TY - JOUR

T1 - A New Method of Solving Noisy Abel-Type Equations

AU - Garza, J.

AU - Hall, P.

AU - Ruymgaart, F. H.

PY - 2001/5/15

Y1 - 2001/5/15

N2 - A new approach to solving noisy integral equations of the first kind is applied to the family of Abel equations. Such equations play a role in stereology (Wicksell's unfolding problem), medicine, engineering, and astronomy. The method is based on an expansion in an arbitrary orthonormal basis, coupled with exact inversion of the integral operator. The inverse appears in the Fourier coefficients of the expansion, where it can be carried over to the usually well-behaved basis elements in the form of the adjoint. This method is an alternative to Tikhonov regularization, regularization of the inverse of the operator itself, or a wavelet-vaguelette/singular-value decomposition. The method is particularly interesting in irregularity of the kernel, the input, or both. Because knowledge of the spectral properties of the operator is not required, the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with. For smooth input functions, the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of Abel operators. This can be shown when smooth wavelets are used for Abel operators with index smaller than 1/2, and when the Haar wavelet is used for operators with index larger than 1/2.

AB - A new approach to solving noisy integral equations of the first kind is applied to the family of Abel equations. Such equations play a role in stereology (Wicksell's unfolding problem), medicine, engineering, and astronomy. The method is based on an expansion in an arbitrary orthonormal basis, coupled with exact inversion of the integral operator. The inverse appears in the Fourier coefficients of the expansion, where it can be carried over to the usually well-behaved basis elements in the form of the adjoint. This method is an alternative to Tikhonov regularization, regularization of the inverse of the operator itself, or a wavelet-vaguelette/singular-value decomposition. The method is particularly interesting in irregularity of the kernel, the input, or both. Because knowledge of the spectral properties of the operator is not required, the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with. For smooth input functions, the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of Abel operators. This can be shown when smooth wavelets are used for Abel operators with index smaller than 1/2, and when the Haar wavelet is used for operators with index larger than 1/2.

UR - http://www.scopus.com/inward/record.url?scp=0011103548&partnerID=8YFLogxK

U2 - 10.1006/jmaa.2000.7371

DO - 10.1006/jmaa.2000.7371

M3 - Article

SN - 0022-247X

VL - 257

SP - 403

EP - 419

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -