Special functions as subordinated semigroups on the real line. (English) Zbl 1368.33008
Summary: We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.
MSC:
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 34A30 | Linear ordinary differential equations and systems |
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
| 46N20 | Applications of functional analysis to differential and integral equations |
| 47D03 | Groups and semigroups of linear operators |
Keywords:
convolution semigroups; special functions; Sobolev algebras; confluent hypergeometric functions; Hankel functions; modified Bessel functions; McDonald functions; Hermite functions; Poisson semigroup; Gauss-Weierstrass semigroup; fractional semigroup; backwards heat semigroupReferences:
| [1] | Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Math., vol. 96. Birkhäuser, Basel (2001) · Zbl 0978.34001 |
| [2] | Davies, E.B.: Integral Transforms and Their Applications, 2nd edn. Springer, Berlin (1985) · Zbl 0628.44001 |
| [3] | Galé, J.E., Miana, P.J.: One parameter groups of regular quasimultipliers. J. Funct. Anal. 237, 1–53 (2006) · Zbl 1104.47041 · doi:10.1016/j.jfa.2006.03.021 |
| [4] | Galé, J.E., Miana, P.J., Peña, A.: Hermite matrix-valued functions associated to matrix differential equations. Constr. Approx. 26, 93–113 (2007) · Zbl 1129.34019 · doi:10.1007/s00365-006-0664-1 |
| [5] | Galé, J.E., Miana, P.J., Pytlik, T.: Spectral properties and norm estimates associated to the $C\^{(k)}_{c}$ -functional calculus. J. Oper. Theory 48, 385–418 (2002) · Zbl 1019.47021 |
| [6] | Galé, J.E., Miana, P.J., Royo, J.J.: Nyman type theorem in convolution Sobolev algebras. Rev. Mat. Complut. (to appear). doi: 10.1007/s13163-010-0051-6 · Zbl 1273.46025 |
| [7] | Galé, J.E., Miana, P.J., Royo, J.J.: Estimates of the Laplace transform on convolution Sobolev algebras. J. Approx. Theory (to appear) · Zbl 1237.44001 |
| [8] | Galé, J.E., Pytlik, T.: Functional calculus for infinitesimal generators of holomorphic semigroups. J. Funct. Anal. 150(2), 307–355 (1997) · Zbl 0897.43002 · doi:10.1006/jfan.1997.3136 |
| [9] | Galé, J.E., Wrawrzyńczyk, A.: Standard ideals in weighted algebras of Korenblyum and Wiener types. Math. Scand. 108, 291–319 (2011) · Zbl 1228.46050 |
| [10] | Galé, J.E., Wrawrzyńczyk, A.: Standard ideals in convolution Sobolev algebras on the half-line. Colloq. Math. 124, 23–34 (2011) · Zbl 1238.46045 |
| [11] | Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics. Birkhäuser, Basel (1988) · Zbl 0624.33001 |
| [12] | Sinclair, A.M.: Continuous Semigroups in Banach Algebras. London Mathematica Society Lecture Note Series, vol. 63. Cambridge University Press, Cambridge (1982) · Zbl 0493.46042 |
| [13] | Titchmarsh, E.C.: Introduction to the theory of Fourier Integrals, 2nd edn. Oxford University Press, London (1948) · Zbl 0031.03202 |
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