### Abstract

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_{k}sin kx - b_{k}cos 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 C^{2}[0, 1] nondecreasing σ(•) satisfying ∫^{1}
_{1/N}σ(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.

Original language | English (US) |
---|---|

Pages (from-to) | 101-135 |

Number of pages | 35 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 7 |

Issue number | 1 |

State | Published - Jul 1999 |

### Fingerprint

### Keywords

- Concentration factors
- Conjugate partial sums
- Fourier expansion
- Piecewise smoothness

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Applied and Computational Harmonic Analysis*,

*7*(1), 101-135.

**Detection of edges in spectral data.** / Gelb, Anne; Tadmor, Eitan.

Research output: Contribution to journal › Article

*Applied and Computational Harmonic Analysis*, vol. 7, no. 1, pp. 101-135.

}

TY - JOUR

T1 - Detection of edges in spectral data

AU - Gelb, Anne

AU - Tadmor, Eitan

PY - 1999/7

Y1 - 1999/7

N2 - 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, {f̂k = ak + ibk}N k=1, we form the generalized conjugate partial sum N S̃σ N[f](x) = ∑ σ(k/N)(aksin kx - bkcos kx). The classical conjugate partial sum, k=1 the jump function [f](x) := f(x+) - f(x-); thus, -π/log NS̃N[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," σk,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, S̃σ N[f](x)→[f](x), with convergence rate, \S̃σ 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 σN(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, σF α(x) = -π/Si(α) sin αx, and polynomial factors, σp(x) = -pπxp. These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered.

AB - 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, {f̂k = ak + ibk}N k=1, we form the generalized conjugate partial sum N S̃σ N[f](x) = ∑ σ(k/N)(aksin kx - bkcos kx). The classical conjugate partial sum, k=1 the jump function [f](x) := f(x+) - f(x-); thus, -π/log NS̃N[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," σk,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, S̃σ N[f](x)→[f](x), with convergence rate, \S̃σ 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 σN(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, σF α(x) = -π/Si(α) sin αx, and polynomial factors, σp(x) = -pπxp. These yield effective detectors of (one or more) edges, where both the location and the amplitude of the discontinuities are recovered.

KW - Concentration factors

KW - Conjugate partial sums

KW - Fourier expansion

KW - Piecewise smoothness

UR - http://www.scopus.com/inward/record.url?scp=0000994152&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000994152&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000994152

VL - 7

SP - 101

EP - 135

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

IS - 1

ER -