On Recovering the Coefficients of a Haar Series Convergent Along a Subsequence of Partial Sums

Key Points

• To establish conditions under which a Haar series converging along a subsequence can be recovered as a Fourier–Haar series.
• Proved existence of a specific sequence of natural numbers with zero density.
• Analyzed the convergence properties of subsequence of partial sums of Haar series.
• Applied integrability conditions based on Perron to validate findings.
• Confirmed that certain subsequences of partial sums converge to a finite function.
• Demonstrated that these conditions ensure the Haar series represents a Fourier–Haar series.