In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
Definition A function ψ ∈ L 2 ( R ) {\displaystyle \psi \,\in \,L^{2}(\mathbb {R} )} is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is, a complete orthonormal system for the Hilbert space of square-integrable functions on the real line. The Hilbert basis is constructed as the family of functions { ψ j k : j , k ∈ Z } {\displaystyle \{\psi _{jk}:\,j,\,k\,\in \,\mathbb {Z} \}} by means of dyadic translations and dilations of ψ {\displaystyle \psi \,} ,
ψ j k ( x ) = 2 j 2 ψ ( 2 j x − k ) , {\displaystyle \psi _{jk}(x)=2^{\frac {j}{2}}\psi \left(2^{j}x-k\right),}
for integers j , k ∈ Z {\displaystyle j,\,k\,\in \,\mathbb {Z} } . If, under the standard inner product on L 2 ( R ) {\displaystyle L^{2}\left(\mathbb {R} \right)} ,
⟨ f , g ⟩ = ∫ − ∞ ∞ f ( x ) g ( x ) ¯ d x , {\displaystyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}dx,}
this family is orthonormal, then it is an orthonormal system:
⟨ ψ j k , ψ l m ⟩ = ∫ − ∞ ∞ ψ j k ( x ) ψ l m ( x ) ¯ d x , = δ j l δ k m , {\displaystyle {\begin{aligned}\langle \psi _{jk},\psi _{lm}\rangle &=\int _{-\infty }^{\infty }\psi _{jk}(x){\overline {\psi _{lm}(x)}}dx,\\&=\delta _{jl}\delta _{km},\end{aligned}}}
where δ j l {\displaystyle \delta _{jl}\,} is the Kronecker delta. Completeness is satisfied if every function f ∈ L 2 ( R ) {\displaystyle f\,\in \,L^{2}\left(\mathbb {R} \right)} may be expanded in the basis as
f ( x ) = ∑ j , k = − ∞ ∞ c j k ψ j k ( x ) {\displaystyle f(x)=\sum _{j,k=-\infty }^{\infty }c_{jk}\psi _{jk}(x)}
with convergence of the series understood to be convergence in norm. Such a representation of f {\displaystyle f} is known as a wavelet series. This implies that an orthonormal wavelet is self-dual. The integral wavelet transform is the integral transform defined as
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