It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

Nous reprenons la construction des bases orthonormées d'ondelettes à partir des filtres miroirs en quadrature tel qu'elle apparaît dans [4]. Nous montrons que leur régularité est liée à une mesure invariante pour la transformation ω → 2ω mod-2π. Cette méthode permet d'obtenir le facteur exact qui relie asymptotiquement la régularité des ondelettes constriutes dans [4] à la taille de leur support.

The relationship between the spectral properties of the transfer operator corresponding to a wavelet refinement equation and the $L^p$-Sobolev regularity of solution for the equation is established.

We prove that ridgelet transform $R:𝒮\left({ℝ}^2\right)\to 𝒮\left(𝕐\right)$ and adjoint ridgelet transform $R^{*}:𝒮\left(𝕐\right)\to 𝒮\left({ℝ}^2\right)$ are continuous, where $𝕐={ℝ}^{+}\times ℝ\times [0,2\pi ]$. We also define the ridgelet transform $ℛ$ on the space ${𝒮}^{\text{'}}\left({ℝ}^2\right)$ of tempered distributions on ${ℝ}^2$, adjoint ridgelet transform ${ℛ}^{*}$ on ${𝒮}^{\text{'}}\left(𝕐\right)$ and establish that they are linear, continuous with respect to the weak${}^{*}$-topology, consistent with $R$, $R^{*}$ respectively, and they satisfy the identity $({ℛ}^{*}\circ ℛ)\left(u\right)=u$, $u\in {𝒮}^{\text{'}}\left({ℝ}^2\right)$.

Following Beurling's ideas concerning sampling and interpolation in the Paley-Wiener space Lτ∞, we find necessary and sufficient density conditions for sets of sampling and interpolation in the Paley-Wiener spaces Lτp for 0 < p ≤ 1.

We extend results on constructing semiorthogonal linear spline prewavelet systems in one and two dimensions to the case of irregular dyadic refinement. In the one-dimensional case, we obtain sharp two-sided inequalities for the $L_p$-condition, 1 < p < ∞, of such systems.

Solution of a boundary value problem is often realized as the application of the Galerkin method to the weak formulation of given problem. It is possible to generate a trial space by means of splines or by means of functions that are not polynomial and have compact support. We restrict our attention only to RKP shape functions and compactly supported wavelets. Common features and comparison of approximation properties of these functions will be studied in the contribution.