Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms
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Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms S.B. Damelin Department of Mathematics, Georgia Southern University, PO Box 8093, Statesboro, GA 30460, USA E-mail: [email protected]
Received 25 July 2002; accepted 13 December 2004 Communicated by L. Reichel
We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest. Keywords: Fourier series, Freud weight, Hilbert transform, pointwise convergence, polynomial approximation, orthonormal expansions, weighted polynomial approximation Mathematics subject classifications (2000): 41A10, 42C05
1.
Introduction and statement of results
1.1. Background: Fourier series/orthogonal expansions In this paper, we study pointwise approximation of measurable functions f : R → R, by orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest. To set the scene for our investigations, a weight w will be a positive function on R with x n w(x) ∈ L1 (R) := L1 , n = 0, 1, . . . . Given w as above, we may form an orthonormal/Fourier expansion f →
∞ j =0
bj pj ,
bj :=
R
fpj w2 , j 0,
for any measurable function f : R → R for which f (x)x j w2 (x) dx < ∞, j = 0, 1, . . . . R
(1.1)
452
S.B. Damelin / Pointwise bounds of orthogonal expansions
Here, see [13], pn := pn (w2 ), n 0, are the unique orthonormal polynomials of degree n satisfying pn (x)pm (x)w2 (x) dx = δm,n , m, n 0, (1.2) R
where
δm,n :=
0, 1,
m = n, m = n.
For n 1 and f satisfying (1.1), we set: Sn [f ] :=
n−1
bj pj .
(1.3)
j =0
Our focus in this paper is to study pointwise bounds for the partial sums given by (1.3) in suitable weighted spaces on the line, which in turn, allows for further investigations concerning pointwise convergence with rates of convergence. For orthonormal expansions on finite intervals, there are many well-known mean convergence results starting with those of Riesz and continuing with results on Chebyshev, Jacobi and generalized Jacobi weights. We do not review these aforementioned results here but refer the reader to [16,22,29] and the many references cited therein for a comprehensive account of this vast and interesting subject. The first significant results dealing with mean convergence of orthonormal expansions on the line are due to Askey and Wainger for the Hermite weight w(x) = exp(−x 2 ), see [1]. Thereafter, followed related results of Muckenhoupt, see [24,25], Mhaskar and Xu, see [23] and Jha and Lubinsky, see [14]. The subject of pointwise convergence of orthonormal expansions on the line is not cited much in the literature. Indeed, the only results that are known to this author are
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