Pointwise convergence and summability

For every $f\in L^2(\mathbb T)$, symmetric Fourier projections converge to $f$ in L² norm under normalized circle measure. A sequence converging in this norm has a subsequence converging almost everywhere, but the norm statement alone does not prove pointwise convergence of its entire sequence. Altering a function at one point leaves every coefficient unchanged. Therefore no procedure based only on these coefficients can reconstruct an arbitrary reassignment at that point. This is a structural issue, not a missing improvement in numerical precision.

Uniform convergence is stronger because one cutoff controls the error at every point simultaneously. Absolute summability of Fourier coefficients gives uniform convergence of the series by the uniform comparison test. To identify the sum with a given integrable function, use uniqueness of Fourier coefficients, and then continuity if equality at every point is desired. A twice continuously differentiable periodic function has summable coefficients: integrating by parts twice bounds nonzero coefficients by its second derivative's L¹ norm divided by the squared frequency.

For a periodic function of bounded variation, the symmetric Fourier partial sums converge at each point to the average of its left and right limits. Bounded variation supplies the existence of those limits and controls oscillation sufficiently for the Dirichlet–Jordan theorem. A periodic function with finitely many pieces that is continuously differentiable up to each one-sided endpoint has bounded variation; more generally, it suffices to have continuous differentiability on each open piece, an integrable absolute derivative there, and finite one-sided limits. At a continuity point the average equals the function value if the representative is chosen continuously there; at a jump it need not equal the assigned value.

This theorem is a sufficient condition, not a characterization of every convergent Fourier series. The oscillatory Dirichlet kernel and the local difference between the function and its one-sided limits drive a cancellation argument; a full proof requires that oscillatory-integral estimate, rather than just bounding the kernel by its maximum. Near a jump, partial sums can overshoot even while converging at each fixed point away from it. Pointwise convergence therefore does not imply monotone error improvement or uniform convergence across a discontinuity.

Fejér means converge uniformly for continuous periodic functions and converge in Lᵖ norm for each Lᵖ input with finite exponent at least one. They also recover an integrable function at its Lebesgue points, hence almost everywhere. These are results about averaged sums, not a proof that ordinary partial sums behave identically. The positive approximate-identity kernel makes the norm and continuity arguments accessible, while the almost-everywhere assertion uses local average control. An exceptional null set remains permitted.

Ordinary partial sums can fail to converge at a point even for a continuous periodic function, and can fail in L¹ norm for an integrable one. Stronger almost-everywhere results for ordinary partial sums in Lᵖ with exponent strictly above one are deep theorems, not consequences of Parseval alone. When communicating a reconstruction guarantee, state whether the claim is a theorem under a verified regularity assumption, an observed finite-resolution pattern, or a summability result for a different operator. This discipline prevents a convincing graph from silently replacing the missing convergence argument.