Basisbands

Basisbands are a mathematical construct used in functional analysis and signal processing to describe sets of basis functions whose spectral content is confined to predefined frequency bands. They arise in the study of band-limited representations and filter bank theory, and they are used to enable selective reconstruction and structured representations of signals.

Formally, a Basisband B = {φ_k} is a basis for a vector space V (for example a subspace of L2(R) or a finite-dimensional R^n) such that there exists a partition of the frequency axis into bands {Ω_j} with the property that the Fourier transform Φ_k(ω) of each φ_k is supported in some Ω_j. If the φ_k are mutually orthogonal, B is an orthonormal Basisband.

Construction methods include applying band-pass filters to an existing basis (such as a Fourier basis or a wavelet basis), forming filter banks, and re-orthogonalizing to restore basis properties after band-limiting. In discrete settings, Basisbands correspond to columns of a filter-bank matrix or to a polyphase representation.

Applications include communications channelization, multiresolution analysis, compressed sensing with structured sparsity, and audio processing. They support efficient, band-limited reconstructions and can help reduce redundancy in representations of signals with natural spectral groupings.

Relation to existing concepts: Basisbands generalize standard bases by enforcing band-localization and relate to filter banks, Gabor frames, and multirate signal processing. Limitations include potential loss of sparsity when signals do not align with the chosen band structure and numerical conditioning challenges in high dimensions.