Lebesgue inequalities for Poisson integrals (Q2739811)

Let \(f(x)\) be a \(2\pi\)-periodic function, let \({\mathcal T}_{2n-1}\) be the set of all trigonometric polynomials \(f_{n-1}(\cdot)\) of the order \(\leq n-1\), and let \(E_n(f)_p=\inf_{t_{n-1}\in{\mathcal T}_{2n-1}} \|f(\cdot)-f_{n-1}(\cdot)\|_p\) be the best approximation of the function \(f(\cdot)\) by trigonometric polynomials \(f_{n-1}(\cdot)\) from the set \({\mathcal T}_{2n-1}\) in the space \(L_p\). Let, for \(f\in L\), \(S_n(f;x)\) be a partial sum of the Fourier series and let \(\rho_n(f;x)=f(x)-S_n(f;x)\). The authors prove the following theorem.NEWLINENEWLINENEWLINELet \(q\in(0,1)\), \(\beta\in\mathbb R\) and \(p\geq 1\). Then for any function \(f\in L_{\beta}^qL_p\) the asymptotic estimate holds true: NEWLINE\[NEWLINE \|\rho_n(f;x)\|_p\leq\left( {{8q^n}\over{\pi^2}}K(q)+ {{O(1)g^n}\over{(1-q)^2n}}\right) E_n(f_{\beta}^q)_p,NEWLINE\]NEWLINE where \(O(1)\) is bounded uniformly on \(n,q,p,\beta\) and \(f\in L_{\beta}^qL_p\), \(K(q)\) is the elliptic integral of the first kind. NEWLINENEWLINENEWLINEA similar theorem is proved in the case of \(L_{\beta}^q C\) space.