In this work, we look at problems in the theory of approximation of functions on the space \(L_{q,\alpha }^{2}(\mathbb {R}_{q})\) with power weight. We prove analogues of the direct Jackson’s theorems for the modulus of smoothness (of arbitrary order) constructed using the generalized q-Dunkl translation operator. We also show that the modulus of smoothness and the K-functional constructed from the Sobolev-type space corresponding to the q-Dunkl operator are equivalent.

中文翻译：

---

关于具有空间权重的空间$$ L ^ {2} _ {q，\ alpha}（\ mathbb {R} _ {q}）$$ L q，具有幂权重的α2（R q）的光滑度模和定理

在这项工作中，我们研究了具有权重的空间\（L_ {q，\ alpha} ^ {2}（\ mathbb {R} _ {q}）\）上的函数逼近理论中的问题。我们证明了使用广义q -Dunkl平移算子构造的（任意阶数）平滑模量的直接Jackson定理的类似物。我们还表明，光滑度模量和从对应于q -Dunkl算子的Sobolev型空间构造的K函数是等效的。