Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line
© Babenko et al.; licensee Springer. 2014
Received: 11 August 2014
Accepted: 2 December 2014
Published: 17 December 2014
In this paper we establish some new Kolmogorov type inequalities for the Marchaud and Hadamard fractional derivatives of the functions defined on a real axis or semi-axis. Simultaneously we solve two related problems: the Stechkin problem on the best approximation of unbounded operators by bounded ones on a given class of elements and the problem of optimal recovery of an operator on elements from some class given with prescribed error.
Inequalities estimating the norm of an intermediate derivative of a function in terms of the norm of the function itself and the norm of its higher-order derivative (inequalities of Kolmogorov type) are important in many areas of mathematics and its applications. Due to the efforts of many mathematicians, nowadays, a great number of sharp Kolmogorov type inequalities are known (see, for instance, surveys [1–3] and monographs [4–6]). In many questions of analysis and its applications the study of fractional-order derivatives is also important (see, for instance, ). For some known results on the Kolmogorov type inequalities for derivatives of fractional order we refer the reader to [8–15], the book [, Chapter 2] and references therein.
In this paper we shall obtain some new Kolmogorov type inequalities for fractional derivatives. Simultaneously, we consider two closely related problems: the Stechkin problem on approximation of unbounded operators by bounded ones on a given class of elements Q, and the problem of optimal recovery of unbounded operator on the class Q under assumption that elements in Q are given with known error (for more information see [1, 2] and [, Section 7.1]).
1.1 The Kolmogorov type inequalities
For and , by we denote the space of the functions having locally absolutely continuous on G derivative and such that .
Naturally, inequalities with the lowest possible (sharp) constant K are of the most interest. We refer the reader to [1, 2] and the books [4, 6] for the detailed survey on the Kolmogorov type inequalities and discussion of related questions.
Together with inequalities (1) the study of inequalities between the norms of intermediate function derivative, the function itself and its higher-order derivative in spaces more general than are also important. In Sections 2-4 we shall obtain several inequalities between the norms of derivatives in ideal lattices (see [, Chapter 2, Section 2]).
For a function , the right hand sided derivatives and are defined by formulas (4) and (5), respectively. The left hand sided derivatives and are defined with the help of slightly different constructions (see [, Sections 5.1, 5.5]), and we shall not study these derivatives here.
It is well known (see ) that for ‘good’ functions . However, construction (5) is also suitable for a wider class of the functions, e.g. constant functions or functions whose power growth at infinity has order lower than k.
Similarly to inequalities (1) for derivatives of integer order, it is easy to see that the constant K in (7) is finite only if parameters λ and μ satisfy equalities (2).
, , , and , - Geĭsberg ;
, , , and , - Arestov  (for );
, , , and , - Arestov  (for );
or , , , , - Buslaev and Tihomirov  (for the Weyl derivative);
or , , , , , - Babenko and Churilova ;
or , , , and , the norm of is considered in an ideal lattice;
, , , and ;
, , , and , the norm of is considered in an ideal lattice;
, , , , and ;
or , , , , and .
1.2 The Stechkin problem
The problem of the best approximation of unbounded operators by linear bounded ones is close to the problem of finding sharp constants in inequalities (1), (7) and, furthermore, presents an independent interest. We follow  (see also surveys [1, 2]) to set the problem rigorously.
is called the modulus of continuity of the operator A on the set Q.
The Stechkin problem on the best approximation of the operator A by linear bounded operators on set Q consists in evaluating quantity (9) and finding extremal operators (if any exists) delivering an infimum in the right hand part of (9).
Consequently, the operator is extremal in problem (9) for , and the element in problem (8) for .
Therefore, in all cases when the sharp constant K in inequality (7) for is found, we immediately know the exact value of the quantity of the best approximation of the operator by linear bounded operators on the class .
1.3 The problem of optimal recovery of operators on elements given with an error
Another problem that is closely related to the Stechkin problem and sharp Kolmogorov type inequalities is the problem of optimal recovery of an operator with the help of the set of linear operators (or mappings in general) on elements of some set that are given with an error. We follow  to set the problem rigorously.
called the best recovery of the operator A with the help of mappings from ℛ on elements Q given with prescribed error δ. The detailed survey of existing results and further references can be found, for instance, in . The following statement is a corollary of the result by Arestov [, Theorem 2.1] that indicates the close relations between this problem and the Stechkin problem.
So once the sharp constant in inequality (7) is found, we immediately know the value of the error of optimal recovery of the operator by operators from (or ℒ) on elements of the class given with error δ.
1.4 Organization of the paper
The paper is organized in the following way. Section 2 is devoted to auxiliary results concerning properties of the Marchaud fractional derivatives: existence, continuity, and integral representation in terms of the higher-order function derivative. Then we establish some sufficient conditions when sharp Kolmogorov type inequalities (7) can be written and derive some consequences from these conditions for in Section 4. Finally, in Section 5 we present applications of the main results: the Kolmogorov problem for three numbers consisting in finding necessary and sufficient conditions on the triple of real positive numbers that guarantee the existence of a function attaining these numbers as the norms of its three consecutive derivatives, and sharp Kolmogorov type inequalities for the weighted norms of the Hadamard fractional derivatives.
2 Auxiliary results
In this section we formulate auxiliary propositions on the existence and continuity of the Marchaud fractional derivative and its integral representation in terms of the higher-order derivative. These and similar questions were studied by many mathematicians. For an overview of known results we refer the reader to the books [7, 25] and references therein.
2.1 Definitions and results
It is clear that is the ideal lattice on G and is a subspace in the space dual to E. Ideal lattices generalize many important spaces e.g. spaces , , the Orlicz spaces , the Lorentz spaces , the Marcinkiewicz spaces , etc.
In what follows we would also say that an ideal lattice E is semi shift invariant if, for every and , we have and either if or if .
Let , , and F let be an ideal lattice. By and we denote the spaces of the functions and , respectively, such that is locally absolutely continuous on G and . In addition, let stand for the characteristic (indicator) function of a measurable set .
where is the associated space to E. Then exists and is continuous on G, for every function .
Proposition 3 Let or , , , E be a semi-shift invariant lattice on G satisfying condition (12) and F be a semi-shift invariant lattice on G such that . Then exists, for every and , and the integral representation (14) for holds true.
In particular, when , , both conditions (12) and (13) are equivalent to the inequality . So the following corollaries hold true.
Proposition 4 Let or , , , and . Then, for every , exists and is continuous on G, and (14) holds true.
Proposition 5 Let or , , , and . Then exists, for every and , and (14) holds true.
2.2 The proofs of auxiliary results
which proves the existence and uniform boundedness of derivative at an arbitrary point .
Therefore, the is continuous on G. □
which finishes the proof. □
Hence, exists, for every . Finally, we remark that equality (14) immediately holds true if exists. The proof is finished. □
3 Main results
Let us present results on some general sufficient conditions allowing one to write a sharp Kolmogorov type inequality in various situations. We start with the Kolmogorov type inequality between the uniform norms of the Marchaud fractional derivative of a function, the function itself and its higher-order derivative. In Section 3.2 we extend this result on the case of inequalities between the norms of the function and its derivatives in an ideal lattice. Then in Section 3.3 we give another extension of results of Section 3.1 on the case of inequalities between the uniform norms of the Marchaud fractional derivative of a function, the uniform norm of the function itself, and the norm of the higher-order derivative in an ideal lattice.
which is an additive form of the Kolmogorov type inequality (7). If for some operators T and R there exists a function turning the above inequality into an equality then the corresponding Kolmogorov type inequality is sharp.
We remark that this idea is not new and is already contained in  by Stechkin. Besides, some similar ideas were even in the papers by Landau and Hadamard. The corresponding operators T and R as well as the extremal function f were found in many cases (see [28–30], and surveys [1, 2, 4] for more details).
3.1 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case of uniform norms
The following results hold true.
then (18) is sharp and the function turns (18) into an equality.
Minimizing the right hand part of (18) by h we obtain the next consequence.
- 2.For , , and , the extremal function Φ, and the corresponding function Ω that satisfy the conditions of Corollary 1 were found by Arestov [, Theorem 3]:
- 3.For , , and , the extremal function Φ and the corresponding function Ω that satisfy the conditions of Corollary 1 were also found by Arestov [, p.32]:
For integer values of k, the extremal function Φ on ℝ in inequality (21) was found by Kolmogorov  (see also ), for every . In the surveys [1, 2, 4] the reader could find more references and a detailed history of the subject and overview of cases when the extremal function Φ in inequality (21) on is known.
In addition, for integer values of k, the function Ω on for which inequality (18) is sharp was explicitly constructed by Stechkin  in the case . In the case the existence of such a function Ω was proved by Domar  and explicitly it was constructed by Stechkin  for , Arestov  for , and Buslaev  for .
Therefore, the statement of the theorem is proved in the case .
Now, we let and be arbitrary, and consider the function , . Evidently, and by substituting into (22) we derive inequality (18). Clearly, turns (18) into an equality. □
3.2 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case of norms in an ideal lattice
Let us generalize Theorem 1 to the case of Kolmogorov type inequalities between the norms of the Marchaud fractional derivative of a function, the function itself, and its higher-order derivative in an ideal lattice.
An immediate consequence of Theorem 2 is the following.
Evidently, inequality (23) is sharp for . In Section 4.1 we shall show that this inequality is also sharp when , , and . For integer values of k and , inequality (23) is known as the Stein inequality  (see also [36, 37]).
The proof is finished. □
3.3 The Kolmogorov type inequalities for the Marchaud fractional derivatives: case when the norm of the higher-order derivative is considered in an ideal lattice
In this subsection we generalize the results of Section 3.1 on the case when the norm of the higher-order derivative is taken in an ideal lattice. For convenience, we split the subsection into two parts: first we present results concerning the case when extremal function in the Kolmogorov type inequality (i.e. turning it into an equality) exists and then we present results concerning the case when the extremal function in the Kolmogorov type inequality does not exist. For integral-order derivatives, the existence of an extremal function in the Kolmogorov type inequalities (1) was proved in the case when , , and inequality (3) is strict. For the corresponding results, we refer the reader to [24, 38, 39].
3.3.1 Case of existence of extremal function in the Kolmogorov type inequality
then inequality (24) is sharp and Φ turns (24) into an equality.
We remark that Theorem 3 can be generalized as follows.
For the spaces , , we obtain the following consequence.
hold true. Moreover, the function turns (26) and (27) into equalities.
We remark that Theorems 3.1.2 and 3.2.2  are concretizations of Corollary 3. In addition, in the case of integer values of k and and , the functions Φ and Ω satisfying conditions of Corollary 26 were explicitly constructed by Arestov in .
The extremity of the function Φ can be proved in a similar way to Theorem 1. □
Hence, by Theorem 3 the desired inequality (26) holds true and the function turns (26) into an equality. Finally, minimizing the right hand part of (26) by the variable h, we arrive at inequality (27). The proof is finished. □
3.3.2 Case of non-existence of extremal function in the Kolmogorov type inequality
Let us present two results showing when conditions (19) and (25) can be relaxed.
are valid. Then inequality (24) holds true and is sharp.
In the case we obtain the following.
We remark that for integer values of k and , the functions Φ and Ω satisfying conditions of Corollary 4 were constructed by Arestov in .
The proof is finished. □
The proof is finished. □
4 Consequences of main results
In this section we deduce new sharp Kolmogorov type inequalities from the results of the previous section when the order of the higher-order derivative is 1 or 2.
4.1 Case and
The following proposition is the consequence of Theorem 5.
It is easy to check that (17) holds true, for every , , and .
Therefore, the function Ω and the family of the functions satisfy the assumptions of Theorem 5. Hence, inequality (29) holds true and is sharp. □
Next, we formulate the following Stein type inequality.
Plugging the latter relations into the first of inequality (31) we turn it into an equality. The proof is finished. □
4.2 Case , , and
The following consequence of Theorem 5 holds true.
Therefore, the function Ω and the family of the functions satisfy assumptions of Theorem 5. Hence, inequality (32) holds true and is sharp. □
4.3 Case , , and
The next proposition is a consequence of Corollary 3.
We start with the proof of an auxiliary lemma.
We remark that in some cases the pair can be found explicitly, e.g. and .
Hence, there exists such that . The latter implies that and satisfies system (33). □
Moreover, , , , for every , decreases on and increases on . Hence, the functions Ω and satisfy the conditions of Corollary 3. □