The Lebesgue constant \(\mathcal {L}_n\) is classically interpreted as the norm of Sn, the n-th partial Taylor sum, on the disc algebra or the Lebesgue space \(L^1(\mathbb {T})\) . Although numerous integral and summation formulas for \(\mathcal {L}_n\) exist, an exact, closed-form expression remains elusive. This concept can be extended by considering Sn on various Banach spaces of functions on the open unit disc \(\mathbb {D}\) , thereby defining the corresponding Lebesgue constant for each space. Lebesgue constants for local Dirichlet spaces were comprehensively studied and fully characterized in [8]. In this brief note, we revisit these results, with a particular emphasis on the striking emergence of the golden ratio in the derived formulas.

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Golden Ratio and Lebesgue Constant in Local Dirichlet Spaces

  • Mostafa Nasri

摘要

The Lebesgue constant \(\mathcal {L}_n\) is classically interpreted as the norm of Sn, the n-th partial Taylor sum, on the disc algebra or the Lebesgue space \(L^1(\mathbb {T})\) . Although numerous integral and summation formulas for \(\mathcal {L}_n\) exist, an exact, closed-form expression remains elusive. This concept can be extended by considering Sn on various Banach spaces of functions on the open unit disc \(\mathbb {D}\) , thereby defining the corresponding Lebesgue constant for each space. Lebesgue constants for local Dirichlet spaces were comprehensively studied and fully characterized in [8]. In this brief note, we revisit these results, with a particular emphasis on the striking emergence of the golden ratio in the derived formulas.