On strong Euler absolute \(\phi\)-summability a.e. of orthonormal series. (English) Zbl 0719.42024
Constructive theory of functions, Proc. Int. Conf., Varna/Bulg. 1987, 343-347 (1988).
[For the entire collection see Zbl 0695.00019.]
The author proves a result about Euler means \(E^ q_ n(x)\) of order \(q>0\) of a real orthonormal series \(\sum c_ k\phi_ k(x)\). She shows under suitable growth conditions that \(\sum^{\infty}_{n=0}\phi_ n(\lambda | E^ q_ n(x)-E^ q_{n-1}(x)|)\) a.e. is finite for any sequence \(\phi_ 0,\phi_ 1,..\). of increasing \(\phi\)-functions and any \(\lambda >0\). This generalizes one of her earlier results [Funct. Approximation, Comment. Math. 14, 155-158 (1984; Zbl 0591.40004)] which identified conditions sufficient that \(\sum^{\infty}_{n=0}n^ p| E^ q_ n(x)-E^ q_{n-1}(x)|^{\alpha}\) be finite a.e.
The author proves a result about Euler means \(E^ q_ n(x)\) of order \(q>0\) of a real orthonormal series \(\sum c_ k\phi_ k(x)\). She shows under suitable growth conditions that \(\sum^{\infty}_{n=0}\phi_ n(\lambda | E^ q_ n(x)-E^ q_{n-1}(x)|)\) a.e. is finite for any sequence \(\phi_ 0,\phi_ 1,..\). of increasing \(\phi\)-functions and any \(\lambda >0\). This generalizes one of her earlier results [Funct. Approximation, Comment. Math. 14, 155-158 (1984; Zbl 0591.40004)] which identified conditions sufficient that \(\sum^{\infty}_{n=0}n^ p| E^ q_ n(x)-E^ q_{n-1}(x)|^{\alpha}\) be finite a.e.
Reviewer: W.R.Wade (Knoxville)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
