Abstract
We consider relations between the Friedrichs operator and constructive aspects of the Dirichlet problems for the Laplace and \({\overline \partial ^{2}}\)-operator. Then we investigate the Fourier expansions in the eigenfunctions of the Friedrichs operator. A link between a generalized Friedrichs operator and minimal nodes quadratures for complex polynomials of a fixed degree is explained. We initiate a discussion of the boundary Friedrichs operator on the Hardy space of a domain. The transformation law of the Friedrichs operator under conformal mappings leads to a modified version of it, based on a symbol function; this object will turn out to be closely related to Hankel operators. We obtain some results concerning which symbols correspond to compact operators.
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References
S. Axler, I. D. Berg, N. P. Jewell and A. Shields, Approximation by compact operators and the space H ∞ + C, Ann. Math., 109 (1979), 601–612.
M. B. Balk, Polyanalytic Functions, Akademie Verlag, Berlin, 1991.
S. Bergman, The Kernel Function and Conformal Mapping, Amer. Math. Soc., Providence, R. I., 1970.
J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math., 53 (1986), 315–332.
J. J. Carmona, K. Yu. Fedorovski and P. V. Paramonov, On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, preprint, 1999.
P. Duren, Theory of H p Spaces, Academic Press, New York, 1970.
K. Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc., 41 (1937), 321–364.
T. Gamelin, Uniform Algebras, Prentice Hall, Englewood Cliffs New Jersey, 1969.
B. Gustafsson, Quadrature identities and the Schottky double, Acta Appl. Math., 1 (1983), 209–240.
B. Gustafsson, On mother bodies of convex polyhedra, SIAM J. Math. Analysis, 29 (1998), 1106–1117.
V. Havin, Approximation in the mean by analytic functions, Dokl. Akad. Nauk. SSSR, 178 (1968), 1025–1028.
T. kato, Perturbation Theory for Linear Operators, Springer Verlag, Berlin, 1995.
D. Khavinson, J. Mccarthy and H. S. Shapiro, Best approximation in the mean by analytic and harmonic functions, Indiana Univ. Math. J., to appear.
D. Khavinson and H. S. Shapiro, Best approximation in the supremum norm by analytic and harmonic functions, Ark. För Mat., to appear.
A. Korányi and L. Pukánsky, Holomorphic functions with positive real part on polycylinders, Trans. Amer. Math. Soc., 108 (1963), 449–456.
A. Lenard, The numerical range of a pair of projections, J. Funct. Analysis, 10 (1972), 410–413.
P. Lin and R. Rochberg, On the Friedrichs operator, Proc. Amer. Math. Soc., 123 (1995), 3335–3342.
D. Luecking, The compact Hankel operators form an M-ideal in the space of Hankel operators, Proc. Amer. Math. Soc., 79 (1980), 222–224.
I. P. Mysovskikh, Interpolatory Cubature Formulas, Nauka, Moscow, 1981 (in Russian).
Z. Nehari, On bounded bilinear forms, Ann. Math., 65 (1957), 153–162.
D. J. Newman, Pseudo-uniform convexity in H1, Proc. Amer. Math. Soc., 14 (1963), 676–679.
N. K. Nikolskii, Treatise on the Shift Operator, Springer, Berlin, 1986.
V. V. Peller, Estimates of functions of power bounded operators on Hilbert spaces, J. Operator Theory, 7 (1982), 341–472.
V. V. Peller and S. V. Hruscev, Hankel operators, best approximations and stationary Gaussian processes, Uspehi Math. Nauk, 37 (1982), 53–124.
M. Putinar and H. S. Shapiro, The Friedrichs operator of a planar domain, S. A. Vinogradov Memorial Volume, Springer Basel AG, Basel, 2000, 303–330.
F. Riesz and B. Sz.-Nacy, Functional Analysis, Dover, New York, 1990.
M. Sakai, Quadrature Domains, Lect. Notes Math. Vol. 934, Springer, Berlin, 1982.
H. S. Shapiro, Some inequalities for analytic functions integrable over a plane domain, Approximation and Function Spaces, Proc. Conf. Gdansk 1979, North Holland, 1981, 645–666.
H. S. Shapiro, On some Fourier and distribution-theoretic methods in approximation theory, Approximation Theory. III, Proc. Conf. Austin, Texas, 1980, Academic Press, San Diego, 1980, 87–124.
H. S. Shapiro, The Schwarz Function and its Generalization to Higher Dimensions, Wiley-Interscience, New York, 1992.
P. K. Suetin, Polynomials Orthogonal over a Region and Bieberbach Polynomials, Proc. Steklov Inst. Vol. 100 (1971), Amer. Math. Soc., Providence, R.I., 1974
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Putinar, M., Shapiro, H.S. (2001). The Friedrichs operator of a planar domain. II. In: Kérchy, L., Gohberg, I., Foias, C.I., Langer, H. (eds) Recent Advances in Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol 127. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8374-0_29
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DOI: https://doi.org/10.1007/978-3-0348-8374-0_29
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