On Pointwise Approximation of Functions by Some Matrix Means of Conjugate Fourier Series

The results corresponding to some theorems of S. Lal [Tamkang J. Math., 31(4)(2000), 279-288] and the results of the authors [Banach Center Publ. 92(2011), 237-247] are shown. The same degrees of pointwise approximation as in mentioned papers by significantly weaker assumptions on considered functions are obtained. From presented pointwise results the estimation on norm approximation with essentialy better degrees are derived. Some special cases as corollaries for iteration of the Nörlund or the Riesz method with the Euler one are also formulated.

GENERALIZED RIESZ METHOD AND CONVERGENCE ACCELERATION

Several propositions on A‐boundedness for generalized Riesz method (5ft, Pn), where Pn are linear bounded operators from Banach space X into X, are proved. These results are applied to study convergence acceleration using generalized Riesz method and generalized Zygmund method. Apibendrintas Riesz metodas ir konvergavimo pagreitis Santrauka Straipsnyje irodyta keletas teiginiu, susijusiu su A aprežtumu apibendrintam Rieszo metodui, kai Pn yra tiesinis aprežtasis operatorius Banacho erdveje X. Šie rezultatai taikomi tiriant Rieszo ir apibendrinto Zygmundo metodu konvergavimo pagreiti.

A Tauberian Theorem Concerning Borel-Type and Riesz Summability Methods

It is proved that the summability of a series by the Borel-type summability method (B,α,β) together with a certain Tauberian condition implies its summability by the Riesz method (R, log(n + l),p).

Inclusion Relations for General Riesz Typical Means

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Writewhere s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) ifas ω→∞.

Some inclusion theorems

1. A number of inclusion theorems have been given in connection with methods of summation which include the Riesz method (R, λ, κ). Lorentz [4, Theorem 10] gives necessary and sufficient conditions for a sequence to sequence regular matrix A = (an, v) to be such that A ⊃ (R, λ, 1)†. He imposes restrictions on the sequence { λn}, so that A does not include all Riesz methods of order 1. In Theorem 1 below, we generalize the Lorentz theorem by giving a condition without restriction on λn, If the matrix A is a series to sequence or series to function regular matrix, there do not appear to be any results concerning the general inclusionA ⊃ (R, λ, κ).However, when A is the Riemann method (ℜ, λ, μ), Russell [7], generalizing earlier results, has given sufficient conditions for (ℜ, λ, μ) ⊃ (R, λ, κ). Our Theorem 2 gives necessary and sufficient conditions for A ⊃ (R, λ, 1), where A satisfies the condition an, v → 1 (n →co, ν fixed). Thus Theorem 2 applies to any series to sequence regular matrix A. In Theorem 3 we give a further representation for matrices A which include (R, λ, 1), and finally make some remarks on the problem of characterizing matrices which include Riesz methods of any positive order κ.

A note on the Riesz method and the method of residues

It is shown that the Riesz method of analytic continuation and the method of residues give the same results in the classical electromagnetic theory of a point source.