Application of the Rigged Hilbert Spaces into the Generalized Signals and Systems Theory: Practical example

Groups, Jacobi functions, and rigged Hilbert spaces

“…This proves that K + f ∈ Φ m,q . In addition, the inequality (26) shows the continuity of K + on Φ m,q , after (19). A similar proof works for K − .…”

“…This shows that, for all f H (x, φ, χ) ∈ Φ H , we have that(A + f H )(x, φ, χ) ∈ Φ H . Moreover, this discussion also concludes that [p H r,s (A + f H )] 2 ≤ 2 2r [p H r+1,s (f H )] 2 , ∀f H ∈ Φ H ,which, after(19), shows the continuity of A + on Φ H . A similar argument is valid on Φ F and the same considerations are in order for A − , B ± and K ± on Φ H and Φ F .…”

“…where ||−|| denotes the Hilbert norm on L 2 [−1, 1]. Then, the conclusion comes from a simple generalization of (19). Thus,…”

“…The action of F ∈ Ψ × on each vector ϕ ∈ Ψ gives a complex number often denoted as ϕ|F , following the Dirac notation. It is not our intention here to review properties and applications of RHS, the reader may visit for this purpose the extensive literature on the subject [5][6][7][8][9][10][11][12][13], but we would like to mention that recently they have received renewed attention, see [14][15][16][17][18][19], including their relations with partial inner products [20] and frames [21]. We limit us to remember that our primary purpose is to provide of mathematical sense to the Dirac formulation of quantum mechanics and its continuous basis of eigenvectors of observables.…”

Groups, Special Functions and Rigged Hilbert Spaces

“…In the last years the RHS have appeared associated to time-frequency analysis and Gabor analysis that have many applications in physics and engineering related to signal proceesing [43][44][45][46][47][48][49][50]. In particular, applications in electrical engineering have been introduced in [52][53][54].…”