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Abstract Bernstein inequality is an essential inequality for Besov spaces. Smoothness based approaches are widely used in establishing the inequality. Yet, despite numerous studies over the last two decades, there is still little research focusing on decay-based approaches. However, motivating authors to establish inequality poses challenges, many of which can be overcome by means of the completeness of wavelet bi-frames in Lebesgue spaces and the stability of wavelet coefficients. The research has shown how wavelets with decay conditions enable descriptions of Lebesgue spaces, and in particular, the Bernstein inequality.