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The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.
A list of six desiderata1 is proposed that in our opinion would justify calling an operator (or ) a fractional derivative (or a fractional integral) of non-integer order . Derivatives and integrals of fractional order have a long history and, up until the recent proliferation of novel fractional derivatives, most definitions and interpretations of fractional operators seem to implicitly assume the desiderata of an operational calculus as formulated in this article.
Mathematical terms used in the formulation of our desiderata are defined in Appendix A. A family of operators with or is proposed to be called a family of fractional derivatives and integrals of order α (with )2 if and only if it satisfies the following six desiderata:
Integrals and derivatives of fractional order should be linear operators on linear spaces3.
On some subset4 the index law (semigroup property)
holds true for and , where denotes the domain of , and ∘ denotes composition of operators.
Restricted to a suitable subset of the domain of the fractional derivatives of order operate as left inverses
for all with , where is the identity on .
There is a subset of the domain of such that the limits
exist in some sense and define linear maps resp. .
The limiting map is the identity on , i.e., ;
The limiting map is a derivation on . This means it is possible to define a multiplication on such that the Leibniz rule
holds for all .
If the semigroup law (1) can be extended to all or , we propose to speak of fractional calculus. Our desiderata are obviously inspired by operational calculus. Recall that an operational calculus is a continuous one-to-one mapping between an algebra of functions and an algebra of operators such that the neutral elements match and algebraic relations are preserved. Extending the algebra from polynomial functions to convergent power series suffices for an operational calculus. More singular functions, namely non-analytic power functions, are required for fractional calculus.
Desiderata differing substantially from those above have been formulated in  (p. 5) and  (p. 5). Envisaging exclusively analytic functions the criteria given in  (p. 5) are extremely restrictive. In theory and applications it is nowadays imperative to include more general functions, measures and also distributions into the purview.
Given the extreme restrictions in  (p. 5) a more recent proposal  went to opposite extremes. Little or no attention is given to a domain of definition for the fractional derivatives in . Our desiderata for fractional derivatives again differ substantially from those in . Rather than requiring some form of the generalized Leibniz rule P5 in  for all we desire the Leibniz rule only for , and that differs not only from P5, but also from P3. In addition the identity rule P2 in  does not restrict the admissible operators at all. As long as there is no continuity in or a well defined limit, the identity rule can always be fulfilled, simply by setting . More generally, Ref.  seems to neglect parameters other than , or the topological and operator-theoretic implications of the limit in Equation (2) .
Our desiderata do not include non-locality of fractional derivatives. Fractional derivatives, that are local operators, were introduced in [3,4] and are discussed further in  and Section 7 of . Contrary to P3 in  we do not constrain the limits with for , because we wish to allow more generality.
To illustrate our desiderata in a simple case consider fractional operators of Riemann-Liouville type for complex valued functions on a compact interval with when the fractional order is real and restricted to . The (right-sided) Riemann-Liouville fractional integrals of order with lower limit a are defined by setting for and
for and . The (right-sided) Riemann-Liouville-type fractional derivatives of order are defined as  (p. 434)
where the number parametrizes different types of fractional derivatives. The classical Riemann-Liouville derivative is of type , while the popular Liouville-Caputo derivative has type  (p. 10).
Both operator families are linear and desideratum (a) can be fulfilled on numerous linear spaces due to the compactness of . Examples are the Lebesgue spaces with or Hölder spaces with . Desideratum (c) holds e.g. for with and . The Riemann-Liouville fractional integrals (extended from to ) are a strongly continuous semigroup of operators with respect to the parameter , and obey the index law (1) in desideratum (b) for all on or suitable subspaces. The desiderata (d), (e) and (f) can then be derived with the help of the semigroup property. For the Riemann-Liouville operators they hold e.g. for smooth (infinitely often differentiable) functions .
For infinite intervals or for generalized functions the problem of domains may become more involved and our desiderata may become more restrictive. As an example consider the family of symmetric Riesz operators
on the real line. In this case the limiting operator is again well defined, but does not fulfill desideratum (f). Instead, it fulfills Leibniz’ formula for with , i.e.,
We propose to call such operators obeying Leibniz formula for with pseudofractional derivatives or fractional pseudoderivatives.
Of course, our desiderata do not define fractional derivatives and integrals in a unique way. Still, they considerably restrict the set of admissible operators as seen above. In our opinion the above desiderata formulate crucial constraints for the development of a meaningful mathematical theory of fractional calculus and its reasonable applications.
Much work has been done on mathematical interpretations of fractional derivatives and integrals. The results are documented in numerous texts and treatises (see [6,8] for recent reviews). It seems however, that the connection (or not) of classical and recent fractional calculi with historical and contemporary forms of operational and functional calculi such as Heaviside-Mikusinski calculus, Dunford-Schwarz calculus, or Hille-Phillips calculus is a rich source of numerous open problems whose speedy solution would seem pertinent to advance and ultimately consolidate the field. We hope that the desiderata above are sufficiently restrictive to initiate a discussion of these pressing problems, and thereby stimulate readers and contributors to address some of these open problems in their areas of expertise and interest.
For the convenience of readers from non-mathematical disciplines we recall some definitions:
A real (or complex) linear space (or vector space) over the field (or ) of real (or complex) numbers is a non-empty set with two operations called addition and scalar multiplication fulfilling the usual rules of vector addition and multiplication of vectors with numbers5.
Let be linear spaces (vector spaces). A linear operator is a linear subspace of the direct sum , where
is the linear space of pairs with , and addition defined as for all and . The identity operator is defined as
The domain and range of a linear operator are
The inverse of is defined as
with domain . For their sum is defined as
with . For the composition is the linear operator defined as
Let be a fixed real number. The Lebesgue space consists of those Lebesgue measurable functions on the intervall for which the norm
is finite. For the space is the set of all Lebesgue measurable functions such that is finite where denotes supremum up to sets of Lebesgue measure zero (called essential supremum). The space consists of all bounded functions on . Its norm is . The space consists of all continuous functions. Its norm is again because continuous functions on a compact interval are also bounded. For and the number
is called Hölder constant of f on of Hölder order γ. The Hölder space is defined as
and is a Banach space for the norm
For with and it is defined as
where is the k-th derivative of f.
A family , of bounded linear operators on a Banach space is called a strongly continuous one-parameter semigroup if it satisfies:
for all .
For every the orbit maps are continuous from into .
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properties to be desired.
It is common to use only one of the symbols or in the sense that either or . In this paper we keep the distinction between and by assuming unless otherwise specified. This entails discussing the case separately whenever necessary.
Dependencies of and on other parameters are usually present, but notationally suppressed.
Here the index (b) refers to desideratum (b). The same applies in desiderata (d)–(f) below.
(a) for all also , (b) , (c) , (d) there exists an element (called origin ) such that for all , (e) for all there is an element such that (f) for all (or ) and an element is defined, (g) for all (or ) and one has , (h) for all (or ) and one has , (i) for all (or ) and one has , and (j) for all .
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