TY - JOUR A1 - Fink, Thomas A1 - Forster-Heinlein, Brigitte A1 - Heinrich, Florian T1 - Rebricking frames and bases JF - Journal of Mathematical Analysis and Applications N2 - In 1949, Denis Gabor introduced the “complex signal” (nowadays called “analytic signal”) by combining a real function fwith its Hilbert transform Hfto a complex function f+iHf. His aim was to extract phase information, an idea that has inspired techniques as the monogenic signal and the complex dual tree wavelet transform. In this manuscript, we consider two questions: When do two real-valued bases or frames {fn:n∈N} and {gn:n∈N} form a complex basis or frame of the form {fn+ign:n∈N}? And for which bounded linear operators Adoes {fn+iAfn:n∈N} form a complex-valued orthonormal basis, Riesz basis or frame, when {fn:n ∈N} is a real-valued orthonormal basis, Riesz basis or frame? We call this approach rebricking. It is well-known that the analytic signals don’t span the complex vector space L2(R; C), hence H is not a rebricking operator. We give a full characterization of rebricking operators for bases, in particular orthonormal and Riesz bases, Parseval frames, and frames in general. We also examine the special case of finite dimensional vector spaces and show that we can use any real, invertible matrix for rebricking if we allow for permutations in the imaginary part. KW - Real and imaginary parts of bases KW - frames and operators KW - Orthogonal and self-adjoint operators KW - Point spectrum KW - Approximate spectrum Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-19227 SN - 1096-0813 VL - 2025 PB - Elsevier CY - Amsterdam ER -