Mercerian Theorems Research Articles

Mercerian theorem for four dimensional matrices

Mercerian theorem for four dimensional matrices

On the Fine Spectrum of the Operator Defined by the Lambda Matrix over the Spaces of Null and Convergent Sequences

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces andc. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator on the sequence spaces andc. Finally, we present a Mercerian theorem. Since the matrix is reduced to a regular matrix depending on the choice of the sequence having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.

Mercerian theorems involving Cesàro and Hölder means of positive order

For a sequence–to–sequence transformation defined by a matrix

On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c

We determine the fine spectrum of the generalized difference operator B(r, s) defined by a band matrix over the sequence spaces c0 and c, and derive a Mercerian theorem. This generalizes our earlier work (2004) for the difference operator Δ, and includes as other special cases the right shift and the Zweier matrices.

On the fine spectrum of the difference operator Δ on c0 and c

In the present paper, the fine spectrum of the difference operator on the sequence spaces c 0 and c have been examined and a Mercerian theorem has also been given.

On the fine spectrum of the Cesàro operator in c0

In the present paper, the fine spectrum of the Cesaro operator in the sequence space c0 has been examined. Although the discussion is made for determination of the spectrum of the Cesaro operator in the sequence space c0 by Reade (14) and the others, our consequences are more refinement and include a remark concerning with the previous works. Further, a Mercerian theorem has also been given. Finally, the fine spectrum of the Cesaro operator in the sequence space c has been given, without proof.

Tauberian and Mercerian Theorems for Systems of Kernels

In [BI3], we introduced the notion of ratio Mercerian theorem, to improve the Mercerian theorem for Fourier and Hankel transforms first proved in [BI1]. In the first half of this paper, we extend (and correct) the ratio Mercerian theorem, and apply it to the proofs of Tauberian theorems for kernels of Korenblum type. The latter can be applied to analytic number theory; this was in fact the motivation for the present paper. The application to analytic number theory will be given in a separate paper [BI5]. In the second half of this paper, we prove a Mercerian counterpart to one of the Tauberian theorems – one in the boundary case. This is done via a further extension of our previous extension of the Drasin–Shea–Jordan theorem (see e.g. [BGT, Chap. 5]). Throughout the paper, the idea of proving assertions for a system of kernels, rather than a single kernel, plays a key role. For measurable functions f, g : (0,∞) → R, the Mellin convolution f ∗ g is the function defined by

Abelian, Tauberian, and Mercerian Theorems for Arithmetic Sums

We consider Abelian results, passing from behaviour of f to that of single or double sums as above, Tauberian theorems converse implications under additional conditions (Tauberian conditions), and Mercerian theorems, in which we pass from some comparison statement between f and a sum to a conclusion on f alone. This line of work may be traced to a seminal paper of Polya in 1917 ([P]; see also Polya & Szego [PS, Part II, Ch. 4, No. 156]). Our two main tools are the Karamata theory of regular variation (originated in 1930: see [BGT]) and the Wiener Tauberian theory (originated in 1932: see Hardy [H], Widder [W]). Polya’s achievement is all the more striking in that neither of these tools was available to him. This study arises from a fusion of two recent lines of work. On the arithmetic sums side, our interest was stimulated by a series of studies by De Koninck & Ivic [DeKI1,2,3]. On the Abelian-Tauberian-Mercerian side, we make use of recent work of our own [BI5],

Ratio Mercerian Theorems with Applications to Hankel and Fourier Transforms

We complete and complement our recent work on Drasin-Shea-Jordan theorems for Fourier and Hankel transforms. In improving on the methods of our previous work, we were led to certain ratio Mercerian theorems for general kernels; these yield definitive versions of our earlier results for Fourier and Hankel transforms. 1991 Mathematics Subject Classification: primary 44A15; secondary 47B38.

A note on Mathieu's inequality

The function defined by the real infinite series $$S(t): = \sum\limits_{n = 1}^\infty {2n(n^2 + t^2 )^{ - 2} } $$ has been the subject of a number of estimates, starting with an 1890 conjecture by Mathieu (in connection with work on elasticity of solid bodies) thatS(t) < t −2. More recently it appeared in work on Mercerian theorems for Cesaro summability. A combination of the best of the various published inequalities so far (for larget) is $$\frac{1}{{t^2 }} - \frac{5}{{16t^4 }}< S(t) = \frac{1}{{t^2 }} - \frac{1}{{6t^2 }} + O\left( {\frac{1}{{t^6 }}} \right).$$ However, one has only to observe that $$g(z): = \sum\limits_{n = 0}^\infty {\frac{1}{{(n + z)^2 }} = \left( {\frac{d}{{dz}}} \right)^2 } \log \Gamma (z) z \ne 0, - 1, - 2,...)$$ in order to see that the original series satisfies $$ - tS(t) = \operatorname{Im} g(it)$$ and one therefore has access to all the theory of the Gamma-function. While there are some difficulties due to the fact that Re(it) = 0, so that the remainder estimates in the standard asymptotic expansion require modification, nonetheless we can prove that $$S(t) = \frac{1}{{t^2 }} - \frac{{B_1 }}{{t^4 }} - ... - \frac{{B_r }}{{t^{2r + 2} }} + \frac{{( - 1)^r }}{{t^{2r + 2} }}\int_0^\infty {f^{(2r + 1)} (x)\cos tx dx} $$ whereB 1,B 2,⋯ are Bernoulli numbers andf(x): = x/(e x − 1). Sincef (2r+1) (x) = O(xe −x) asx → + ∞, this certainly shows that the remainder term isO(t −2r-2). However, more delicate analysis allows us to place bounds on the remainder term.

Characterization of a class of infinite matrices with applications

In this paper, K denotes a complete, non-trivially valued, non-archimedean field. The class (lα, lα) of infininite matrices transforming sequences over K in lα to sequences in lα is characterized. Further a Mercerian theorem is proved in the context of the Banach algebra (lα, lα), α ≥ 1 and finally a Steinhaus type result is proved for the space lα. In the case of ℝ or ℀, on the other hand, the best known result so far seems to be a characterization of positive matrix transformations of the class (lα, lβ), ∞ &gt; α ≥ β &gt; 1.

Mercerian theorems involving Ces�ro means of positive order

IfCα denotes the Cesaro matrix of arbitrary positive order α, we investigate the setDα of complex numbers μ for which the sequence-to-sequence transformation defined by the matrixI−μCα is equivalent to convergence. In particular, we show thatDα is strictly decreasing as α increases. Most previous investigations have taken μ real and/or α an integer; we make no such restrictions.

Mercerian theorems via spectral theory

Given a regular matrix A, Mercerian theorems are concerned with determining the real or complex values of a for which al + (1 — a)A is equivalent to convergence. For aΦl, the problem is equivalent to determining the resolvent set for A, or, determining the spectrum σ(A) of A, where σ(A) = {λ IA — λl is not invertible}. This paper treats the problem of determining the spectra of weighted mean methods; i.e., triangular matrices A = (ank) with ank pk/Pn, where p0 > 0, Pn 0, Σ&=o Pk ~ Pn It is shown that the spectrum of every weighted mean method is contained in the disc {λ\\λ 1/2} (Theorem 1), and, if lim pJPn exists,

A Mercerian theorem for N�rlund means

A Mercerian theorem for N�rlund means

Spectra of conservative matrices

In this paper we study the spectra of conservative matrices and show that the spectrum of any Hausdorff method is either uncountable or finite. In the latter case it is shown that the spectrum consists of either one point or two points. We obtain the snarpest possible Mercerian theorem for Euler methods. We also get some results about the location of conservative matrices with respect to the maximal group of invertible operators.

On Some Mercerian Theorems in Summability

On Some Mercerian Theorems in Summability

On some Mercerian theorems in summability

On some Mercerian theorems in summability

Equivalence of methods for evaluation of sequences

For expositions of the subject, see Hardy [4] and Cooke [3]; our terminology agrees with that of Hardy. Unfortunately, there are no such simple necessary and sufficient conditions that (1.1) be included by convergence, that is, be such that lim Sn = lim an whenever lim an exists. Some Mercerian theorems, which are treated briefly by Hardy [4], solve the problem for special classes of transformations involving a parameter in a simple way, but the general problem never has been and perhaps cannot be successfully attacked. It is the object of this note to prove the following theorem which gives simple sufficient conditions that (1.1) be included by convergence, and to show that the conclusion will fail to follow if the condition (1.31) or any one of the three conditions for regularity is removed from the hypothesis.

An extension of Mercer’s theorem

Some of the known generalisations [4, 7, 8, 9] replace oa by other kinds of summation means. We propose to show that Oa may be replaced by the means of any regular Toeplitz method of summation; and even by those of any convergence-preserving method (if the limits (1+q)s and s are left unspecified). Some reduction in the domain of values of q is not unexpected. No other restriction is required, however, than that Sn should be bounded; and this is, in the generality of Toeplitz methods, necessary for the existence of the means. Mercer's theorem takes the form summability implies convergence (without Tauberian condition) when expressed in terms of the summation means rn= (sn+qun)/(A +q); this is the character of the very general Mercerian theorems of Pitt [8]. They refer to certain integral means, and to Hausdorff means not usually restricted to have the form of rn; so our theorem may not be closely related to them. Agnew [6] gave three theorems of the same type, referring to Toeplitz means. We show that two of these together compose exactly a special case of our Corollary 1 on positive regular summation matrices, apart from our hypothesis of boundedness which Agnew shows to be unnecessary in his more special context. His other theorem of this kind is slightly more general, but only in that it permits the matrix to differ negligibly from one which is positive; it is included in our Corollary 2. Our restrictions on q may perhaps be heavier than necessary, at any rate in the contexts of particular methods of summation. For instance, Hardy [2] showed that Mercer's original theorem with arithmetic means a-n holds throughout the half-plane '(q > -1 (and nowhere