Bases Consisting of Rational Functions of Uniformly Bounded Degrees or More General Functions

We prove in this paper the existence of a Schauder basis for C[0, 1] consisting of rational functions of uniformly bounded degrees. This solves an open question of some years standing concerning the possible existence of such bases. This result follows from a more general construction of bases on R and [0, 1]. We prove that the new bases are unconditional bases for Lp, 1<p<∞, and Besov spaces. On [0, 1], they are Schauder bases for C[0, 1] as well. The new bases are utilized for nonlinear approximation.

[1]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[2]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[3]  George G. Lorentz,et al.  Problems in approximation theory , 1977 .

[4]  Z. Ciesielski,et al.  Construction of an orthonormal basis in $C^{m}(I^{d})$ and $W^{m}_{p}(I^{d})$ , 1972 .

[5]  Estimates of the derivatives of rational functions in LP[−1, 1] , 1986 .

[6]  Charles Fefferman,et al.  Some Maximal Inequalities , 1971 .

[7]  J. R. D. Francia Some Maximal Inequalities , 1985 .

[8]  D. Wulbert The Rational Approximation of Real Functions , 1978 .

[9]  A. Privalov Growth of degrees of polynomial bases , 1990 .

[10]  George C. Kyriazis Wavelet Coefficients Measuring Smoothness in H p (open face R d ) , 1996 .

[11]  P. Petrushev,et al.  Rational Approximation of Real Functions , 1988 .

[12]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[13]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[14]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[15]  Z. Ciesielski Constructive function theory and spline systems , 1975 .

[16]  C. Chui,et al.  Approximation Theory VI , 1990 .

[17]  P. Lemarié,et al.  Base d'ondelettes sur les groupes de Lie stratifiés , 1989 .

[18]  A. Privalov,et al.  Growth of the degrees of polynomial bases and approximation of trigonometric projectors , 1987 .

[19]  R. A. Lorentz,et al.  Orthogonal Trigonometric Schauder Bases of Optimal Degree for C(K) , 1994 .

[20]  J. Peetre New thoughts on Besov spaces , 1976 .

[21]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[22]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[23]  Przemysław Wojtaszczyk,et al.  Orthonormal polynomial bases in function spaces , 1991 .

[24]  G. Weiss,et al.  A First Course on Wavelets , 1996 .