Approximation of continuous functions by F(a,q) means (Q921398)

Let \({\mathbb{C}}[0,2\pi]\) denote the space of continuous \(2\pi\)-periodic functions. The transform \(\sigma (p,f,x)=\sum^{\infty}_{k=0}c_ k(p)s_ k(x)\) of the Fourier series of \(f\in C[0,2\pi]\) at x, where \(s_ k(x)\) is the k-th partial sum of the Fourier series of f belongs to the F(a,q) family provided for some fixed \(\gamma\), \(1/2<\gamma <2/3\), \[ c_ k(p)=g(q,k)\{1+O((| k-q| +1)/q)+O(| k-q|^ 3/q^ 2\} \] as \(q\to \infty\) uniformly in k for \(| k-q| <q^{\gamma}\), where \(g(q,k)=\sqrt{a/\pi q} \exp (-aq^{-1}(k-q)^ 2),\) \(a\geq 0\), \(q=q(p)\) is a positive increasing function of a continuous or a discrete parameter p, q tends to infinity as \(p\to \infty\), and \[ \sum_{| k-q| >q}\gamma (k+1)c_ k(p)=O(\exp (-q^{\mu})) \] as \(q\to \infty\) for some number \(\mu\) independent of p. In this paper the following results are proved. Theorem 1: Let \(f\in C[0,2\Pi]\) and \(\omega\) (t,f) denote the modulus of continuity of f. Let [q] denote the integral part of \(q=q(p)\). Set \(m=[q]+1\). Suppose that \(\omega\) (t,f)/t is a non-increasing function of t. Then \[ \sigma (p,f,x)=1/\pi^ 2\sum^{[m]}_{k=0}m/(k+1)\int^{u(k+1)}_{u(k)}\psi (t)\sin (mt)dt+O(\omega (1/\omega,f)), \] where \(\psi (t)=1/2\{f(x+t)+f(x-t)-2f(x)\}\) and \(u(k)=(4k+1)\pi /2m.\) Let \(\omega\) (t) \((t>0)\) be a continuous function satisfying the conditions \(\omega (0)=0\), \(0\leq \omega (t_ 2)-\omega (t_ 1)\leq \omega (t_ 2-t_ 1),\) \(0\leq t_ 1\leq t_ 2\) and \(\omega\) (t)/\(\sqrt{t}\) be a non-increasing function of t. \(\omega\) (t) is assumed to be convex, i.e., \(\omega (t_ 1)+\omega (t_ 2)\leq 2\omega ((t_ 1+t_ 2)/2),\) \(0\leq t_ 1\leq t_ 2\). We say \(f\in H(\omega)\) if \(f\in C[0,2\pi]\) and its modulus of continuity satisfies \(\omega\) (t,f)\(\leq \omega (t)\), \(t>0\). Theorem 2: The following asymptotic equality (with m as in Theorem 1) holds: \[ \sup_{f\in H(\omega)}\| \sigma (p,f,x)- f(x)\| =\frac{\log (m)}{\pi^ 2}\int^{\pi /2}_{0}\frac{\omega (4t)}{m} \sin (t)dt+O(\omega (1/m)). \] The results are extensions of results proved by \textit{A. A. Efimov} [ibid. 12, 97-113 (1986; Zbl 0607.42002)] for the Euler (E,1) method. The author gives new results for \(Euler(E,q)(q>0)\), generalised Borel, Taylor, \(S_{\alpha}\), and Valiron means which are members of the family F(a,q).