Wavelet transform

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. Applied the following discretization of frequency and time:As apparent from wavelet-transformation representation (shown below) 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. A function ψ ∈ L 2 ( R ) {displaystyle scriptstyle 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 L 2 ( R ) {displaystyle scriptstyle L^{2}left(mathbb {R} ight)} of square integrable functions. The Hilbert basis is constructed as the family of functions { ψ j k : j , k ∈ Z } {displaystyle scriptstyle {psi _{jk}:,j,,k,in ,mathbb {Z} }} by means of dyadic translations and dilations of ψ {displaystyle scriptstyle psi ,} , for integers j , k ∈ Z {displaystyle scriptstyle j,,k,in ,mathbb {Z} } . If under the standard inner product on L 2 ( R ) {displaystyle scriptstyle L^{2}left(mathbb {R} ight)} ,