The concept of μ-statistical convergence at a point for measurable functions in measurable space with a measure is introduced in this work. This concept is a generalization of a similar idea about the sequence of numbers. We also introduce the concept of μ-statistical fundamentality at a point, and the equivalence of these two concepts is proved. The concept of μ-statistical convergence at a point generalizes the usual one of the limit of a function at a point.
A part of an exponential system with degenerate coefficients is considered. The frame properties (completeness, minimality, basicity, atomic decomposition) of this system in Hardy classes are studied in the case where the coefficients may not satisfy the Muckenhoupt condition.
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