Convergence of Fourier series in classical systems

The following results are proved:

• there exists an integrable function such that any subsequence of the Cesàro means of negative order of the Fourier series of this function diverges almost everywhere;
  
• the values of an arbitrary integrable function can be changed on a set (independent of this function) of arbitrarily small measure so that the Fourier series with respect to both the Franklin system and the Haar system of the ‘modified’ function will be absolutely convergent almost everywhere on $[0,1]$;
  
• there exists a continuous function which features an unremovable absolute divergence.

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