the operator D α + is bijective. Thus, domain and kernel are given by Equation (3.23) and (3.24a).

Remark 2
The normal form (4.4c) exists as well for composite operators of the form Equation (4.7) holds also when K ∈ D Y,O(K ) . However, even for K ∈ A [Y R ] a similar description of composites of the form R γ 1 ,...,γ n • C K leads to a more complicated expression that involves multiple case distinctions. Let us remark, that for future studies of the operators C K • R β 1 ,...,β n , it seems plausible to generalize the eliminators as well.

Restriction and extrapolation of distributions
Let I ⊆ R be an interval and J ⊆ R an open interval. The restriction of a distribution f : The restriction of a Radon measure μ : K (R) → C to I is defined by μ| I (ϕ) := μ( ϕ| zero R ) with ϕ| zero R (x) equal to 0 for x ∈ I and equal to ϕ(x) for x / ∈ I . Here, the Borel measurable function ϕ| zero R is the zero extrapolation of ϕ ∈ K (I ).
The restriction of a distribution f ∈ D 0+,δ to R 0+ is defined as the linear form where μ and g are defined as in (3.6), The zero extrapolation f | zero where μ and g are defined as in (3.6) and with μ resp. g as in the representation formula (3.6) and their zero extrapolations defined by Equations (4.10) resp. (4.11). The continuous constant extrapolation f | c.c.
R of f is defined as where 1 R − denotes the indicator function for the open left half axis. Restricting a distribution from D Y,0 to R + and extending to R by zero afterwards has the same effect as applying the eliminator. Restricting to R 0+ and zero-extrapolating to R has no effect. This can be expressed as Because R reduces to the identity on D 0+,δ / , the mappings define mutually inverse isomorphisms that restrict to isomorphmisms between D Y,α and D Y,α for α > 0.

Examples
Numerous operators can be reinterpreted as generalized sequential fractional derivatives on subspaces of D 0+ in the context of restriction to the right half axis and extrapolation to the full real axis. This section collects some examples.

Distributional derivatives on the right half axis
The distributional derivative is well defined as an operator acting on distributions defined on the open right half axis R + . Using restriction and zero extrapolation operators the sequential derivative D 1|1 0+ can be related to the operator D : The sequential derivative D 1|1 0+ f of f ∈ D Y,1 (R) is a modification of the distributional derivative of f that ignores jumps at the origin.
According to (4.16b), this is equivalent to interpreting f as a distribution on the open right half axis and extrapolating by zero after the calculation of the derivative.
Unfortunately, the case of the closed right half axis is more involved, despite the existence of the isomorphism D Y,1 (R + ) → D Y,1 (R 0+ ). The reason is, that the definition of the derivative of a function f ∈ D Y,1 (R 0+ ) at the origin depends on its extrapolation to a neighborhood of the origin. In particular, it holds for all f ∈ D Y,1 (R 0+ ). These relations reflect the fact that it is always necessary give a precise interpretation of the derivatives when a boundary is involved.

Riemann-Liouville fractional integrals
Riemann-Liouville fractional integrals with orders α ∈ H, that act on functions on the closed or open right half axis, can be represented using the sequential derivatives D −α| Note, that the Riemann-Liouville integral RL I α 0+ μ(t) of a measure μ ∈ M (R) or μ ∈ M (R + ) can be defined, in the almost everywhere sense for the variable t, by applying the classical formula to μ. (In the latter case, the domains of integration must include the point zero.) This extends most common definitions of the Riemann-Liouville integral (compare [24,Remark 4.4]).
With these definitions one obtains the relations However, the action of the sequential derivative D −α| ,δ can not be described using Riemann-Liouville integrals defined on the closed or open right half axis, because neither the corresponding restriction operators nor the corresponding extrapolation operators are well defined for such functions f .

Generalized Riemann-Liouville fractional derivatives [7]
The fractional derivatives of Riemann-Liouville and their generalizations can be reinterpreted by replacing the first order derivatives with D  (4.20) With the notations from Equation (4.19) the well-known relation [7, p.434] between two derivatives of same order α ∈ ]0, 1], but distinct types 0 ≤ μ 1 < μ 2 ≤ 1 reads The relation is immediate from the fact that

nth-level fractional derivatives [5,24]
The sequential fractional derivatives from Definition 5 extend earlier definitions as studied in [5], or later ones, in [24]. In the following, the operators from Definition 5 are compared with the "nth-level derivatives" from Definition 3.6 in [24]. Similar remarks apply to the sequential derivatives given in [31]. For the purpose of a more convenient comparison the domains X 1 nL of nth-level derivatives are defined as spaces of functions on the whole real line. This definition becomes equal to the definition in Equation (3.41) from [24] when the functions are restricted to ]0, 1[. More precisely, let 0 < α ≤ 1, γ = (γ 1 , . . . , γ n ) such that γ k ≥ 0 and α+s k ≤ k for all k = 1, . . . , n, with the notation s k := k i=1 γ i . The indices for the corresponding generalized sequential fractional derivatives are defined as δ n−k+1 := α + s k − k + 1. These satisfy δ 1 < · · · < δ n if and only if γ k < 1 for k = 2, . . . , n − 1. Further, it holds 0 < δ 1 if and only if α + s n > n + 1.
Consider now, the space X 1 nL,+ (R), defined as The space X 1 nL from Definition 3.6 in [24] can be characterized as the set of restrictions f | ]0,1[ with f ∈ X 1 nL,+ (R) by using the following lemma.

Generalized fundamental theorem of fractional calculus
The composition law from Proposition 5 implies: Corollary 3 Let α ∈ R and γ 1 < · · · < γ n . Then D α|γ 1 ,...,γ n 0+ • I α The domain D 0+,δ / contains all domains that are commonly used to define Riemann-Liouville operators on the right half axis, as discussed on page 17. In particular, it

Generalized derivatives with Sonine kernels
Generalizations of fractional derivatives and integrals where the kernels Y α are replaced by pairs of kernels K and L from L 1 loc ∩ D 0+ that satisfy the Sonine relations K * L = Y 1 were introduced in [18] and recently discussed in [21,29]. The Sonine relations imply (4.29) The convolution operator C K and the composite L D 1 0+ := C L • R 1 (4.30) correspond to the generalized integral and the generalized derivative of Caputo type in [29,Eq. (26), (20)]. Equations (4.29) and (4.7) imply From similar considerations as for Equations (4.27) it is found that the Equations (4.31) imply Theorem 1 from [29].

Two counterexamples
The The formulas follow immediately from the form of the derivative operators in Equations (1) and (3) in [4] and the formulas for the corresponding operators, CF J α 0+ = C CF K α and ABC J α 0+ = C ABC K α , from Equations (5) and (9) in [4]. An application of Equation (4.7) yields the relations that correspond to Equations (6) and (10) from [4]. This shows, once again, that these operators do not satisfy a Fundamental Theorem such as (4.31).

Generalized sequential fractional initial value problems
The main purpose of this section is to study the kernels of generalized sequential fractional differential operators. Subsection 5.1 provides an algorithm (Theorem 2) that can be used to simplify linear combinations of sequential derivatives to an eventually simpler normal form. The result is used in Subsection 5.2 to derive a representation formula for kernels (Theorem 3) and a characterization of the maximal injective domains (Theorem 4). The case of sequential derivatives with distinct order is considered in Subsection 5.3 in more detail. This yields a simplified criterion for kernel functions in general (Theorem 5) and a full characterization for important special cases (Theorem 6). Last, in Subsection 5.4, all possible cases with two types are described.

Simplifying linear combinations of sequential derivatives
Due to annihilation effects evaluating linear combinations of partially defined linear operators can be cumbersome. The theorem to be established below provides an algorithm that reduces sums of composites C K • R Γ of convolution operators C K and composite eliminators R Γ to a simple expression. Due to the theorem, the operators to be investigated can be assumed to be given in the form (5.3). Note, that every finite subset Γ ⊆ R can be represented as Γ = {γ 1 , . . . , γ n } with unique γ 1 < · · · < γ n and n ∈ N 0 . Thus, the notations are well defined with the conventions R ∅ = E and P ∅ = 0.
Proof Reducibility to the normal form: Let n ∈ N 0 . Define the total number of eliminators arising in expression (5.2) as Σ := n k=1 #Γ k . The reducibility will be proved via of induction over Σ. If Σ = 0 or n = 0 the expression (5.2) obviously has the form (5.3a). Thus, the Lemma holds for Σ = 0 or n = 0.
Assume now, that Σ > 0 and n > 0. The induction hypothesis is, that the statement of the Lemma holds for all expressions of the form (5.2) that have a total number of eliminators Σ < Σ. Consider the order parameter In the expression (5.2), all operators R γ with γ < γ 1 can be canceled from the right due to (2.14), (3.14) and Corollary 1. A new expression of the form (5.2) emerges with a reduced total number of eliminators Σ < Σ. Thus, the induction hypothesis applies to the new expression and it thus reduces to the form (5.3a). If no eliminator can be canceled in this way, then for all k = 1, . . . , n either γ 1 = max Γ k or Γ k = ∅. Then, the operator R γ 1 can be factored out as The expression inside the brackets has the form (5.2) and satisfies Σ < Σ. Thus, the induction hypothesis applies and the whole expression (5.5) can be written in the form and therefore Y γ ∈ dom C for all γ ∈ R with γ ≥ γ m . Further, (3.23) and (4.3) imply Thus, the orders γ 1 < · · · < γ m and the number m are uniquely determined by C. For γ ∈ R one calculates From this equations it is clear how to represent the convolution kernels U l , l = 0, . . . , m as linear combinations of expressions Y −γ * C(Y γ ). Therefore, the U l are uniquely determined by the operator C.

The structure of the kernel
Consider an operator C in the form of Equation (5.3a). Changing to a different normal form makes it easy to obtain certain structural results on its kernel ker C. Using Lemma 5 one calculates directly that With these notations, the structure of ker C is characterized by the following.

Theorem 3 Every distribution K ∈ dom C satisfies
where K l and L l are defined as In particular, the kernel satisfies the inclusions The right hand side of (5.10) can be rewritten as Let K ∈ dom C and setK = R γ 1 ,...,γ l 0 (K ). Applying the operator from (5.14) to K and setting the result to zero yields Solving forK in Eq. (5.15) and noting that the distributions Y γ 1 , . . . , Y γ l 0 belong to ker C, due to (5.14), proves the proposition.
The following theorem provides the maximal domains for admissible inhomogeneities g of the equation C f = g.

Theorem 4
The operator C from (5.10) restricts to a bijection Proof This is immediate from the formula (4.3) for kernels of sequential derivatives and Proposition 4.
Using Theorem 3 the kernel of the operator D can be expressed in terms of the functions K l and the composed coefficient operators L l . The distributions K l are given by and the distributions W l are given by with the notation Z k := μ k Y β k and the parameters β k := α n − α n−k and μ k := λ n−k /λ n for k = 0, . . . , n.
with the order parameter where β k was defined just after Eq. (5.18b).

Example 5
Linear combinations of a first order time derivative and a generalized Riemann-Liouville fractional derivative were used in [10] as infinitesimal generators for composite fractional time evolutions. Reinterpreting the operator D α,μ 0+ in [10] along the lines of eq. (4.19) as D α;μ 0+ the solution f (t), t ≥ 0 of the initial value problem (32) in [10] can be represented as the restriction f := K | R 0+ of a distribution K ∈ D 0+ , where the distribution K is the solution to the following generalized initial value problem: Define the operator where λ 1 = (τ α ) α , λ 2 = τ 1 and the parameters τ α , τ 1 > 0, 0 < α < 1 and 0 ≤ μ ≤ 1. According to Theorem 2 the equality holds if and only if μ = 1 (corresponding to Equation (34) in [10]). In any case, there exists a unique distribution K ∈ D 0+ such that K ∈ ker D, The relation R = S * Y 1 between the normalized relaxation function R = K +Y 1 and the time-domain representation of the corresponding normalized susceptibility S was verified using convolutional calculus in [17]. Here,Y 1 denotes the reflection of Y 1 . The Laplace-transform of S (in the sense of [33]) defines the susceptibility function ε(u) in the frequency domain. Another interpretation of the infinitesimal generator D from [10] was recently given in [11]. It is obtained from observing D α;μ 0+ = R μ−μα • D α| 0+ and shifting R μ−μα from D α| 0+ to the infinitesimal generator D 1|1 0+ of translations in eq. (5.41). The physical motivation for this are relaxation processes that are too fast to be resolved [11]. The modified interpretation leads to a sequential first order derivative where λ 1 = (τ α ) α , λ 2 = τ 1 , δ = 1 + μ(1 − α) with the parameters τ α , τ 1 > 0, 0 < α < 1 and 0 < μ < 1. Note, that 1 − δ < 1 − α. Therefore, Theorem 6 guarantees the existence of a unique distributionK ∈ D 0+ such that K ∈ kerD, V 1K = 1, V 1,δK = −ν/λ 2 . As above, the relaxation motionR satisfiesR =K +Y 1 =S * Y 1 , with the normalized susceptibilityS given by the convolution quotient The normalized susceptibility function ε(u) from Equation (3) in [11] coincides with the Laplace-transform of the distributionS. Thus,S is the time-domain representation ofε(u) and it follows that the function t →K (t) for t ≥ 0 is the normalized relaxation motion corresponding toε(u). As shown in [11,17] the solution (5.45) agrees over a range of 12 decades in time or frequency with a physical relaxation experiment.
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