On the acceleration of convergence by regular matrix methods

Regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. The results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.


INTRODUCTION AND PRELIMINARIES
In recent years the most significant results in convergence acceleration have been proved for nonlinear methods of acceleration (see, for example, [ 1,2 ]).The present paper deals with regular matrix methods that improve and accelerate the convergence of sequences and series.The research has been inspired by the papers [ 3−8 ] where this problem is considered.Some data on the improvement of convergence by regular matrix methods are available also in [ 9 ].Let us describe the main results of the mentioned papers more precisely.For this purpose we need some notions.Let M = (m nk ) be a matrix with real or complex entries.Throughout the paper we assume that indices and summation indices change from 0 to ∞ if not specified otherwise.A sequence x = (x k ) is said to be M-summable if the sequence Mx = (M n x), where We denote the set of all M-summable sequences by c M .Thus, a matrix M determines the summability method on c M , which we also denote by M. A method M is called sequence-to-sequence regular (shortly, Sq-Sq regular) if lim for each convergent sequence x = (x k ) .Classically the following concepts are used to estimate and compare the speeds of convergence of sequences.
Definition 1.1 ([ 1,5 ]).A method M is called accelerating the convergence if the relation holds for every convergent sequence x = (x n ) .If relation (1.1) holds for particular x, then it is said that M accelerates the convergence of this x.If M accelerates the convergence of x, then it is said that Mx converges faster than x.
Definition 1.2 (cf.[ 5 ], p. 310).A matrix method M is said to be accelerating with respect to a matrix method A if Mx converges faster than Ax for every x ∈ c A .If Mx converges faster than Ax for particular x ∈ c A , then M is said to be accelerating with respect to A for this x.
Weakened criteria are used to estimate and compare the speeds of convergence of sequences in [ 3,4 ] and [ 6−9 ].Let λ = (λ k ) be a sequence with 0 < λ k .Definition 1.3 ([ 3,4 ]).A convergent sequence x = (x k ) with is called bounded with the speed λ (shortly, λ -bounded) if l k = O (1) and convergent with the speed λ (shortly, λ -convergent) if there exists the finite limit lim k l k .
We denote the set of all λ -bounded sequences by m λ and the set of all λ -convergent sequences by c λ .For λ k = O (1) we get c λ = m λ = c, where c is the set of all convergent sequences.A sequence respectively.We denote the set of all A λbounded sequences by m λ A and the set of all A λ -summable sequences by 1), then M is called improving the convergence of sequences.

Definition 1.5 ([ 9 ]). We say that M improves A
we say that M improves A-summability.It is not difficult to see that if A is the identity method, i.e.A = I = (δ nk ), where δ nn = 1 and δ nk = 0 for n = k, then Definition 1.5 coincides with Definition 1.4.
Kornfeld [ 5 ] proved that any Sq-Sq regular method M cannot accelerate the convergence and cannot be accelerating with respect to another Sq-Sq regular method A. Kangro ([ 3 ], pp.139-140) proved that an Sq-Sq regular triangular method M = (m nk ) (i.e.m nk = 0 for k > n) cannot improve the λ -convergence.In [ 9 ] it is proved that any Sq-Sq regular triangular method improves neither the convergence nor the λboundedness for an unbounded speed λ .In [ 9 ] it is also shown that any triangular method M improves neither A-summability nor A λ -boundedness for a normal method A = (a nk ) (i.e.A is triangular and ] generalized the concepts of A λ -summability and A λ -boundedness, considering instead of a matrix with real or complex entries a matrix whose elements are bounded linear operators from a Banach space X into a Banach space Y .He proved that a triangular Sq-Sq regular method cannot improve the λ -boundedness ([ 8 ], pp.370-371) and the λ -convergence ([ 7 ], p. 91).
The aim of the present paper is to inquire into the properties of nontriangular regular matrix methods improving and accelerating the convergence.The question is whether there exist such regular methods that improve the convergence, the λ -boundedness or the A λ -boundedness.The results of the papers [ 3−9 ]  show that the answer to this question is always negative for triangular Sq-Sq regular methods.The results giving a positive answer to that question are proved in the present paper for nontriangular regular methods (Propositions 3.1-3.3,Corollary 3.2, Theorem 4.2).Moreover, the convergence acceleration on the subsets of convergent sequences and series is studied.In addition to Sq-Sq regular methods, series-to-sequence regular (shortly, Sr-Sq regular) methods are used, and it is proved that for some cases these methods have better convergence improving and accelerating properties than Sq-Sq regular methods (compare, for example, Theorem 3.2 and Proposition 3.1).A method M is called Sr-Sq regular if It is easy to see that the set of sequences cs is equivalent to the set of convergent series.We note that Sr-Sq regular methods play an important role in the approximation theory (see, for example, [ 10,11 ]).In the present paper Sr-Sq regular methods are used for increasing the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.
The paper is organized as follows.In Section 2 some notions and auxiliary results are presented, which are needed later.In Section 3 the improvement of the convergence and λ -boundedness, and convergence acceleration by nontriangular Sq-Sq and Sr-Sq regular methods are studied.Also some examples of Sr-Sq regular methods improving the convergence of series, and some examples of Sq-Sq regular methods improving λ -boundedness of sequences are presented.Besides, the sufficient conditions for a nontriangular method M to be accelerating for all elements from a certain subset of c or cs are specified.In Section 4 it is shown that using some nontriangular regular method M, it is possible to improve A λ -boundedness for some unbounded speed λ if A is an Sr-Sq regular Zygmund method Z r (r > 1).Also the sufficient conditions for M to be accelerating with respect to Z r for all elements from a certain subset of c Z r are found.In Section 5 the obtained results are used for increasing the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

AUXILIARY RESULTS
Throughout this paper we assume that λ = (λ k ) and µ = (µ k ) are sequences with 0 < λ k , µ k ∞; A is a normal matrix with its inverse matrix A −1 = (η kl ); B = (b nk ) is a triangular matrix, and M = (m nk ) is an arbitrary matrix.We say that M transforms m and t l = A l x hold for each x ∈ c A .Therefore we get the following results by Theorem 20.2 of [ 12 ] (see also [ 4 Then there exist f inite limits lim ) We say that methods A and B are M-consistent on m λ A or on c A if the transformation Mx exists and (2.10)

ACCELERATION OF CONVERGENCE AND IMPROVEMENT OF CONVERGENCE AND λ -BOUNDEDNESS WITH REGULAR MATRIX METHODS
First we consider the relation between the improvement of λ -boundedness and the acceleration of convergence in m λ .
Theorem 3.1.If M improves λ -boundedness of sequences, then M accelerates the convergence of all sequences from the subset m λ of m λ , defined as follows: Proof.As M improves λ -boundedness of sequences, there exists µ Hence relation (1.1) holds for every x ∈ m λ .Thus M accelerates the convergence of all sequences from m λ .
It is well known that a conull method cannot be Sq-Sq regular (see [ 13 ], p. 49).Now we prove the following auxiliary result.
Thus Γ is a conull method.
As Γ = M for A = B = I and any conull method cannot be Sq-Sq regular, with the help of Lemma 3.1 we get the following result.Note that it is possible to prove Theorems 3.2 and 3.3 with the help of Theorem 1 of [ 4 ].Now we inquire into the properties of nontriangular Sr-Sq regular methods improving and accelerating the convergence.For this purpose we first introduce some necessary notions.A series is said to be λ -bounded if the sequence of partial sums of this series is λ -bounded.Let It is easy to see that bs λ ⊆ cs.Definition 3.1.We say that M accelerates the convergence of a series ∑ k x k if the sequence Mx (where x = (x k )) converges faster than the sequence of partial sums (X n ) of this series.If Mx converges faster than (X n ) for every x ∈ cs, then we say that M accelerates the convergence of series.1), we say that M improves the convergence of sequences.
From Definitions 3.1 and 3.2 and Theorem 3.1 we immediately get Corollary 3.1.If M improves λ -boundedness of series, then M accelerates the convergence of all sequences from the subset bs λ of bs λ , defined as follows: The assertion of Theorem 3.2 cannot be extended to Sr-Sq regular methods.Indeed, it is not difficult to see that the Sr-Sq regular method M = (m nk ), where m nk = 1 for all n and k, improves the convergence of series.For getting more complicated examples we first prove Similarly it is sufficient to show for cs ⊆ m Therefore we have γ 0 nl = H 0 nl = m n0 and It immediately follows from these relations that conditions (2.Let us define M = (m nk ) as follows: where s < 0 and s + t > 0. As lim n m nk = 1 and M is an Sr-Sq regular method by Proposition 14 of [ 15 ].Also lim k m nk = 1 = 0. Consequently, the M defined in this way is equivalent to Σ, i.e. c M = cs (see [ 16 ], pp.199-200).
Proposition 3.1.The Sr-Sq regular method M = (m nk ) defined by (3.3), where s < 0 and s +t > 0, improves the convergence of series.Proof.It is sufficient to show by Lemma 3.2 that condition (3.2) holds for some unbounded sequence µ.We define µ = (µ k ) by the equalities As from relation (3.4) we get if and only if β ≤ s + t.Thus condition (3.2) is valid and consequently, cs ⊆ m µ M for β = s + t.Further, we notice that condition (3.1) follows from (3.2) for every unbounded sequence λ .Hence from the proof of Proposition 3.1 we imply the following result with the help of Corollary 3.1 and Lemma 3.2.
Corollary 3.2.The Sr-Sq regular method M = (m nk ) defined by (3.3), where s < 0 and s + t > 0, improves λ -boundedness of series and accelerates the convergence of all sequences from bs λ for λ = (λ k ), defined by equalities Let now M = (m nk ) be defined by the relation where N is a fixed positive integer and c, β , σ > 0. Using Proposition 14 of [ 15 ], it is not difficult to check that the method defined in this way is Sr-Sq regular.
Then M improves λ -boundedness of series and accelerates the convergence of all sequences from bs λ for λ = (λ k ) , defined by the equalities λ k = (k + 1) ck if β ≤ c and σ ≥ 1.
Proof.Let µ = (µ k ) be defined with the help of equalities where As the sequence λ is monotonically increasing and we have and where and γ nl = H nl = m nl , we have γ l = 0 and ρ n = 1.Consequently, conditions (2.6)-(2.8)are satisfied.Further, we can write This completes the proof.
The Sq-Sq regular methods improving λ -boundedness are rather specific.We show that the following result holds.Proof.As γ nl = m nl and γ l = 0 (see the proof of Proposition 3.3), Thus condition (2.9) of Lemma 2.3 is not satisfied and therefore m λ m µ M .

IMPROVEMENT OF A λ -BOUNDEDNESS USING NONTRIANGULAR REGULAR MATRIX METHODS
First we explain the relationship between Definitions 1.2 and 1.5.
Theorem 4.1.If M improves A λ -boundedness, then M is accelerating with respect to A for all sequences from the subset m λ A of m λ A , defined as follows: Proof is similar to the proof of Theorem 3.1.
It is proved in [ 9 ] that any triangular method M cannot improve A λ -boundedness for an unbounded speed λ and a normal method A if M is consistent with A on m λ A .We show that this assertion cannot be extended to nontriangular methods M. Let us prove that a nontriangular Sr-Sq regular method M improving A λ -boundedness exists for some normal Sr-Sq regular method A. For this purpose we consider the case where A is a Riesz method.Let (p n ) be a sequence of nonzero complex numbers, P n = p 0 + ... + p n = 0, P −1 = 0 and let P = (R, p n ) = (a nk ) be a Riesz method generated by (p n ), i.e. (see [ 17 ], p. 113) It is easy to see that P is a normal method.
Lemma 4.1 (see [ 14 ], pp.59-61).Let P be a Riesz method satisfying the properties bs λ ⊆ m λ P , P n = O (P n−1 ) , A matrix M = (m nk ) transforms m λ P into m µ B if and only if the following conditions hold: there exist the f inite limits lim n g nl = g l , ( Now we consider (R, p n ) in the special case where p n is defined by the relation p n = (n + 1) r −n r (r > 1).The Riesz method defined in this way is called the Zygmund method and is denoted by Z r (see [ 17 ], p. 112).Thus Z r = (a nk ) is defined by the relation It is not difficult to verify that Z r is an Sr-Sq regular method.We also have cs m λ Z r for each unbounded sequence λ .Indeed, let Zr = (∆a nk ) for Z r = (a nk ).Then (see [ 17 ], pp.51-52) for every x = (x k ) ∈ c Z r , where X = (X k ) is the sequence of partial sums of series ∑ k x k .Hence Zr is an Sq-Sq regular method, because Z r is Sr-Sq regular.As Zr cannot improve the convergence of sequences by Corollary 3 of [ 9 ], Z r cannot improve the convergence of series by (4.8), i.e., cs m λ Z r for each unbounded speed λ .
Using Lemma 4.1 for B = I, M = Σ, and In addition, M = (m nk ) defined by (3.3), where s < 0 and s + t > 0, is Sr-Sq regular.Therefore we immediately get Proposition 4.1.Let M = (m nk ) be defined by (3.3), where s < 0 and s + t > 0, and λ = (λ k ) by the relation λ k = (k + 1) α , α > 1.Then Z r (r > 1) and M are consistent on m λ Z r .Now we prove the main result of this section.
Theorem 4.2.The Sr-Sq regular method M defined by (3.3), where s < 0 and s + t > 1, improves (Z r ) λboundedness and is accelerating with respect to Z r for all sequences from m λ Z r if λ = (λ k ) is defined by equalities (3.6), where 1 < α < r and α < s + t.Proof.It is sufficient to show by Definition 1.5 and Theorem 4.1 that m λ Z r ⊆ m µ M for some speed µ = (µ k ), satisfying the property µ k /λ k −→ ∞.To show it, we prove that the conditions of Lemma 4.1 are fulfilled for P = Z r , B = (δ nk ), for M = (m nk ) defined by (3.3), and for µ = (µ k ) defined by the relation First we note that conditions (4.1) are satisfied (see [ 14 ], p. 62).Further, we can write where We subsequently get with the help of the mean-value theorem of Cauchy that where 0 < θ l , θ l+1 < 1, and 0 < θ 1 l < 2. As we have because s < α.Therefore L = O n (1), i.e. condition (4.2) is satisfied.We can write with the help of the mean-value theorem of Lagrange that since 0 ≤ m nk < 1 and α > 1.Thus condition (4.3) is satisfied.

SOME REMARKS ON INCREASING THE ORDER OF APPROXIMATION OF FOURIER EXPANSIONS BY REGULAR NONTRIANGULAR MATRIX METHODS
Let X be a Banach space with norm • , and c (X), cs (X), and c A (X) be the spaces of convergent sequences, convergent series, and A-summable sequences, respectively.Moreover, let Remark 5.1.All results of this paper are valid if scalar-valued sequences or sequence sets are replaced by corresponding X-valued sequences or sequence spaces (see [ 14 ], pp.58-59; [ 4 ], p. 139).
Considering Remark 5.1, we can use the results of our paper for increasing the order of approximation of Fourier expansions and Z r -means of Fourier expansions in Banach spaces.We assume that a total sequence of mutually orthogonal continuous projections (T k ) (k = 0, 1, ...) on X exists, i.e., T k is a bounded linear operator of X into itself, T k x = 0 for all k implies x = 0, and T j T k = δ jk T k .Then we may associate formal Fourier expansion to each x from X.It is known (see [ 11 ], pp.74-75, 85-86) that the sequence of projections (T k ) exists if, for example, X = C 2π is the set of all 2π-periodic functions, which are uniformly continuous and bounded on R, X = L p 2π (1 ≤ p < ∞) is the set of all 2π-periodic functions, Lebesgue integrable to the pth power over is the set of all functions, Lebesgue integrable to the pth power over R.
Let M = (m nk ) be defined by (3.3), where s < 0 and s + t > 0. Then we put for every x ∈ X if the series in (5.1) are convergent.Using Remark 5.1, we immediately get the following result from Corollary 3.2.Note that several comparison theorems for the orders of approximation of Fourier expansions, similar to Corollary 5.3, were proved in [ 14,18,19 ].However, in all above-mentioned results the order of approximation of Fourier expansions by M-means was not higher than the corresponding order of approximation by Zygmund means.

Theorem 3 . 2 .
Any Sq-Sq regular method cannot improve the convergence.As c ⊆ c A for an Sq-Sq regular method A, Theorem 3.2 implies Theorem 3.3.Any Sq-Sq regular method M cannot improve A λ -summability of any other Sq-Sq regular method A.

Lemma 3 . 2 .
Let M = (m nk ) be such an Sr-Sq regular method where m n0 = 1.Then bs λ ⊆ m µ M if and only if µ n ∑ l |∆m nl | λ l = O (1) (3.1) and cs ⊆ m µ M if and only if µ n ∑ l |∆m nl | = O (1) .(3.2) Proof.It is sufficient to show for bs λ ⊆ m µ M that condition (3.1) is equivalent to the conditions of Lemmas 2.1 and 2.3 if B = I and A = Σ = (a nk ), where

µM
that condition (3.2) is equivalent to the conditions of Lemmas 2.2 and 2.4 if A = Σ and B = I.For the proof of the above-mentioned equivalences we first note that γ r nl = H r nl , Γ = (γ nl ) = (H nl ) for A = Σ and B = I, where the inverse matrix Σ −1 = (η nk ) of Σ is defined by the equalities it is sufficient to show by Definition 3.2 and Corollary 3.1 that bs λ ⊆ m µ M .For this purpose we prove that condition (3.1) is fulfilled.Let us write 1) , since β ≤ c.Consequently, T = O (1) , i.e condition (3.1) is satisfied.As M is Sr-Sq regular and m n0 = 1, bs λ ⊆ m µ M by Lemma 3.2.Now we show that there exist nontriangular Sq-Sq regular methods, improving λ -boundedness for an unbounded speed λ .Let M = (m nk ) be defined by the relation

> 1 ,Proposition 3 . 3 .
and the symbol x denotes the integer part of number x.According to Theorem 2.3.7 of [ 13 ], the M defined in this way is Sq-Sq regular.Let M = (m nk ) be defined by(3.8),where b > 1.Then M improves λ -boundedness and accelerates the convergence of all sequences from m λ for λ = (λ k ) defined by equalities(3.6).Proof.It is sufficient to prove by Definition 1.4 and Theorem 3.1 that the conditions of Lemma 2.3 are fulfilled for A = B = I and for some µ = (µ k ) satisfying the property µ k /λ k −→ ∞.We define such µ = (µ k ) by the relation µ k = (k + 1) α+σ ; σ > 0 and notice that the transformation y = Mx exists for each x ∈ c.Hence conditions (2.1)-(2.4)are fulfilled.As

Corollary 5 . 1 . 1 ) 2 . 5 . 2 .Corollary 5 . 3 .
Let M n be defined by (5.1) and x 0 ∈ X.If the estimation m < (n + 1) α n ∑ k=0 T k x 0 − x 0 < K holds for some numbers m, K > 0, and for 0 < α < s + t, then(n + 1) s+t M n x 0 − x 0 = O (1) , (5.2)i.e., M-means increase the order of approximation of Fourier expansion of x 0 .Let M be defined by (3.7), where N is a positive integer and c, β , σ > 0. Then we set cn+β +σ T k x (5.3) for every x ∈ X.Using Remark 5.1, we immediately get the following result from Proposition 3.Corollary Let M n be defined by (5.3) and x 0 ∈ X.If the estimationm < (n + 1) cn n ∑ k=0 T k x 0 − x 0 < Kholds for c > 0 and for some numbers m, K > 0, then(n + 1) cn+β M n x 0 − x 0 = O (1)for 0 < β ≤ c and σ > 1, i.e., M-means increase the order of approximation of Fourier expansion of x 0 .method Z r and for every x ∈ X.Using Remark 5.1, Proposition 4.1, and Theorem 4.2, we immediately get Let M n and Z r n be defined by (5.1) and (5.4), respectively, and x 0 ∈ X.If the estimation m < (n + 1) α Z r n x 0 − x 0 < K holds for α ∈ (1, r) and for some numbers m, K > 0, then estimation (5.2) for s +t > α and s < 0 also holds, i.e., M-means increase the order of approximation of Z r -means of x 0 .
consistency of A and I on m λ A or on c A coincides with the usual consistency of A and M respectively on m λ A or on c A .By Lemmas 2 and 4 of [ 9 ] and Lemma 2.2 we immediately get the following results.