waveletfunktio

Waveletfunktio, in wavelet theory, refers to the mother wavelet, a localized oscillatory function in L2(R) used to generate a family of wavelets by dilation and translation. The standard construction defines psi_{a,b}(t) = (1/√a) psi((t−b)/a) for a > 0 and b ∈ R. This family analyzes a signal at different scales and positions, providing a time–frequency representation.

Key properties include finite energy (psi ∈ L2(R)) and zero mean (∫ psi(t) dt = 0), which ensure the wavelet captures changes rather than constants. Localization in time and frequency is a central feature. The wavelet is called admissible if its Fourier transform Psi(ω) satisfies ∫ |Psi(ω)|^2 / |ω| dω < ∞, enabling reconstruction of the original signal from its wavelet coefficients.

Common families of waveletfunktioe include Haar (simple, piecewise-constant), Daubechies (orthogonal with compact support and vanishing moments), Morlet (Gaussian-windowed sinusoid, good frequency localization), and the Mexican hat (second derivative of a Gaussian). The choice affects smoothness, support size, and vanishing moments, influencing performance in applications.

Applications are wide, including time–frequency analysis, edge detection, denoising, compression, and feature extraction. The continuous wavelet transform (CWT) uses psi_{a,b} across continuous scales and translations, while the discrete wavelet transform (DWT) samples a and b on a discrete grid for efficient computation.

History notes that the wavelet framework emerged in the late 20th century, rapidly becoming a foundational tool in signal processing, image analysis, and data science. See also wavelet transform, mother wavelet, and admissibility condition.