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Linear algebraic foundations of the operational calculi. (English) Zbl 1223.44003

Let [p k ;k[ be a group under the operation p k p n =p k+n and let 𝔉 be the algebra of formal Laurent series generated by the elements p k over the complex numbers. The elements of 𝔉 are of the form a k p k , where the coefficients a k are complex numbers and only a finite number of them with k<0 are nonzero. The left-shift on 𝔉 is the operator S -1 that corresponds to the multiplication with p -1 . The modified shift L is defined by Lp k =p k-1 , if k0 and Lp 0 =0.

With the operational calculus developed here the authors are finding solutions of w(L)f=g, where w is a polynomial, g is a given element in 𝔉, and f is the unknown. If g can be written as an element of 𝔉 , a solution in the abstract space 𝔉 can be written as d w g, where d w is the solution of w(L)d w =p 0

Examples of this method are presented for the case of L being a difference operator Δ that acts on sequences, the case of L being a differential operator a(t)D+b(t), and the case of L being the fractional differential operator D α .


MSC:
44A45Classical operational calculus
44A40Calculus of Mikusiński and other operational calculi
13F25Formal power series rings
44A55Discrete operational calculus
26A33Fractional derivatives and integrals (real functions)
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