On the Weyl Derivative of Positive Order


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Let f be a real-valued function of a real variable, and v a real number. Unless v is a nonnegative integer, the (left) Weyl derivative of f at a point t can exist only if f(u) is close to 0 as u approaches minus infinity, loosely speaking. Oldham and Spanier write pessimistically: "Functions other than exponentials, however, seldom yield finite differintegrals when the lower limit is minus infinity, and we shall have no occasion again to make use of these 'Weyl differintegrals'."

In similar vein, Lavoie, Osler and Tremblay remark: "Because the integrals defining these Weyl derivatives are improper, greater restrictions must be placed on the function F(x) than are needed when x sub 0 is finite. Also, general theorems about Weyl derivatives are often more difficult to formulate and prove than are corresponding theorems about Reimann-Liouville derivatives. Because of this difficulty, the authors suspect that the Weyl fractional derivative is an inferior tool for exploring the special functions, and thus it is given little consideration in this paper".

This paper introduces the modified Weyl derivative, which is the same as the Weyl derivative per se if v is negative or an integer. In the case of positive noninteger v, it usually agrees with the Weyl derivative if the Weyl derivative exists, but the restriction on the behavior of f(u) as u approaches minus infinity is substantially alleviated. Other variants of the Weyl derivative are subsumed in the same way, including certain definitions attributed to Marchaud (1927). Again the modified Weyl derivative exists under substantially broader conditions on f. At the same time it is more straightforwardly conceived and defined in terms of the Reimann-Louville derivative. Properties and representations of the modified Weyl derivative are discussed.



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