Gram determinant

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Gram determinant

[′gram di′tərm·ə·nənt]
(mathematics)
The Gram determinant of vectorsv1, …,vn from an inner product space is the determinant of the n × n matrix with the inner product ofvi andvj as entry in the i th column and j th row; its vanishing is a necessary and sufficient condition for linear dependence.
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Using Claim 1, we observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [sigma](M) denotes the spectrum of the Gramian matrix M as an operator on [l.
In this case, the Gramian matrix M is a convolution operator, having some sequence a as kernel.
Finally, it should be noted that the techniques used in this article could be applied to a general polynomially self-localized basis in the abstract sense of [5], where a basis is called self-localized if its Gramian matrix presents certain off-diagonal decay.
Their conditions are expressed in terms of the eigenvalues of the Gramian matrix.
The N x N Gramian matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to plays a crucial role in the recovery of missing samples; its elements are the h-periodic functions
the Gramian matrix is in the Sjostrand algebra (see Theorem A.