Unique minimality of Fourier projections

Fisher, S. D.; Morris, P. D.; Wulbert, D. E.

doi:10.1090/S0002-9947-1981-0607118-1

Unique minimality of Fourier projections
HTML articles powered by AMS MathViewer

by S. D. Fisher, P. D. Morris and D. E. Wulbert PDF

Trans. Amer. Math. Soc. 265 (1981), 235-246 Request permission

Abstract:

The question of when the Fourier projection is the only one of least norm from a space of continuous functions on the circle onto spaces spanned by trigonometric polynomials is studied in two settings. In the first the domain space is the disc algebra and the range is finite-dimensional. In the second the domain space consists of all real continuous functions and the range has finite codimension.

References

D. L. Berman, On the impossibility of constructing a linear polynomial operator furnishing an approximation of the order of the best approximation, Dokl. Akad. Nauk SSSR 120 (1958), 1175–1177 (Russian). MR 0098941
E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
E. W. Cheney, C. R. Hobby, P. D. Morris, F. Schurer, and D. E. Wulbert, On the minimal property of the Fourier projection, Trans. Amer. Math. Soc. 143 (1969), 249–258. MR 256044, DOI 10.1090/S0002-9947-1969-0256044-3
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655

The theory of matrices

Pol V. Lambert, Minimum norm property of the Fourier projection in spaces of continuous functions, Bull. Soc. Math. Belg. 21 (1969), 359–369. MR 273292
Pol V. Lambert, On the minimum norm property of the Fourier projection in $L^{1}$-spaces, Bull. Soc. Math. Belg. 21 (1969), 370–391. MR 273293

On a class of linear operators

61

Quelques remarques sur l’interpolation

8

Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972