Definition and Simplest Properties of Generalized Functions

Solution of Generalized Fractional Reaction-Diffusion Equations

“…D ′ is the dual of the space D which consists of all smooth functions with compact supports (Brychkov and Prudnikov, 1989 p. 3). For further details , the reader is referred to the monographs by Gelfand and Shilov (1964) and Brychkov and Prudnikov (1989), if we replace the Laplace and Fourier transforms in (15) and (17) by the corresponding Laplace and Fourier transform of generalized functions.…”

“…The Fourier transforms in S ′ and D ′ are introduced by Gelfand and Shilov (1964). S ′ is the dual of the space S , which is the space of all infinitely differentiable functions which, together with their derivatives approach zero more rapidly than any power of 1/|x| as |x| → ∞ (Gelfand and Shilov, 1964, p.16).…”

The Fourier Transform on the Group $$\mathrm {GL}_2({\mathbb R })$$ GL 2 ( R ) and the Action of the

“…As usual (see, e.g., [3], §I.3), the integral admits a meromorphic continuation to the whole plane (μ 1 , μ 2 ) ∈ C 2 with poles at μ 1 − μ 2 = 0, 1, 2, …. The operators A μ 1 ,ε 1 ;μ 2 ,ε 2 are intertwining, Discrete series.…”

“…For fixed ε 1 , ε 2 it has a meromorphic continuation to the whole complex plane (μ 1 , μ 2 ) with poles on the hyperplanes μ 1 = −1/2 − k, μ 2 = −1/2 − k, where k = 0, 1, 2, …(see, e.g., [3], §I.…”

“…Notice that canonical overgroups exist for all 10 series of real classical groups. 3 Moreover overgroups exist for all 52 series of classical semisimple symmetric spaces G/H , see [8,15], see also [18], Addendum D.6. So the problem makes sense for all classical symmetric spaces.…”