Detection of edges in spectral data

We are interested in the detection of jump discontinuities in piecewise smooth functions which are realized by their spectral data. Specifically, given the Fourier coefficients, {f̂_k = a_k + ib_k}^N_k=1, we form the generalized conjugate partial sum N S̃^σ_N[f](x) = ∑ σ(k/N)(a_ksin kx - b_kcos kx). The classical conjugate partial sum, k=1 the jump function [f](x) := f(x+) - f(x-); thus, -π/log NS̃_N[f](x) tends to "concentrate" near the edges of f. The convergence, however, is at the unacceptably slow rate of order script O sign(1/log N). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called "concentration factors," σ_k,N = σ(k/N). Our main result shows that an arbitrary C²[0, 1] nondecreasing σ(•) satisfying ∫¹_1/Nσ(x)/x dx→-π leads N→∞ to the summability kernel which admits the desired concentration property, S̃^σ_N[f](x)→[f](x), with convergence rate, \S̃^σ_N[f](x)\ ≤ Const(log N/N + |σ(1/N)|) N→∞ for x's away from the jump discontinuities. To improve over the slowly convergent conjugate Dirichlet kernel (corresponding to the admissible σ_N(x) ≡ -π/log N), we demonstrate the examples of two families of concentration functions (depending on free parameters p and α): the so-called Fourier factors, σ^F_α(x) = -π/Si(α) sin αx, and polynomial factors, σ^p(x) = -pπx^p. These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered.