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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 .

44A45Classical operational calculus
44A40Calculus of Mikusiński and other operational calculi
13F25Formal power series rings
44A55Discrete operational calculus
26A33Fractional derivatives and integrals (real functions)
[1]Berg, L.: General operational calculus, Linear algebra appl. 84, 79-97 (1986) · Zbl 0562.44006 · doi:10.1016/0024-3795(86)90308-3
[2]Deakin, M. A. B.: The ascendancy of the Laplace transform and how it came about, Arch. hist. Exact sci. 44, 265-286 (1992) · Zbl 0763.01018 · doi:10.1007/BF00377050
[3]Dimovski, I. H.: Convolutional calculus, (1990)
[4]Flegg, H. G.: A survey of the development of operational calculus, Internat. J. Math. ed. Sci. tech. 2, 329-335 (1971) · Zbl 0219.44009 · doi:10.1080/0020739710020402
[5]Fuhrmann, P. A.: Functional models in linear algebra, Linear algebra appl. 162-164, 107-151 (1992) · Zbl 0756.15001 · doi:10.1016/0024-3795(92)90373-I
[6]Glaeske, H. -J.; Prudnikov, A. P.; Skórnik, K. A.: Operational calculus and related topics, (2006)
[7]Lützen, J.: Heaviside’s operational calculus and the attempts to rigorise it, Arch. hist. Exact sci. 21, 161-200 (1979) · Zbl 0419.01016 · doi:10.1007/BF00330405
[8]Mieloszyk, E.: Application of non-classical operational calculus to solving some boundary value problem, Integral transforms spec. Funct. 9, 287-292 (2000) · Zbl 0965.44003 · doi:10.1080/10652460008819262
[9]Mikusiński, J.: Operational calculus, (1959) · Zbl 0088.33002
[10]Péraire, Y.: Heaviside calculus with no Laplace transform, Integral transforms spec. Funct. 17, 221-230 (2006)
[11]Poularikas, A. D.: The transforms and applications handbook, (2000)
[12]Rota, G. -C.: On models for linear operators, Comm. pure appl. Math. 13, 469-472 (1960) · Zbl 0097.31604 · doi:10.1002/cpa.3160130309
[13]Rubel, L. A.: An operational calculus in miniature, Appl. anal. 6, 299-304 (1977) · Zbl 0362.44007 · doi:10.1080/00036817708839162
[14]Van Der Put, M.; Singer, M. F.: Galois theory of linear differential equations, Grundlehren math. Wiss. 328 (2003)
[15]Van Der Put, M.; Singer, M. F.: Galois theory of difference equations, Lecture notes in math. 1666 (1997)
[16]Verde-Star, L.: Inverses of generalized Vandermonde matrices, J. math. Anal. appl. 131, 341-353 (1988) · Zbl 0642.15005 · doi:10.1016/0022-247X(88)90210-7
[17]Verde-Star, L.: An algebraic approach to convolutions and transform methods, Adv. in appl. Math. 19, 117-143 (1997) · Zbl 0880.47004 · doi:10.1006/aama.1997.0530
[18]Verde-Star, L.: Functions of matrices, Linear algebra appl. 406, 285-300 (2005) · Zbl 1083.15035 · doi:10.1016/j.laa.2005.04.016
[19]Verde-Star, L.: Rational functions, Amer. math. Monthly 116, 804-827 (2009) · Zbl 1229.12001 · doi:10.4169/000298909X474873