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, {(f) over cap(k) = a(k) + ib(k)}(k=1)(N), we form the generalized conjugate partial sum [GRAPHICS] (a(k)sin kx - b(k)cos kx). The classical conjugate partial sum, ((S) over tilde(N)[f](x), corresponds to sigma = 1 and it is known that -pi/log N (S) over tilde(N)[f] (x) converges to the jump function [f](x) := f(x+) - f(x-); thus, -pi/log N (S) over tilde(N)[f](x) tends to "concentrate" near the edges of f. The convergence, however, is at the unacceptably slow rate of order O(1/log N). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called "concentration factors," sigma(k,N) = sigma(k/N) Our main result shows that an arbitrary C-2[0, 1] nondecreasing sigma(.) satisfying [GRAPHICS] leads to the summability kernel which admits the desired concentration property, [GRAPHICS] with convergence rate, \(S) over tilde(N)(sigma)[f](x)\ less than or equal to Const(log N/N + \sigma(1/N)\) for x's away from the jump discontinuities. To improve over the slowly convergent conjugate Dirichlet kernel (corresponding to the admissible sigma(N)(x) equivalent to -pi/log N), we demonstrate the examples of two families of concentration functions (depending on free parameters p and alpha): the so-called Fourier factors, sigma(alpha)(F)(x) = -pi/Si(alpha) sin alpha x, and polynomial factors, sigma(p)(x) = -p pi x(p). These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered, (C) 1999 Academic Press.