Modification of functions (Q795259)

The defect, that the Fourier series of a continuous function need not be convergent everywhere, can be removed in several ways. The author considers and discusses some consequences of the Bohr-Pal theorem (i.e. for each continuous function f that is periodic of the period 2\(\pi\) there is a homeomorphism g:[-\(\pi\),\(\pi]\to [-\pi,\pi]\) such that the Fourier series of \(f\circ g\) converges uniformly) and the Menchov method (for every continuous function f:[-\(\pi\),\(\pi]\to {\mathbb{R}}\) and each \(\epsilon>0\) there is a function g:[-\(\pi\),\(\pi]\to {\mathbb{R}}\) such that \(f=g\) except on a set of measure less than \(\epsilon\) and whose Fourier series converges uniformly on [-\(\pi\),\(\pi]\)).