Necessary and sufficient conditions to perform Spectral Tetris

Weighted fusion frame construction via spectral tetris

“…Note the rows of A(x, a ℓ , a ℓ+1 ) are orthogonal, the first row square sums to x, and now the columns have norms a ℓ and a ℓ+1 . Also in [7], the authors show such blocks A(x, a ℓ , a ℓ+1 ) exist if and only if…”

“…The authors in [8] adapt the algorithm in the complex case to use discrete fourier transform matrices as blocks in the construction in order to extend the range of possible eigenvalues. Most recently [7] modified the algorithm to produce sparse frames with prescribed frame operator spectra and prescribed frame vector norms, naming this new process prescribed norm sprectral tetris construction (PNSTC). As PNSTC is a generalized version of the original spectral tetris construction (STC) and includes STC a special case, we will refer to PNSTC simply as STC.…”

“…Theorem 2.1. (Theorem 3.6 in [7]) STC can construct a frame with spectrum {λ m } M m=1 ⊆ (0, ∞) and vector norms {a n } N n=1 ⊆ (0, ∞) if and only if there exists a permutation of these sequences such that they are spectral tetris ready.…”

The Fundamentals of Spectral Tetris Frame Constructions

“…Example 5.2. We would like to use Spectral Tetris to construct a sparse, unit norm, tight frame with 11 elements in H 4 , so our tight frame bound will be 11 4 . To do this we will create a 4 × 11 matrix T * , which satisfies the following conditions:…”

“…Then in Section (10), we see how the original Spectral Tetris construction method is generalized to construct 2-sparse, equidimensional, unit-weighted fusion frames. This method is further generalized in Section (11) to construct unit weighted Spectral Tetris fusion frames. Our final construction method for fusion frames occurs in Section (12) where we provide a generalized Spectral Tetris fusion frame construction algorithm as well as necessary and sufficient conditions for when this method is applicable.…”