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

Abstract

A pair of dual frames with almost exponentially localized elements (needlets) are constructed on R d + based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients.

Sub-exponentially localized kernels and frames induced by orthogonal expansions

“…Note first that from Lemma 1.5.3 in [22] where γ > 0 is a constant and N := 4n + 2α + 2. These immediately lead to the estimates (see also [9]):…”

“…Analogous results for {L α n } n≥0 and {M α n } n≥0 follow immediately as in [9]. The d-dimensional tensor product Laguerre functions associated to {F α n } are defined by As elsewhere in this paper, we are interested in constructing sup-exponential localized kernels of the form…”

“…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.…”

“…The rapid decay of needlets makes them a powerful tool for decomposition of spaces of functions and distributions in various settings. The above scheme has already been utilized for construction of needlets and needlet decomposition of L p , Sobolev, and the more general TriebelLizorkin and Besov spaces in the frameworks of spherical harmonics [15,16], Jacobi polynomials [17,12], orthogonal polynomials on the ball [18,13], and Hermite and Laguerre functions [4,19,9]. …”

“…Note that in [9] it is proved that for admissible cutoff functionsâ the kernels L α n (x, y) have faster than the reciprocal of any polynomial decay away from the main diagonal in R…”