Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

Abstract

A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( δ ) ( t ) = E W ( q γ t ) where the eigenvalue E ∈ R is independent of the advancing parameter q > 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs.

Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

“…The right-most equality in (33) follows from the fact that for each p ∈ f0, 1, ⋯, M − 1g one has ω p b is a root of x M − b M , and hence, x − ω p b is a factor. To obtain (34), one divides the right two expressions in (33) by ðx − bÞ. The lemma is now proven.…”

“…For an integer M ≥ 2 let ω = e 2πi/M be an M th root of unity. One has (34) to obtain (35). The lemma is shown.…”

“…This is highlighted by works of L. Di Vizio [10][11][12]; C. Hardouin [11]; T. Dreyfus [13,14]; A. Lastra [14], [15][16][17][18][19][20], [21][22][23]; S. Malek [14], [15][16][17][18][19][20], [21][22][23], [24][25][26][27]; J. Sanz [21][22][23]; H. Tahara [28]; and C. Zhang [12,29], along with further references by these researchers and others. Also, for good background references to the current work, consult [2][3][4][5][6][30][31][32][33][34] (especially [2,4]). These last references also exhibit a number of various applications of global solutions of MADEs.…”

“…Observe that in (339) for fixed δ one has ω γ = e 2πiγ/δ = e 2πib γ /δ = ω b γ . Thus, 34 Abstract and Applied Analysis the set f+g associated to γ, δ is identical to the set f+g associated to b γ, δ.…”