This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and let $H$ be a closed subgroup of $G$. Let ${G}/{H}\;$ be the left coset space of $H$ in $G$ and let $\mu$ be the normalized $G$-invariant measure on ${G}/{H}\;$ associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space ${{L}^{2}}\left( {G}/{H,\,\mu }\; \right)$.

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach.

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations.We focus on mortar finite element methods on non-matching triangulations.In particular, we discuss and analyze dual Lagrange multiplier spacesfor lowest order finite elements.These non standard Lagrange multiplier spaces yield optimal discretizationschemes and a locally supported basis for the associatedconstrained mortar spaces. As a consequence,standard efficient iterative solvers as multigrid methods or domain decomposition techniques can be easily adapted to the nonconformingsituation.Here, we introduce new dual Lagrange multiplier spaces. We concentrateon the construction of locally supported and continuous dualbasis functions.The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.

H. O. Kim has shown that contrary to the case of ${{H}^{p}}$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, i.e. they have the same dual spaces and the same Fréchet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into ${{H}^{p}},\,0\,<\,p\,\le \,\infty $.