Bases and Frames for Signal Representations
نویسنده
چکیده
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 make them easy to deal with. We begin by pointing some properties for finite sets of orthonormal vectors. Lemma 1. Let {ei}ni=1 be an orthonormal set in an IPS X . Then: 1. {ei}ni=1 is linearly independent. 2. The orthogonal projection of X onto the subspace S spanned by {ei}ni=1 is given by
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