Comparison of speeds of convergence in some families of summability methods for functions
Küçük Resim Yok
Tarih
2009
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Maltepe Üniversitesi
Erişim Hakkı
CC0 1.0 Universal
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Özet
Speeds of convergence in certain family of summability methods for functions are compared in the talk. The results introduced here extend the results proved in [1] for ”matrix case” to ”integral case” and are partly published in [2]. 1. Let us denote by X the set of all functions x = x(u) defined for u ? 0, bounded and measurable by Lebesgue on every finite interval [0, u0]. Suppose that A is a transformation of functions x = x(u) (or, in particular, of sequences x = (xn)) into functions Ax = y = y(u) ? X. If the limit limu?? y(u) = s exists then we say that x = x(u) is convergent to s with respect to the summability method A, and write x(u) ? s(A). One of the basic notions in our talk is the notion of speed of convergence. Let ? = ?(u) be a positive function from X such that ?(u) ? ? as u ? ?. We say that a function x = x(u) is convergent to s with speed ? if the finite limit limu?? ?(u) [x(u) ? s] exists. We say that x is convergent with speed ? with respect to the summability method A if the function Ax = y = y(u) ? X is convergent with speed ?. 2. We discuss a Riesz-type family {A?} of summability methods A? where ? > ?0 and ?0 is some fixed number and which transform functions x = x(u) into functions A?x = y?(u). This family is defined with the help of relation A? = C?,? ? A? (? > ? > ?0), where C?,? is certain integral transformation (see e.g. [2]). For example, the Riesz methods (R, ?) and certain generalized N¨orlund methods (N, p?(u), q(u)) form Riesz-type families. It is important to be able to compare the speed of convergence of x = x(u) with respect to different methods in family {A?} For a given speed ? = ?(u) and a fixed number ? > ?0 the speeds ?? = ??(u) and ?? = ??(u) can be found (see [2]) such that for all ? > ? > ? the next implications are true: ?(u) [y? (u) ? s] ? t =? ??(u) [y?(u) ? s] ? t, ?(u) [y? (u) ? s] = O(1), ??(u) [y?(u) ? s] ? t =? ??(u) [y?(u) ? s] ? t.
Açıklama
Anahtar Kelimeler
Kaynak
International Conference of Mathematical Sciences
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Sheletski, A. (2009). Comparison of speeds of convergence in some families of summability methods for functions. Maltepe Üniversitesi. s. 105.
