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Frames in Hilbert space and matrices of special structure
In this thesis, properties of matrices of special structure are studied. These include uniform translate of skew-symmetric, unitary, and diagonal matrices. The main focus is the matrix I(–1) of sinc methods of approximation. Several properties of this matrix are discovered, This matrix is used for sinc quadrature and convolution and it is a uniform translate of a skew-symmetric Toeplitz matrix, S, by 12 . It is proved that all the eigenvalues of the matrix S of I(–1) are simple and no eigenvector of S is symmetric or skew-symmetric. Also, it is proved that the determinant of I(–1) is equal to the determinant of S when the dimension is even and this determinant is nonzero. Two computational proofs which show that all eigenvalues of I(–1) have positive real parts are represented. A formula is given for the sum of the squares of the singular values of I(–1). Many other results about I(–1), S, skew-symmetric matrices, the counteridentity matrix, and uniform translate of skew-symmetric, unitary, and diagonal matrices are proved. Such results are focused on determinants, eigen-structure, and singular values. The counteridentity matrix is used to derive many of the theorems, propositions, lemmas, and corollaries. ^ Also, frames in Hilbert space are studied. The studied aspects include: relationships between frames and orthonormal bases, relationships between different frames, generating new frames from old ones, properties of frames, and eigenvalues of the frame operator and related subjects. Discovered results include: properties of operators which map orthonormal bases to frames and vice versa, relationships between biorthogonal frames, similarities and differences between frames and orthonormal bases, properties of the frame operator, and approximation of the Hilbert space elements using a combination of frames and orthonormal bases. Several other aspects are studied. For example, it is known that any element of a Hilbert space can be reconstructed using the frame elements, but the coefficients are not necessarily unique. Examples which illustrate that the coefficients do not have to be unique are given. ^
Abu-Jeib, Iyad Talal, "Frames in Hilbert space and matrices of special structure" (2000). ETD collection for University of Nebraska - Lincoln. AAI9976973.