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Demonstrates convergence properties of orthonormal systems in L2 spaces, suggesting connections between different function bases.
For every orthonormal system of functions \φₙ\ on $$(0,\,1)$$ , pointwise bounded by a function f∈ L²(0,\,1) , that is, |φₙ(x)|≤ f(x) , it is possible to construct an orthonormal system \ψₙ\ with |ψₙ(x)|≡ 1 , such that the series ∑ aₙφₙ and ∑ aₙψₙ converge almost everywhere simultaneously, for every ₙ\∈² .
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V. G. Dilanyan (2026) studied this question.
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