Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

Parseval transforms for finite frames

The relationship between frames and Parseval frames is an important topic in frame theory. In this paper, we investigate Parseval transforms, which are linear transforms turning general finite frames into Parseval frames. We introduce two classes of transforms in terms of the right regular and left Parseval transform matrices (RRPTMs and LPTMs). We give representations of all the RRPTMs and LPTMs of any finite frame. Two important LPTMs are discussed in this paper, the canonical LPTM (square root of the inverse frame operator) and the RGS matrix, which are obtained by using row’s Gram–Schmidt orthogonalization. We also investigate the relationship between the Parseval frames generated by these two LPTMs. Meanwhile, for RRPTMs, we verify the existence of invertible RRPTMs for any given finite frame. Finally, we discuss the existence of block diagonal RRPTMs by taking the graph structure of the frame elements into consideration.