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In this work, we study some asymptotic expansion of the q-dilogarithm at q = 1 and q = 0 by using the Mellin transform and an adequate decomposition allowed by the Lerch functional equation.

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For a given function ϕ ∈ H ∞ with ϕ (z) < 1 (z ∈ D), we associate some special operators subspace and study some properties of these operators including behavior of their Berezin symbols. It turns that such boundary behavior is closely related to the Blaschke condition of sequences in the unit disk D of the complex plane. In terms of Berezin symbols the trace of some nuclear truncated Toeplitz operator is also calculated. Following the definition of [12], we say that a RKHS H is standard, if k H,λ → 0 weakly as λ → ξ for any point ξ ∈ ∂Ω. For example, the Hardy Hilbert space is a standard RKHS. Recall that if B (H) denotes the space of all bounded and linear operators on H, then the Berezin symbol A of any operator A ∈ B (H) is the function defined on Ω by A(λ) := A k H,λ , k H,λ , λ ∈ Ω.

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets.