The relay fusion frame proposed by Hong and Li is an extension of a fusion frame that has many applications in science. In this study, we introduce relay fusion frames in Hilbert C ∗ -modules very naturally and shift some common attributes of fusion frames and relay fusion frames in Hilbert spaces to relay fusion frames in Hilbert C ∗ -modules. In addition, we generalize some perturbation results in frame theory to relay fusion frames in Hilbert C ∗ -modules. Finally, we introduce a class of F , G -relay fusion frames as a generalization of K -frames and present some perturbation results for F , G -relay fusion frames in Hilbert C ∗ -modules.

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In this paper, we study the functions with values in (β, p)-Banach spaces which can be approximated by a quadratic mapping with a given error.

<abstract><p>In this paper, the notion of an $ \mathcal{A} $-group-like unitary system on a Hilbert $ C^{\ast} $-module is introduced and some basic properties are studied, where $ \mathcal{A} $ is a unital $ C^{\ast} $-algebra. Let $ \mathcal{U} $ be such a unitary system. We prove that a complete Parseval frame vector for $ \mathcal{U} $ can be dilated to a complete wandering vector. Also, it is shown that the set of all the complete Bessel vectors for $ \mathcal{U} $ can be parameterized by the set of the adjointable operators in the double commutant of $ \mathcal{U} $, and that the frame multiplicity of $ \mathcal{U} $ is always finite.</p></abstract>