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Norm Properties

There are many other possible choices of norm. To qualify as a norm on $ \mathbb{C}^N$ , a real-valued signal-function $ f(\underline{x})$ must satisfy the following three properties:

  1. $ f(\underline{x})\ge 0$ , with $ 0\Leftrightarrow \underline{x}=\underline{0}$
  2. $ f(\underline{x}+\underline{y})\leq f(\underline{x})+f(\underline{y})$
  3. $ f(c\underline{x}) = \left\vert c\right\vert f(\underline{x})$ , $ \forall c\in\mathbb{C}$
The first property, ``positivity,'' says the norm is nonnegative, and only the zero vector has norm zero. The second property is ``subadditivity'' and is sometimes called the ``triangle inequality'' for reasons that can be seen by studying Fig.5.6. The third property says the norm is ``absolutely homogeneous'' with respect to scalar multiplication. (The scalar $ c$ can be complex, in which case the angle of $ c$ has no effect).

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University