Decomposition of Spaces of Distributions Induced by Hermite Expansions

Abstract

Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the TriebelLizorkin and Besov spaces on R d induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite-TriebelLizorkin and Besov spaces are, in general, different from the respective classical spaces.

Sub-exponentially localized kernels and frames induced by orthogonal expansions

“…The localization principle put forward in [18] says that for all "natural orthogonal systems" the kernels {L n (x, y)} decay at rates faster than any inverse polynomial rate away from the main diagonal y = x in E × E with respect to the distance in E. This principle is very well-known in the case of the trigonometric system (and the Fourier transform) and not so long ago was established for spherical harmonics [15,16], Jacobi polynomials [1,17], orthogonal polynomials on the ball [18], and Hermite and Laguerre functions [2,4,19,9]. Surprisingly, however, the localization principle as formulated above fails to be true for tensor product Jacobi polynomials and, in particular, for tensor product Legendre or Chebyshev polynomials, as will be shown in §10.…”

“…To this end one should follow the well established scheme from e.g. [16,17,18,19]. The main ingredient of this development are Theorems 7.1 and 7.2 from above, which provide the needed localization results.…”

Decomposition of Triebel–Lizorkin and Besov spaces in the context of Laguerre expansions

“…For such spaces induced by {L α ν }, see [2]. This paper is an integral part of a broader undertaking for needlet characterization of TriebelLizorkin and Besov spaces on nonstandard domains (and with weights) such as the sphere [11], interval [8], ball [9], and in the setting of Hermite expansions [13].…”

On the construction of frames for spaces of distributions

“…The ϕ-transform of Frazier and Jawerth [6][7][8] is an example of frames which have had a significant impact in Harmonic analysis. Orthogonal expansions were recently used for the development of frames of a similar nature in non-standard settings such as on the sphere [19,20], interval [15,23] and ball [16,24] with weights, and in the context of Hermite [25] and Laguerre [12] expansions.…”