Wavelet Transforms and Their Applications by Lokenath Debnath & Firdous Ahmad Shah

Author:Lokenath Debnath & Firdous Ahmad Shah
Language: eng
Format: epub
Publisher: Birkhäuser Boston, Boston, MA

(4.5.1)

where the convergence is defined by the norm.

Definition 4.5.2 (Orthogonal Basis and Orthonormal Basis).

A basis of H is called orthogonal if for n ≠ m.

An orthogonal basis is called orthonormal if for all n.

An orthogonal basis is complete in the sense that if for all n, then x = 0 (see Theorem 2.​10.​4).

Every separable Hilbert space has an orthonormal basis, and for an orthonormal basis the expansion (4.5.1) has the form

(4.5.2)

with

(4.5.3)

More generally, for any x, y ∈ H,

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