θ-Summation and Hardy Spaces

Abstract

A general summability method of Fourier series and Fourier transforms is given with the help of an integrable function % having integrable Fourier transform. Under some weak conditions on % we show that the maximal operator of the %-means of a distribution is bounded from H p (T) to L p (T) ( p 0 <p< ) and is of weak type (1,1), where H p (T) is the classical Hardy space and p 0 <1 is depending only on %. As a consequence we obtain that the %-means of a function f # L 1 (T) converge a.e. to f. For the endpoint p 0 we get that the maximal operator is of weak type (H p0 (T), L p0 (T)). Moreover, we prove that the %-means are uniformly bounded on the spaces H p (T) whenever p 0 <p< and are uniformly of weak type (H p0 (T), H p0 (T)). Thus, in the case f # H p (T), the %-means converge to f in H p (T) norm ( p 0 <p< ). The same results are proved for the conjugate %-means and for Fourier transforms, too. Some special cases of the %-summation are considered, such as the Weierstrass, Picar, Bessel, Feje r, Riemann, de La Valle e-Poussin, Rogosinski and Riesz summations. Academic Press

Local Hardy spaces and summability of Fourier transforms

“…The θ -summation was considered in a great number of papers and books, such as Butzer and Nessel [3], Trigub and Belinsky [21], Bokor, Schipp, Szili and Vértesi [14,2,18,17], Natanson and Zuk [12], Weisz [23][24][25][26] and Feichtinger and Weisz [4,5].…”

“…Butzer and Nessel [3] and recently Bokor, Schipp, Szili and Vértesi [14,2,18,17], Trigub and Belinsky [21] and Weisz [23][24][25][26] considered a general method of summation, the so-called θ -summation, which is generated by a function θ . They proved that ifθ can be estimated by a non-increasing integrable function, then the θ -means σ θ T f of the Fourier transform of a function f ∈ L p (R) (1 p ∞) converge to f a.e.…”

Herz spaces and restricted summability of Fourier transforms and Fourier series

“…The θ -summation was considered in a great number of papers and books, such as Butzer and Nessel [3], Trigub and Belinsky [19], Bokor, Schipp, Szili and Vértesi [13,16,17] and Weisz [5,6,[22][23][24].…”

Weak inequalities for Cesàro and Riesz summability of Walsh–Fourier series

“…This weak type inequality implies the known results mentioned above, namely the two maximal operators are of weak type (1, 1) and are bounded from H p to L p (1/( + 1) < p < ∞). Analogous weak type inequalities for the -summation of the trigonometric Fourier series can be found in Weisz [15].…”