An Approach to Fractional Powers of Operators via Fractional Differences

Abstract

Jo which represents 'Hadamard's finite part' of the divergent 'fractional integral' [F( -a)]-1 ^w^^T^fdu of negative order -a. This method of defining fractional powers by a closure process was also used by Komatsu ([12]); yet he considered the operator J a not on D(A) but on a certain intermediate space of D(A) and X. If, instead of J", one takes the integral -IT( -a)]" 1 f V«-i|7 -T(u)]fdu (f e X)

Unbounded functional calculus for bounded groups with applications

“…Furthermore we prove a generation theorem and a Grünwald-type approximation formula for fractional powers A α with exponents α > 0 being different from an odd integer (Theorems 4.6 and 4.9). These extend results of [21] and [50] and are applied in the last part. The third part consists of a description of the problem in subsurface hydrology that has motivated this research.…”

“…As an example we prove a shifted finite difference formula for the fractional powers based on a shifted Grünwald formula which was developed in [38] for the fractional derivatives of functions. A fractional difference formula was obtained by Westphal in [50] for generators of semigroups which was then used to obtain the one in the L 1 (R + ) setting. Following the philosophy of the present paper we show once again that it is easily done in the other direction, too: the L 1 (R)-result implies the abstract one.…”

Sharp Ul’yanov-type inequalities using fractional smoothness

“…The key concept in the sharp Ul'yanov inequality is the use of the modulus of smoothness of fractional order. For a function f ∈ L p , 1 p ∞, the modulus of smoothness ω α ( f, δ) p of fractional order α > 0 is given by [3,32] ω α ( f, δ) p = sup…”

Fractional Powers of Infinitesimal Generators of Semigroups

“…We want to emphasize, however, that, in principle, the calculus can be reduced to dealing with Laplace transforms of L 1 -functions; cf. [45] and [46]. This is not surprising if one has in mind that an 1K+ -summable distribution convolved with a suitable test function has to belong to Z 1 (K + ).…”

“…Alternatively, one may proceed from the following definition considered in [46], (1.3) which is motivated by the characterization of fractional derivatives due to Liouville, Griinwald and Letnikov. The methods used in both of these papers heavily depend on the Laplace transform which is a very natural tool when dealing with families of operators that satisfy the functional equation of the exponential function.…”