We describe extensions to the \best-basis" method to select orthonormal bases suitable for signal classiica-tion and regression problems from a large collection of orthonormal bases. For classiication problems, we select the basis which maximizes relative entropy of time-frequency energy distributions among classes. For regression problems, we select the basis which tries to minimize the regres...

The most simple (and useful) linear expansions for Hilbert spaces are the ones that are based on orthonormal bases. Recall that a set of vectors {xα} in an inner product space (IPS) X is said to be orthogonal if xα ⊥ xβ whenever α 6= β. In addition, if each vector in the set is normalized to have unit norm then the set is said to be orthonormal. Orthonormal sets have many nice properties that m...

The Balian–Low Theorem is a strong form of the uncertainty principle for Gabor systems which form orthonormal or Riesz bases for L(R). In this paper we investigate the Balian–Low Theorem in the setting of Schauder bases. We prove that new weak versions of the Balian–Low Theorem hold for Gabor Schauder bases, but we constructively demonstrate that several variants of the BLT can fail for Gabor S...

This article discusses the basic properties of finite-dimensional Boolean vector spaces and their linear transformations. We first introduce a natural Boolean-valued inner product and discuss orthonormal bases. We show that all bases are orthonormal and have the same cardinality. It is shown that the set of subspaces form an atomistic orthomodular poset. We then demonstrate that an operator tha...

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L2(0, 1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar me...

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L2(0, 1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar me...

This paper presents a new method for designing two-band orthonormal IIR wavelet "lter banks using allpass "lters. It is well known that orthonormal wavelet bases can be generated by paraunitary "lter banks. Thus, synthesis of orthonormal wavelet bases can be reduced to the design of paraunitary "lter banks. In this paper, two-band orthonormal IIR wavelet "lter banks using a parallel connection ...