Multivariate Wavelet Frames by Aleksandr Krivoshein Vladimir Protasov & Maria Skopina

Author:Aleksandr Krivoshein, Vladimir Protasov & Maria Skopina
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

(4.16)

After the change of variable, the latter integral is reduced to

The integrand tends to

as for each . Repeating the arguments of the proof of (4.11) and taking into account that , it is not difficult to see that the integrand has a summable majorant. Thus, by Lebesgue’s dominated convergence theorem,

Combining this with (4.16) completes the proof of (b).

4.3 Wavelet Expansions

Above, we studied the scaling operators with arbitrary compactly supported functions or distributions .

Assume that are refinable, and dual wavelet systems , are generated from by MEP. Recall that the associated wavelet functions , , are defined by (4.4), and their masks are trigonometric polynomials (see Sect. 4.1). It follows that are in the same function spaces as , respectively. We are interested, if the system is frame-like, i.e., if (4.5) holds for every function f (from an appropriate class) or almost frame-like, i.e., (4.6) holds for every f. Because of the following lemma, which establishes the so-called perfect reconstruction property, (4.6) holds if and only if , and (4.5) holds if and only if and , where the convergence is in the same sense as the convergence of the series in (4.5) and (4.6).

Lemma 4.3.1

Let be compactly supported refinable distributions , , , be the associated wavelet functions, , , . Then

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