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**Wavelet transforms and their applications to turbulence.**
*(English)*
Zbl 0743.76042

Lumley, John L. (ed.) et al., Annual review of fluid mechanics. Vol. 24. Palo Alto, CA: Annual Reviews Inc. (ISBN 0-8243-0724-0/hbk). Annu. Rev. Fluid Mech. 24, 395-457 (1992).

[For the entire collection see Zbl 0742.00036.]

Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allow one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which area localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or synthesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions.

In the context of turbulence, the wavelet transform may yield some elegant decompositions of turbulent flows. The continuous wavelet transform offers a continuous and redundant unfolding in terms of both space and scale, which may enable us to track the dynamics of coherent structures and measure their contributions to the energy spectrum. The discrete wavelet transform allows an orthonormal projection on a minimal number of independent modes which might be used to compute or model the turbulent flow dynamics in a better way than with Fourier modes.

Wavelet transforms are recent mathematical techniques, based on group theory and square integrable representations, which allow one to unfold a signal, or a field, into both space and scale, and possibly directions. They use analyzing functions, called wavelets, which area localized in space. The scale decomposition is obtained by dilating or contracting the chosen analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal at infinity does not play any role. Therefore the wavelet analysis or synthesis can be performed locally on the signal, as opposed to the Fourier transform which is inherently nonlocal due to the space-filling nature of the trigonometric functions.

In the context of turbulence, the wavelet transform may yield some elegant decompositions of turbulent flows. The continuous wavelet transform offers a continuous and redundant unfolding in terms of both space and scale, which may enable us to track the dynamics of coherent structures and measure their contributions to the energy spectrum. The discrete wavelet transform allows an orthonormal projection on a minimal number of independent modes which might be used to compute or model the turbulent flow dynamics in a better way than with Fourier modes.

### MSC:

76F99 | Turbulence |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

42C99 | Nontrigonometric harmonic analysis |