Wannier functions that are maximally localized help in understanding many properties of crystalline materials. In the absence topological obstructions, they at least exponentially localized. some cases such as flat-band Hamiltonians, it is possible to construct even more localized, so compactly supported thus having zero support outside their corresponding locations. Under what general conditio...

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I2×2 in the bivariate setting and show how to construct compactly supported functions ψ1, · · · , ψn with n > 3 such that {2ψj(2x − `, 2ky − m), k, `,m ∈ Z, j = 1, . . . , n} ...

Since the publication, less than ten years ago, of Mallat’s paper on Multiresoltion Analysis [Ma], and Daubechies’ paper on the construction of smooth compactly supported refinable functions [D], wavelets had gained enormous popularity in mathematics and in the application domains. It is sufficient to note that there are currently more than 10,000 subscribers to the monthly Wavelet Digest. At t...

Daubechies 5] showed that, except for the Haar function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation q = 2. Nevertheless, such scaling functions do exist for dilations q > 2 (as evidenced by Chui and Lian's construction 3] for q = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the \...