Lower estimate of Jackson's constant in $L_p$-spaces on the sphere with Dunkl weight function associated with dihedral group

In the late 80ies and in the early 90ies of the past century the framework for a theory of special functions and integral transforms in several variables related with reflection groups was systematically built up in a series of papers of american mathematician C. F. Dunkl. This theory was further developed by many mathematicians. Nowadays, this theory is called Dunkl theory in the literature. Dunkl theory is widely used in probability theory, mathematical physics, approximation theory.
The present paper is devoted to an application of Dunkl harmonic analysis on the Euclidean space $\mathbb{R}^d$ and the unit Euclidean sphere $\mathbb{S}^{d-1}$ with Dunkl weight function invariant under the reflection group associated with some root system to problems of approximation theory.
The problem of finding the sharp constant in Jackson's inequality, or Jackson's constant, between the value of best approximation of a function and its modulus of continuity in $L_p$-spaces is an important extremum problem of approximation theory. In the paper, the problem of Jackson's constant in $L_p$-spaces, $1\leq p<2$, on the unit circle $\mathbb{S}^1$ in the Euclidean plane $\mathbb{R}^2$ with Dunkl weight function invariant under the dihedral group $I_m$, $m\in\mathbb{N}$, is considered. Best approximation is given in terms of linear combinations of $\kappa$-spherical harmonics defined by means of the Dunkl Laplacian. We introduce the modulus of continuity using the generalized translation operator first appeared in the papers of Y. Xu.
In the «weightless» case when the multiplicity function is identically equal to zero on a root system, D. V. Gorbachev proved Jackson's inequality in $L_p$-spaces, $1\leq p<2$, on the unit multidimensional Euclidean sphere $\mathbb{S}^{d-1}$ with the constant $2^{1/p-1}$ coinciding with Jung's constant of the $L_p$-spaces. He also established its sharpness.
Jackson's inequality with the same constant in $L_p$-spaces on the unit multidimensional Euclidean sphere $\mathbb{S}^{d-1}$ with arbitrary Dunkl weight function was established earlier by the author. Now in the paper, we obtain the lower estimate of Jackson's constant in $L_p$-spaces, $1\leq p<2$, on the unit circle $\mathbb{S}^1$ in $\mathbb{R}^2$ with Dunkl weight function invariant under the dihedral group $I_m$, $m\in\mathbb{N}$. The dihedral groups are symmetry groups of regular $m$-gons in $\mathbb{R}^2$ for $m\geq 3$.
To solve the given problem, we essentially use the method developed by V. I. Ivanov in cooperation with Liu Yongping. There are additional difficulties associated with the new modulus of continuity based on the nonsymmetric generalized translation operator in the spaces $L_p[0,\pi]$, $1\leq p<2,$ with the weight function $|\sin(t/2)|^{2\alpha +1}|\cos(t/2)|^{2\beta +1}$, $\alpha\geq\beta\geq-1/2$.
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