Spline approximation of a non-linear Riemann–Hilbert problemFootnote†

Dedicated to Ian H. Sloan on the occasion of his 70th birthday.

References

• Šnirel’man , AI . 1972 . The degree of quasi-ruled mapping and a nonlinear Hilbert problem . Math. USSR-Sbornik , 18 : 373 – 396 .
   Google Scholar
• Efendiev , MA . 1987 . On a property of the conjugate integral and a nonlinear Hilbert problem . Soviet Math. Dokl. , 35 : 535 – 539 .
   Google Scholar
• Efendiev , MA and Wendland , WL . 1996 . Nonlinear Riemann–Hilbert problems for multiply connected domains . Nonlinear Anal. , 27 : 37 – 58 .
   Web of Science ®Google Scholar
• Efendiev , MA and Wendland , WL . 1997 . Nonlinear Riemann–Hilbert problems without transversality . Math. Nachr. , 183 : 73 – 89 .
   Google Scholar
• Efendiev , MA and Wendland , WL . 2001 . Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data . Topol. Methods Nonlinear Anal. , 17 : 111 – 124 .
   Google Scholar
• Efendiev , MA and Wendland , WL . 2001 . Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data without transversality . Nonlinear Anal. , 47 : 457 – 466 .
   Google Scholar
• Efendiev , MA and Wendland , WL . 2003 . Nonlinear Riemann–Hilbert problems with Lipschitz continuous boundary data: Doubly connected domains . Proc. Roy. Soc. London Ser. A , 459 : 945 – 955 .
   Google Scholar
• Wegert , E . 1992 . Nonlinear Boundary Value Problems for Holomorphic Functions and Singular Integral Equations , Berlin : Akademie-Verlag .
   Google Scholar
• Wegert , E . 1993 . Iterative methods for discrete nonlinear Riemann–Hilbert problems . J. Comp. Appl. Math. , 46 : 143 – 163 .
   Google Scholar
• Wegert , E . 1994 . Nonlinear Riemann–Hilbert problems with unbounded restriction curves . Math. Nachr. , 170 : 307 – 313 .
   Google Scholar
• Wegert , E and Efendiev , MA . 2000 . Nonlinear Riemann–Hilbert problems with Lipschitz continuous boundary condition . Proc. Royal Soc. Edinburgh , 130A : 793 – 800 .
   Google Scholar
• Grigorieff , RD and Sloan , Ian H . 2005 . Discrete orthogonal projections on multiple knot periodic splines . J. Approx. Theory , 137 : 201 – 225 .
   Google Scholar
• Sloan , IH and Wendland , WL . 1999 . Spline qualocation methods for variable–coefficient elliptic equations on curves . Numer. Math. , 83 : 497 – 533 .
   Google Scholar
• Tran , T and Sloan , IH . 1998 . Tolerant qualocation–a qualocation method for boundary integral equations with reduced regularity requirement . J. Integral Equations Appl. , 10 : 85 – 115 .
   Google Scholar
• Gakhov , FD . 1996 . Boundary Value Problems , New York : Pergamon Press .
   Google Scholar
• Saranen , J and Wendland , WL . 1987 . The Fourier series representation of pseudodifferential operators on closed curves . Complex Variables: Theory Appl. , 8 : 55 – 64 .
   Google Scholar
• Micula , Gh and Micula , S . 1999 . Handbook of Splines , Dordrecht/Boston/London : Kluwer Academic Publishers .
   Google Scholar
• Prößdorf , S and Silbermann , B . 1991 . Numerical Analysis for Integral and Related Operator Equations , Basel : Birkhäuser Verlag .
   Google Scholar
• Saranen , J and Vainikko , G . 2002 . Periodic Integral and Pseudodifferential Equations with Numerical Approximation , Berlin/Heidelberg/New York : Springer-Verlag .
   Google Scholar
• Arnold , DN and Wendland , WL . 1985 . The convergence of spline collocation for strongly elliptic equations on curves . Numer. Math. , 47 : 317 – 341 .
   Web of Science ®Google Scholar
• Prößdorf , S and Schmidt , G . 1981 . A finite element collocation method for singular integral equations . Math. Nachr. , 100 : 33 – 60 .
   Google Scholar
• Adams , RA . 1975 . Sobolev Spaces , New York/San Francisco/London : Academic Press .
   Google Scholar
• Atkinson , KF . 1972 . The numerical evaluation of the Cauchy transform on a simple closed curve . SIAM J. Numer. Anal. , 9 : 284 – 299 .
   Google Scholar
• Zeidler , E . 1993 . Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems , New York/Berlin/Heidelberg : Springer-Verlag .
   Google Scholar
• Hilbert , D . 1904 . “ Über eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie ” . In Verhandl. des III. Internat , 233 – 204 . Heidelberg : Mathematiker Kongresses .
   Google Scholar