A note on scalable frames

We study the problem of determining whether a given frame is scalable, and when it is, understanding the set of all possible scalings. We show that for most frames this is a relatively simple task in that the frame is either not scalable or is scalable in a unique way, and to find this scaling we just have to solve a linear system. We also provide some insight into the set of all scalings when there is not a unique scaling. In particular, we show that this set is a convex polytope whose vertices correspond to minimal scalings

Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

In this paper, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in [Dykema and Strawn, 2003]. Our central technique involves continuous liftings of paths from the polytope of eigensteps to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and also that generic FUNTFs are full spark.Comment: 33 pages, 4 figure

Frames and projections

In this dissertation we explore several ways in which the concept of projections arise infinite frame theory. In the first chapter we show that the Paulsen problem from frame theory is equivalent to a long standing open problem about orthogonal projections with constant diagonal. In the second chapter we introduce the idea of nonorthogonal fusion frames and derive some conditions for when tight nonorthogonal fusion frames exist. In particular, we give a classifi cation of how to factor a self-adjoint matrix into a product of projections. The third chapter explores the idea that the cross gramian of a dual pair of frames forms a projection. We use this to give a classification of when two tight frames form a dual pair. We also introduce a notion of Naimark complement of dual pairs and derive some of its basic properties. The fourth chapter is devoted to questions that relate to applying an invertible operator to a given frame to get a new frame with some desired properties. The last chapter looks at frames as sets of rank one projections rather than as sets of vectors. In this chapter we discuss two problems: the first is the question of rescaling a given frame in order to get a tight frame, the second is known as phase retrieval