Abstract We investigate basis and approximation properties of Haar type systems introduced by Vilenkin. The direct approximation theorem by polynomials with respect to a such system is proved in weighted nontrivial rearrangement invariant space under some conditions on weight. Also a direct approximation theorem by double Haar type polynomials is obtained in weighted L p spaces with special weights.

中文翻译：

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Haar 型多项式在加权重排不变空间中的逼近

摘要 我们研究了 Vilenkin 引入的 Haar 型系统的基和近似性质。在加权非平凡重排不变空间中，在权重的某些条件下证明了多项式的直接逼近定理关于这样的系统。在具有特殊权重的加权L p 空间中，还获得了双Haar 型多项式的直接逼近定理。