On the semi-regular frames of translates

2019/08/31 by F. Valizadeh, Valizadeh, F., Hamidreza Rahimi +5
Computer Science · Engineering · Mathematics · #43A15 #43A85 #65T60 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #Digital Filter Design and Implementation #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1909.00243

openalex publication_date 2019/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we fix a real invertible d× d matrix A and consider Aℤ^d as an index set. For f∈ L²(ℝ^d), let Φ^A_f:=(1)/(|det A|)∑_{k∈ ℤ^d}|f(A^T)^-1(⋅+k)|² be the periodization of |f|². By using Φ^A_f, among other things, we characterize when the sequence τ_A(f):=\f(⋅-Ak)\_{k∈ ℤ^d} is a Bessel sequence, frame of translates, Riesz basis, or orthonormal basis. And finally, we construct an example, in which τ_A(f) is a Parseval frame of translates, but not a Riesz sequence.

Citations

• Continuous and Discrete Wavelet Transforms

Related

• An operator based approach to irregular frames of translates
• Frames of translates
• Frames of translates with prescribed fine structure in shift invariant\n spaces
• Unconditional structures of translates for L_p(R^d)
• Existence of Unconditional Frames Formed By System of Translates in Modulation Spaces