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, { fk 5 ak 1 ibk}k51 N , we form the generalized conjugate partial sum S N s @ f #~x! 5 ¥ k51 N sS ND~aksin kx 2 bkcos kx!. The classical conjugate partial sum, SN [ f ]( x), corresponds to s [ 1 and it is known that 2p log N SN@ f #~x! converges to the jump function [ f ]( x) :5 f ( x1) 2 f ( x2); thus, 2p log N SN@ f #~x! tends to “concentrate” near the edges of f. The convergence, however, is at the unacceptably slow rate of order 2(1/log N ). To accelerate the convergence, thereby creating an effective edge detector, we introduce the so-called “concentration factors,” sk,N 5 sS ND . Our main result shows that an arbitrary C[0, 1] nondecreasing s(z) satisfying *1/N 1 s~x! x dxOi N3 ` 2p leads