A space of regulated functions whose Fourier series are everywhere convergent (Q1063823)

Let f be a regulated function (i.e. \(f(x)=1/2(f(x+0)+f(x-0)))\) on the unit circle \(T=[0,2\pi)\). If \(I=[a,b]\) we write \(f(I)=f(b)-f(a)\). Let \(x\in T\), \(\delta >0\). For a given n and \(0\leq t<\pi /n\), let \(I^+_ i(t)=[x+t+i\pi /n,x+t+(i+1)\pi /n]\) and \(I^-_ i(t)=[x-t-(i+1)\pi /n,x-t-i\pi_ n],\) \(i=1,2,...,N\), where \(N=[n\delta /\pi]\). Define \[ V_ n(f;x;\delta)=\sup_{0\leq t\leq \pi_ n}\{\max_{0\leq j<N}[\sum^{N-j}_{i=1}| f(I^+_{j+i}(t)| /i, \] \[ \sum^{j+1}_{i=1}| f(I^+_{j+2-i}(t)| /i,\sum^{N- j}_{i=1}| f(I^-_{j+i}(t)| /i,\sum^{j+1}_{i=1}| f(I^-_{j+2-i}(t)| /i]\} \] and \(V(f;x;\delta)= \limsup_{n\to \infty}V_ n(f;x;\delta)\). If \(V(f;x;\delta)\to 0\) as \(\delta\to 0\) for all x, then the \(n^{th}\) partial sum of the Fourier series of f at x, \(s_ n(f_ ix)\to f(x)\) for all x. If f is continuous on a closed interval I, the convergence is uniform on each interval J contained in the interior of I.