Wavelets on the Interval: A Short Survey

The construction of wavelets on intervals has garnered significant attention, and there are currently two primary approaches employed in this area of research. One approach involves obtaining the wavelet on the interval by reconstructing the boundary function using multi-resolution analysis, starting from wavelets defined on the real line R. This approach was initially proposed by Meyer and subsequently refined by Cohen. More recently, Han extended this approach to encompass biorthogonal multi-wavelets. The second approach involves constructing a spline function as a scaling function from a knot sequence, which allows for the definition of the function itself on the interval. Additionally, wavelets on intervals or their extensions, such as non-uniform meshes and manifolds, have been considered in more generalized settings. Our aim is to provide a comprehensive summary of these results, offering a better understanding of the developmental trajectory of wavelets on intervals. This summary will not only facilitate further investigation in this topic but also aid in the practical application of wavelets on intervals. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.