caa a22 4500 477094627 CHVBK 20180405111524.0 cr unu---uuuuu 170330e19960301xx s 000 0 eng 10.1007/BF03322217 doi (NATIONALLICENCE)springer-10.1007/BF03322217 Smoothing Inversion of Fourier Series Using Generalized Cross-Validation [Elektronische Daten] [Manfred Tasche, Norman Weyrich] Let f be a 1-periodic, smooth function and % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\hat f_k(k\in {\rm Z})$ its Fourier coefficients. The reconstruction of f from a given sequence of perturbed values of % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\hat f_k$ is an ill-posed problem. Applying Tihonov's regularization method, this problem can be solved in a stable manner. The efficiency of this method is closely related to the optimal choice of the smoothing parameter. In this paper we compute an optimal smoothing parameter by the method of generalized cross-validation. The related algorithm is tested in some numerical examples. Finally the results are extended to the bivariate case. Birkhäuser Verlag, Basel, 1996 Inversion of Fourier series nationallicence smoothing approximation nationallicence smoothing parameter nationallicence generalized cross-validation nationallicence Tasche Manfred Institut für Mathematik Medizinische, Universität zu Lübeck, Wallstr. 40, D-23560, Lübeck aut Weyrich Norman Synopsys GmbH, Kaiserstr. 100, D-51234, Herzogenrath aut Results in Mathematics Birkhäuser-Verlag 29/1-2(1996-03-01), 183-195 0378-6218 29:1-2<183 1996 29 25 https://doi.org/10.1007/BF03322217 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF03322217 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Tasche Manfred Institut für Mathematik Medizinische, Universität zu Lübeck, Wallstr. 40, D-23560, Lübeck aut NATIONALLICENCE 700 1- Weyrich Norman Synopsys GmbH, Kaiserstr. 100, D-51234, Herzogenrath aut NATIONALLICENCE 773 0- Results in Mathematics Birkhäuser-Verlag 29/1-2(1996-03-01), 183-195 0378-6218 29:1-2<183 1996 29 25 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer