Maximal theorems for some orthogonal series. I
Maximal theorems for some orthogonal series. I
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一些正交级数的极大定理。
DOI：
10.1090/s0002-9947-1969-0252941-3
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发表时间：
1969
影响因子：
1.3
通讯作者：
J. Gilbert
中科院分区：
文献类型：
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作者：
J. Gilbert
注意，在（ii）中，序列不需要相同。在（i）中，Marcel Riesz定理以某种形式或其他形式的使用似乎是不可避免的，因此，不幸的是，（i）不能用作Marcel Riesz定理的替代证明;结果（ii）需要傅立叶级数的Carleson-Hunt定理的全部强度[7]，[12]。所有五个条件都满足，例如，经典的正交序列（0，7 T）：傅里叶-贝塞尔函数，傅里叶-迪尼函数，超球多项式和雅可比多项式。因此，在结果（ii）中，我们可以形成关于Jacobi多项式{vn}的系数{f（n）}，然后用Fourier-Dini函数{un}形成级数Jnf（n）un（x）。Askey-Wainger [4]、Askey [2]和Muckenhoupt-Stein [14]的许多结果也包含在这个公理化方法中。在其他地方，我们将证明这五个条件满足两大类Sturm-Liouville系统。基本上，所需要的是一个足够好的2项渐近估计，比如2项Hilb型估计。在证明这些”移植”定理的每个阶段，我们都试图挑出必须加在{un}，{v，j上的条件，以保证结果的有效性。显然，需要一些条件，因为没有一般的正交级数具有与傅立叶级数如此相似的性质。关于本文中使用的没有正式定义的概念的讨论，请参见Edwards [9]，Zygmund [24]。
Note that in (ii) the sequences need not be identical. In (i) use of the Marcel Riesz theorem in some form or other seems unavoidable, so, unfortunately,(i) cannot be used as an alternative proof of the Marcel Riesz theorem; result (ii) requires the full strength of the Carleson-Hunt theorem for Fourier series [7],[12]. All five conditions are satisfied, for example, by the classical orthonormal sequences on (0, 7T): Fourier-Bessel functions, Fourier-Dini functions, ultraspherical polynomials and Jacobi polynomials. Thus, in result (ii), we could form the coefficients {f (n)} with respect to Jacobi polynomials {vn} and then the series Jnf (n) un (x) with Fourier-Dini functions {un}. Many of the results of Askey-Wainger [4], Askey [2] and Muckenhoupt-Stein [14] also are included in this axiomatic approach. Elsewhere we shall show that the five conditions are satisfied by two large classes of Sturm-Liouville systems [11]. Basically, what is required is a good enough 2-term asymptotic estimate, say a 2-term Hilb-type estimate. At each stage of the proof of these" transplantation" theorems we have attempted to single out the conditions which must be imposed on {un},{v, j to ensure the validity of the results. Obviously some conditions are required since no general orthogonal series has properties so closely analogous to Fourier series. For a discussion of the concepts used in this paper without formal definition see Edwards [9], Zygmund [24].