Hausdorff Summability Research Articles

E-J Summability of Orthogonal Series

In this paper we obtain a sufficient condition for the E-J summability of certain orthogonal series. Our results generalize the corresponding theorems for ordinary Hausdorff summability obtained by Kalaivana and Youvaraj.

Order of magnitude for Hausdorff summability of double orthogonal series

In [3], we established a necessary and sufficient condition for absolute Hausdorff summability of a double orthogonal series. In this paper, we estimate the order of magnitude for Hausdorff summability of a double orthogonal series under appropriate conditions.

On double Hausdorff summability method

Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savas and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savas and Sevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices. MSC:40F05, 40G05.

Generalized absolute Hausdorff summability of orthogonal series

For 1≦k≦2 and a sequence \(\gamma :={\{\gamma(n)\}}_{n=1}^{\infty}\) that is quasi β-power monotone decreasing with \({\beta>1-\frac{1}{k}}\), we prove the |A,γ|k summability of an orthogonal series, where A is either a regular or Hausdorff matrix. For \({\beta>-\frac{3}{4}}\), we give a necessary and sufficient condition for |A,γ|k summability, where A is Hausdorff matrix. Our sufficient condition for \({\beta>-\frac{3}{4}}\) is weaker than that of Kantawala [1], \({\beta>-\frac{1}{k}}\) for |E,q,γ|k summability; and of Leindler [4], β>−1 for |C,α,γ|k, \({\alpha<\frac{1}{4}}\). Also, our result generalizes the result of Spevakov [6] for |E,q,1|1 summability.

On absolute generalized Hausdorff summability of orthogonal series

On absolute generalized Hausdorff summability of orthogonal series

Vector-Valued Hausdorff Summability Methods and Ergodic Theorems

Suppose $X$ and $Y$ are two general Banach spaces. Let $H = ({\Lambda _{n,k}})$ be a general ${\mathbf {B}}[X,Y]$-operator valued Hausdorff summability method: ${\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}$ for $k \leq n$ and ${\Lambda _{n,k}} = {\theta _{X,Y}}$ for $k > n$, where $\{ {U_k}\} _{k = 0}^\infty$ is a sequence of operators in ${\mathbf {B}}[X,Y]$ and $\Delta$ denotes the backward difference (operator) and ${\theta _{X,Y}}(x) = {0_Y}$ (the zero element in $Y$) for all $x \in$. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages \[ \sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]).\]

On strong generalized Hausdorff summability

On strong generalized Hausdorff summability

Vector-valued Hausdorff summability methods and ergodic theorems

Suppose X X and Y Y are two general Banach spaces. Let H = ( Λ n , k ) H = ({\Lambda _{n,k}}) be a general B [ X , Y ] {\mathbf {B}}[X,Y] -operator valued Hausdorff summability method: Λ n , k = ( k n ) Δ n − k U k {\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k} for k ≤ n k \leq n and Λ n , k = θ X , Y {\Lambda _{n,k}} = {\theta _{X,Y}} for k &gt; n k &gt; n , where { U k } k = 0 ∞ \{ {U_k}\} _{k = 0}^\infty is a sequence of operators in B [ X , Y ] {\mathbf {B}}[X,Y] and Δ \Delta denotes the backward difference (operator) and θ X , Y ( x ) = 0 Y {\theta _{X,Y}}(x) = {0_Y} (the zero element in Y Y ) for all x ∈ x \in . Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages \[ ∑ k = 0 n ( k n ) Δ n − k U k ( T k x ) ( x ∈ X , T ∈ B [ X ] ) . \sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]). \]

The spectrum of the Cesàro operator on c0(c0)

1. In a recent paper [6], the spectrum of the Cesàro operator C on c0 (the space of null sequences of complex numbers with the sup norm) was obtained by finding the eigenvalues of the adjoint operator on and showing that the operator (C–λI)−1 lies in B(c0) for all λ outside the closure of this set of eigenvalues. In this paper we apply a similar method to find the spectrum of the two-dimensional Cesàro operator on a space of double sequences c0(c0) (defined in §2). We shall introduce a simplification to the proof in [6] by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of [11] for the single variable case on the space of convergent sequences c), and the crux of our proof is to show that the moment constant associated with the method (C – λI)−1 is regular for the space c0(c0) and the set of λ under consideration. It turns out that c0(c0) ≅ c0c0 (see page 237 of [7]) and that the two-dimensional Cesàro operator on c0(c0) is the tensor product C ⊗ C of the Cesàro operator C on c0. Thus our result gives a direct proof that the spectrum σ(C ⊗ C) equals σ(C)σ(C), which is a special case of the result of Schechter in [8].

A Tauberian Theorem for Hausdorff Methods

Let $H = (H,\chi )$ be a regular Hausdorff summability method defined by the function $\chi \in {\text {BV}}\left [ {0,1} \right ]$. It is shown that if $\chi$ is absolutely continuous on $\left [ {0,1} \right ]$, then the methods $H$ and $V \cdot H$ are equivalent for bounded sequences, where $V$ belongs to a certain class of summability methods which includes the Cesà ro methods ${C_\alpha }(\alpha {\text { > }}0)$, the Abel method $A$, and the methods $A \cdot {C_\alpha }(\alpha {\text { > }} - 1)$.

A Tauberian theorem for Hausdorff methods

Let H = ( H , χ ) H = (H,\chi ) be a regular Hausdorff summability method defined by the function χ ∈ BV [ 0 , 1 ] \chi \in {\text {BV}}\left [ {0,1} \right ] . It is shown that if χ \chi is absolutely continuous on [ 0 , 1 ] \left [ {0,1} \right ] , then the methods H H and V ⋅ H V \cdot H are equivalent for bounded sequences, where V V belongs to a certain class of summability methods which includes the Cesàro methods C α ( α &gt; 0 ) {C_\alpha }(\alpha {\text { &gt; }}0) , the Abel method A A , and the methods A ⋅ C α ( α &gt; − 1 ) A \cdot {C_\alpha }(\alpha {\text { &gt; }} - 1) .

On the absolute Hausdorff summability of a Fourier series

On the absolute Hausdorff summability of a Fourier series

A Best Possible Tauberian Theorem for the Collective Continuous Hausdorff Summability Method

The purpose of this paper is to prove that o(l/x) is the best possible Tauberian condition for the collective continuous Hausdorff method of summation. The analogue of this result for the collective (discrete) Hausdorff method is known [1, pp. 229, ff.; 7, p. 318; 8, p. 254]. Our method involves generalizing a well-known Abelian theorem of Agnew [2] to locally compact spaces and then applying the analogue for integrals of a result Lorentz obtained for series [6, Theorem 1].Let T and X denote locally compact, non compact, σ-compact Hausdorff spaces. Let T′ = T ∪ (∞) and X′ = X ∪ (∞) denote the onepoint compactifications of T and X, respectively. Let B(T) denote the set of locally bounded, complex valued Borel functions on T and let B∞(T) denote the bounded functions in B(T).

On the Hausdorff summability of series associated with a Fourier and its allied series

Récemment, Tripathy - Jour. Ind. Math. Soc., 32 (1960), 141-154 - a démontré certains résultats sur la sommabilité absolue de Hausdorff de séries associées à des séries de Fourier et des séries qui s’y rattachent, résultats qui généralisent ceux démontrés par Mohanty sur la sommabilité absolue de Cesaro.

The Hausdorff Summability of Fourier Series

The Hausdorff Summability of Fourier Series

The Hausdorff summability of Fourier series

tn = ( ('n-v u)s,. It is known (see, e.g., Hardy [1, Theorem 208]) that a necessary and sufficient condition for a Hausdorff transformation generated by a sequence {,An} to be regular is that there exists in the interval [0, 1] a function X(t) of bounded variation such that x(O) =x(+0) = 0, X(1) = 1, and An = fxdX (x), n = 0, 1, 2, Let f(x) be a periodic function with period 27, and integrable L in (-7r, 7r), and let

On the absolute Hausdorff summability factors of a Fourier series

Let be a given infinite series with the sequence of partial sums {sn}. Then the sequence-to-sequence Hausdorff transformation of the sequence {sn} is given by

Inclusion theorems for generalized Hausdorff summability methods

The purpose of this note is to demonstrate how some of the classical Hausdorff inclusion theorems extend to the case where the sequences have their domains and ranges in topological vector spaces. We follow the terminology in [3] and [1] for the topological vector spaces and Hausdorff summability methods respectively. Suppose (X, Tl) and (Y, T2) are locally convex separated topological vector spaces. We denote by L(X, Y) the linear functions from X to Y which are continuous with respect to the topologies T, and T2. DEFINITION 1. If fmn,L(X, Y) (m, n=O, 1, 2, * ), then the matrix M = (fVn) is called a summability method from X to Y. Suppose now that (Z, Ts) is also a locally convex separated topological vector space. DEFINITION 2. If M1 is a summability method from X to Y and M2 is a summability method from X to Z with the property that for each sequence {xn4 of points in X for which {Ym} =M({xn}) is T2-convergent, the sequence {Zm } =M2({xn}) is T3-convergent, then we say M2 includes M1. We indicate this by M2DM1. We wish to consider Hausdorff methods H(j.A) = 6bt6, where ,u =diag (,o, Al, ), a is the differencing matrix, and MiEL(X, Y) (for example). We denote by N(Mi) the null space of ,u,. Suppose we have two such Hausdorff methods Hi = Hi(A) = 6ui6, where Ai =diag(uio, Ail, 1i2, * (i 2), lk EL(X, Y) and M2kEL(X, Z).

Inclusion Theorems for Generalized Hausdorff Summability Methods

Inclusion Theorems for Generalized Hausdorff Summability Methods

A Remark on Completely Monotonic Sequences, with an Application to Summability

A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to "&gt;". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)