Waveletlike

Waveletlike refers to functions or systems that resemble wavelets in their essential usefulness for time-frequency analysis but do not necessarily fulfill all formal criteria of a wavelet. In practice, a waveletlike function ψ is typically an L2(R) function with some degree of localization in time and frequency, and it may possess properties such as compact support, smoothness, and a finite number of vanishing moments. When used to analyze a signal, a waveletlike function is often dilated and translated to form a family ψ_{a,b}(t) = (1/√a) ψ((t-b)/a), enabling multi-scale analysis across a range of scales a>0 and positions b. However, not all waveletlike systems form an orthonormal basis; they may generate frames or dictionaries that provide stable representations rather than a strict basis.

Waveletlike constructions appear in various contexts, including pseudo-wavelets, band-limited variants, and derivatives of splines used as building blocks in numerical schemes. They share the localization advantages of wavelets and can be tailored to specific applications, such as denoising, compression, or feature extraction, where exact wavelet axioms are not required.

In higher dimensions, waveletlike concepts appear as components of more general multi-scale representations such as curvelets and shearlets, which extend the idea of localization beyond one dimension while retaining a wavelet-inspired philosophy.

Because the term is informal, its precise meaning varies by author and context. It generally signals a practical, approximate wavelet-like tool rather than a mathematically formal wavelet.