Abstract
We construct polynomial orthonormal bases in various function spaces. Our bases have linear order of growth of degrees of polynomials. We show that this order is optimal.
Similar content being viewed by others
References
S. V. Bočkariov,Construction using Fejer kernels of interpolating dyadic basis in the space of continuous functions, Trudy Steklov Inst.172 (1985), 29–59 (in Russian).
S. V. Bočkariov,Construction of polynomial bases in finite dimensional spaces of functions analytic in the unit disc, Trudy Steklov Inst.164 (1983), 49–74 (in Russian).
S. V. Bočkariov,Conjugate Franklin system — a basis in the space of continuous functions, Dokl. Akad. Nauk285 (1985), 521–526 (in Russian).
J. Bourgain,Homogeneous polynomials on the ball and polynomial bases, Isr. J. Math.68 (1989), 327–347.
Z. A. Čanturija,On orthogonal polynomial bases in the spaces C and L, Analysis Math5 (1) (1979), 9–17.
Z. A. Chanturia,On unconditional polynomial bases of the space L p , Studia Math.71 (1981), 163–168.
R. R. Coifman and G. Weiss,Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977), 569–645.
G. Faber,Über die interpolatorische Darstelung stetige Funktionen, Jahresber. D. M. V.23 (1914), 192–210.
C. Foias and I. Singer,Some remarks on strongly linearly independent sequences and bases in Banach spaces, Revue Rom. Math. Pures Appl.16(3) (1961), 589–594.
P. P. Korovkin,Linear operators and approximation theory, Fiz.-Mat. Gos. Izdat., Moscow (1959) (in Russian).
M.I. Kadec and A. Pełczyński,Bases, lacunary sequences and complemented subspaces in the spaces L p , Studia Math.21 (1962), 161–176.
Al. A. Privalov,On the growth of degrees of polynomial basis and approximation of trigonometric projections, Mat. Zametki42(2) (1987), 207–214 (in Russian).
Al. A. Privalov,On the growth of degrees of polynomial bases, Mat. Zametki48(4) (1990), 69–78 (in Russian).
B. Shekhtman,On the norms of interpolating operators, Israel J. Math.64(1) (1989), 39–48.
B. Shekhtman,On polynomial “interpolation” in L 1 , preprint.
E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
A. Torchinsky,Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
P. L. Ulianov,On some solved and unsolved problems in the theory of orthogonal series, Proc. IV All Union Math. Congress2, Publishing House AN SSSR, Moscow (1964), 694–704 (in Russian).
P. L. Ulianov,On some results and problems in the theory of bases, Zapiski LOMI, 274–283 (in Russian).
P. Wojtaszczyk,The Franklin system is an unconditional basis in H 1 , Arkiv für Mathematik20(2) (1982), 293–300.
P. Wojtaszczyk,Banach Spaces for Analysts, Cambridge University Press (to appear).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wojtaszczyk, P., Woźniakowski, K. Orthonormal polynomial bases in function spaces. Israel J. Math. 75, 167–191 (1991). https://doi.org/10.1007/BF02776023
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02776023