Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients
- PDF / 441,500 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 8 Downloads / 183 Views
Laguerre and Disk Polynomial Expansions with Nonnegative Coefficients Yuichi Kanjin
Received: 11 June 2012 / Published online: 24 January 2013 © Springer Science+Business Media New York 2013
Abstract We establish Wiener type theorems and Paley type theorems for Laguerre polynomial expansions and disk polynomial expansions with nonnegative coefficients. Keywords Laguerre polynomial expansions · Disk polynomial expansions · Nonnegative coefficients Mathematics Subject Classification (2000) Primary 42C10 · 33C45 · Secondary 46E30
1 Introduction A well-known theorem on functions with positive Fourier coefficients given by Norbert Wiener (see [4, pp. 242–250] and [19, Sects. 1–2]) is the following: ˆ [A] Wiener’s theorem Let f ∈ L1 (−π, π π) be a function satisfying f (n) ≥ 0 for every n ∈ Z, where fˆ(n) = (1/(2π)) −π f (θ )e−inθ dθ . If there exists a constant π δ δ > 0 such that −δ |f (θ )|2 dθ < ∞, then −π |f (θ )|2 dθ < ∞.
To the memory of my mother and father. Communicated by Hans G. Feichtinger. The research was supported by Grant-in-Aid for Science Research (C) (No. 24540167), Japan Society for the Promotion of Science. Y. Kanjin () Mathematics Section, Division of Innovative Technology and Science, Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920-1192, Japan e-mail: [email protected]
496
J Fourier Anal Appl (2013) 19:495–513
On functions with positive Fourier coefficients satisfying ess sup|θ| 0, we have the following which is a part of the results of Paley [18]: ˆ [B] Paley’s theorem Let f ∈ L1 (−π, π) be an even function ∞satisfying f (n) ≥ 0 for ˆ every n. If ess sup|θ| 0, then n=−∞ f (n) < ∞. Recently, Mhaskar and Tikhonov [17] extended these two theorems to the Jacobi (α,β) polynomial expansions. Let us state an essential part of their results. Let Rn (x) (α,β) be the Jacobi polynomials of order α, β > −1 with the normalization Rn (1) = 1, that is, the orthogonal polynomials pn (x) on the interval [−1, 1] with respect to the weight function wα,β (x) = (1 − x)α (1 + x)β satisfying pn (1) = 1. It is known that (−1/2,−1/2) (cos θ ) = cos nθ . A function f on [−1, 1] is formally expanded: f (x) ∼ R n∞ (α,β) ˆ (x). Here, fˆ(n) is the Fourier-Jacobi coefficient of f defined by f (n)R n n=0 fˆ(n) = ρn−1
1
−1
f (x)Rn(α,β) (x)wα,β (x) dx,
ρn =
1
−1
|Rn(α,β) (x)|2 wα,β (x) dx.
[C] [17] Let f ∈ L1 ([−1, 1], wα,β ). Suppose that every Fourier-Jacobi coefficient fˆ(n) is nonnegative. Then the following (i) and (ii) hold. 1 (i) If there exists a constant δ > 0 such that 1−δ |f (x)|2 wα,β (x) dx < ∞, then f ∈ L2 ([−1, 1], wα,β (x)). (ii) If there exists a constant δ > 0 such that ess sup1−δ0 As the above, the weighted norms are denoted by f p without the subscript α. 2.1 Preparations In this subsection, we summarize some facts and results without proofs which are referred mainly to [9], and we shall give two lemmas which will be used for proving our theorems. (α) Let Ln (x) be the Laguerre polynomial of degree n = 0, 1, 2, . . . , which is given by
Data Loading...
