Ordinary Convergence Research Articles

On Some Trigonometric and Exponential Lattice Sums

Certain two-dimensional trigonometric lattice series, the ordinary convergence of which involves unresolved questions of a deep and delicate number-theoretic nature, are shown to be summable by a special Abelian method and their Abelian sums are obtained. This is done by first evaluating an absolutely convergent exponential lattice series and then analytically extending its sum.

On Some Properties of Multiple Conjugate Trigonometric Series

Abstract We have obtained the estimate in the terms of partial and mixed moduli of continuity of deviation of Cesáro (C, ) means of the sequence of rectangular partial sums of n-multiple (n > 1) conjugate trigonometric series from n-multiple truncated conjugate function. This estimate implies the result on the -convergence (⋋ ≥ 1) of (C, ) means , provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown, that the -convergence cannot be replaced by ordinary convergence.

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesaro (C, α) means (α = (α1,...,αn),αi∈ ℝ, α1 > −1, $$i = \overline {1,n} $$ ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them λ-convergence (λ≥1) of (C, α) means (α1 > 0, $$i = \overline {1,n} $$ ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them λ-convergence cannot be replaced by ordinary convergence.

On Banach spaces of strongly convergent trigonometric series

A study of strongly convergent trigonometric and Fourier series was recently introduced by the present author. The obtained results indicate similarities to both absolutely and ordinarily convergent trigonometric series. This is natural in view of the fact that strong convergence lies between absolute and ordinary convergence. Concerning Fourier series, the properties of strong convergence of index λ ⩾ 1, denoted [ I] λ , justify the present investigation of the following classes of functions: S λ , the class of all f ϵ L 1 whose Fourier series is a.e. [ I] λ convergent to f and S λ, the class of all f ϵ C whose Fourier series is uniformly [ I] λ convergent to f. Using earlier results we give other equivalent descriptions of these classes, prove that for all λ ⩾ 1 both S λ and S λ are Banach spaces, and examine other natural questions about these spaces. In particular, we establish the convergence of the Fourier series in the respective norms ∥ ∥ S λ and ∥ ∥ S λ, show that there are functions f, g ϵ S λ such that f · g ∉ S 1 so that S λ are not Banach algebras under pointwise multiplication and observe that they are complex Banach algebras under the convolution.

Tauberian theorems for J p-summability

We prove Tauberian theorems for J p -summability methods of power series type with respect to ordinary convergence. Under a o-Tauberian condition on the underlying sequence our Tauberian conclusion is valid for all J p -methods. Relaxing the Tauberian assumption to a O-type condition, a related result is shown for methods with “moderate” weight sequences.

Asymptotic Integration of a Perturbed Constant Coefficient Differential Equation under Mild Integral Smallness Conditions

The problem of asymptotic behavior of solutions of an nth order linear differential equation is reconsidered, and a result obtained by Hartman and Wintner under integral smallness conditions on the perturbing terms which require absolute integrability is shown to hold under weaker integrability conditions requiring only ordinary (perhaps conditional) convergence of some of the improper integrals that occur. The estimates of the asymptotic behavior of solutions of the perturbed equation are also sharper than in the classical result.

Matrix Summability of Geometrically Dominated Series

One of the common uses of summability theory is found in its applications to power series. A partial listing such as [l]-[5], [13], and [15]-[17] might serve to remind us of the many instances of summability theory applied to power series. In some studies, the summability transformation is applied to the sequence of partial sums of the power series, while in others it is applied to the general term akzk as a series-to-series transformation. In [2] and [5] the transformation is applied to the coefficient sequence of the Taylor series. In the present study we investigate matrix transformations that are applied to the sequence {akzk), but we are not concerned with the usual preservation of convergence or sums. At a point within its disc of convergence, a power series exhibits more than ordinary convergence; it converges very rapidly, being dominated by a convergent geometric series.

General convergence of continued fractions

We introduce a new concept of convergence of continued fractions—general convergence. Moreover, we compare it to the ordinary convergence concept and to strong convergence. Finally, we prove some properties of general convergence.

The Condition of Ordinary Integral Convergence in the Asymptotic Theory of Linear Differential Equations With Almost Constant Coefficients

In this paper, we continue the work started in [2], wherethe asymptotic integration of nth order linear differential equations was considered under smallness conditions, expressed in terms of ordinary integral convergence. The method of integration in [2] was based on using the Banach contraction principle for a linear nonhomogeneous operator in a Banach space. In this paper, we use the same principle for a nonlinear operator which corresponds to the equation for the logarithmic derivative of the original dependent variable. The result proved concerns a class of equations considered in [2] under essential additional restrictions.

Asymptotic Integration of Linear Differential Equations Subject to Mild Integral Conditions

Sufficient conditions are given for a linear differential equation of order n to have a solution which behaves asymptotically like a given polynomial of degree $ < n$. The integral smallness conditions on the coefficient and forcing functions are stated largely in terms of ordinary (rather than absolute) convergence, and the manner in which the solution behaves like the given polynomial is specified precisely.

Asymptotic Integration of Perturbed Linear Differential Equations under Conditions Involving Ordinary Integral Convergence

In this paper, we consider asymptotic integration of nth order linear differential equations with constant coefficients, modified by the addition of small functions. The integral smallness of the perturbation functions is expressed in terms of ordinary convergence instead of the classic conditions which require absolute integrability. The proof of our result is based on the Banach contraction principle.

Convergence of Solutions to the Classic Dual Cosine Equation

Dual orthogonal series are a primary tool in the analysis of mixed boundary value problems for the potential in geometries conforming to separability conditions for the Laplace operator. Different formal solutions to the classic dual cosine equation, in the form of multiple singular integrals, have been developed by many authors, especially in connection with the study of brittle cracks. Recently an algorithmic representation for these solutions was developed to facilitate the writing of computer software. On the basis of this representation a rigorous proof of convergence for input functions of bounded variation is established. The proof utilizes only classical results in the theory of orthogonal expansions. In the first equation of the dual pair there is ordinary convergence, and in the second there is Abel-Poisson convergence which is not, in general, reducible to ordinary convergence.

Asymptotic Integration of Linear Differential Equations Subject to Integral Smallness Conditions Involving Ordinary Convergence

The problem of asymptotic behavior of solutions of an nth order linear differential equation is reconsidered, and a result obtained by Hartman under integral smallness conditions requiring absolute integrability is shown to hold with most of the conditions stated in terms of ordinary integrability. Results of Fubini and Halanay for linear perturbations of nonoscillatory second order equations are similarly extended.

Preservation of rates of convergence under mappings

For appropriate metrics characterizing various modes of stochastic convergence, it is shown that rates of convergence are preserved by a large class of functions. For example, the extensions of a Lipschitz function on a separable metric space S to the space of all probability measures on S with the Prohorov metric and to the space of all S-valued random variables with the usual metric associated with convergence in probability inherit the Lipschitz property. Consequently, just as with the continuous mapping theorem associated with ordinary convergence, new rate of convergence theorems can sometimes be obtained from old ones by applying appropriate mappings.

Associated convergence and accommodative insufficiency.

Nine patients were described with a syndrome consisting of accommodative and convergence insufficiency. All but two had received extensive orthoptic therapy without benefit in the past. Near points of convergence and accommodation, fusional amplitudes and the AC/A ratio were determined and the data compared with those from a group with ordinary convergence insufficiency. The results show that this syndrome is a distinct and separate entity from ordinary convergence insufficiency. Special therapeutic measures include base in prisms and bifocal correction.

Lebesgue Constants for Regular Taylor Summabllity

The nth Taylor mean of order r of a sequence {sn} is given by1.1where1.2Cowling [l] has shown that this method is regular if and only if 0 ≦ r &lt; 1. Since r = 0 corresponds to ordinary convergence, it will be assumed here that 0 &lt; r &lt; 1.

Central Limit Theorems for Families of Sequences of Random Variables

Let $F$ be a (nonempty) set of distribution functions (d.f.'s) of random variables (r.v.'s) with zero means and positive, finite variances. Let $\mathfrak{F}(F)$ be the set of all sequences of independent r.v.'s (independent within each sequence) whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. No assumptions are made on the interrelations between the joint probability spaces of the r.v.'s of different sequences of $\mathfrak{F}(F)$. Accordingly, dependence or independence between r.v.'s of different sequences needs not be specified. A generic member of $\mathfrak{F}(F)$ will be denoted by $\epsilon = \{\epsilon_k; k = 1, 2, \cdots\}$, or, when we discuss sequences of members of $\mathfrak{F}(F)$, by $\epsilon(n) = \{\epsilon_{nk}; k = 1, 2, \cdots\}, n = 1, 2, \cdots$. In the following, $\mathfrak{F}(F)$ plays the role of a parameter space, the parameter points being $\epsilon, \epsilon(1), \epsilon(2), \cdots$. Since only the d.f.'s of a sequence of (independent) r.v.'s are relevant to the central limit theorem, instead of $\mathfrak{F}(F)$, the set of sequences of d.f.'s corresponding to the elements of $\mathfrak{F}(F)$ may also be regarded as the parameter space. It clearly is a map of $\mathfrak{F}(F)$. The inverse mapping subdivides $\mathfrak{F}(F)$ into certain equivalence classes. From elementary set theory it follows that $\mathfrak{F}(F)$ as well as its map has the cardinality of the continuum if $F$ contains more than one element. Further, let $\{a_{nk}; n = 1, 2, \cdots; k = 1, 2, \cdots\}$ be a double sequence of real constants, and let $\{k_n\}$ be a sequence of positive integers such that $a_{nk_n} \neq 0$ and $a_{nk} = 0$ for $k > k_n, n = 1, 2, \cdots$. We denote the variance of $\sum^{k_n}_{k=1} a_{nk}\epsilon_{nk}$ by \begin{equation*}\tag{1}B_n^2 = \sum^{k_n}_{k=1} a^2_{nk}\sigma^2_{nk}, \quad\sigma^2_{nk} = \operatorname{var} \epsilon_{nk},\end{equation*} and put \begin{equation*}\tag{2}\zeta_n = B_n^{-1} \sum^{k_n}_{k=1} a_{nk}\epsilon_{nk},\quad n = 1, 2, \cdots,\end{equation*} where $B_n = + (B_n^2)^{\frac{1}{2}}$. If all sequences $\epsilon(n)$ are identical we denote them by $\epsilon$ and write instead of (2) $\zeta_n(\epsilon) = B_n^{-1}(\epsilon) \sum^{k_n}_{k=1} a_{nk}\epsilon_k, \quad n = 1, 2, \cdots,$ which shows the dependence on the parameter $\epsilon$ more clearly; $B_n^2(\epsilon)$ is defined analogously to (1). This note deals with necessary and sufficient conditions on the set $F$ and on the double sequence $\{a_{nk}\}$ in order that the d.f.'s of the $\zeta_n$ tend to the standard normal d.f. (denoted by $\Phi(x)$) and that the $B^{-1}_na_{nk}\epsilon_{nk}$ are infinitesimal for every sequence of sequences $\epsilon(1), \epsilon(2), \cdots \varepsilon \mathfrak{F}(F)$ (Theorem 1). For the case of identical sequences $\epsilon(n) (= \epsilon)$, which is of particular interest in applications, a normal convergence theorem (Theorem 3) holding uniformly for $\epsilon$ on $\mathfrak{F}(F)$ is obtained under the same necessary and sufficient conditions as in Theorem 1. Theorem 3 yields, e.g., conditions for the asymptotic normality of the least squares estimators of the parameters in a linear regression with independent and not necessarily identically distributed errors whose d.f.'s are unknown but belong to a certain class $F$ (Eicker (1963)). From the necessity of the conditions it follows that these conditions are the best possible ones under the limited information about the error terms provided by the model assumptions. Frequently, when limit theorems for families of sequences of random variables are met in statistics and probability theory, the emphasis is on the uniformity of the convergence of the sequences with respect to the family parameter which assumes values in a given a priori set. (For an example, compare Parzen (1954), p. 38. That paper also cites some of the earlier publications on the subject.) The present note emphasizes not primarily the uniformity of the convergence, but the necessity of the conditions (including one on the parameter space $\mathfrak{F}(F))$ for the convergence on the parameter space. Accordingly, the set $F$ cannot be prescribed arbitrarily. The uniformity of the convergence on $\mathfrak{F}(F)$ is shown without difficulty to be implied by the ordinary convergence on $\mathfrak{F}(F)$. Another difference from earlier work may be seen in the particular structure of $\mathfrak{F}(F)$, which is comparable to an infinite-dimensional vector space. Although many of the earlier convergence theorems are (or can be) formulated for abstract parameter spaces, in the applications such as estimation theory these spaces are usually specialized to intervals on the line or in a finite-dimensional vector space.

Reciprocal Convergence Classes for Fourier Series and Integrals

The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L2(0, ∞) states that (see (7) for references)converges in mean square to a function g(x) which also belongs to L2(0, ∞), and furthermoreSome years ago in a series of papers (1; 2; 3) on summation formulae I showed that a similar symmetrical theory for narrower classes of functions and ordinary convergence of the integrals can also be developed.

On the domain of absolute convergence of Dirichlet series in several variables

with complex Am and Umk. In ?1 we show that the domain of absolute convergence of this series is convex. Conversely, given a convex domain in E2n=E2n(xl, Yl, , * Xn, yn) and a preassigned sequence of exponents {mk } we construct a Dirichlet series of the given type with the given domain as its domain of absolute convergence, provided the sequence { 0mk } satisfies a certain set of conditions. In ?2 we construct a maximal domain of ordinary convergence. For one complex variable this reduces to a simplified construction of the maximal domain of convergence as given by Hille. We also find a relationship between the domain of absolute convergence and the maximal domain of convergence. For one complex variable this relationship reduces to a refinement of a result by Gallie [2] and for one variable with positive increasing exponents it reduces to the classical theorem regarding the relationship between the abscissas of absolute and ordinary convergence.

A note on summability methods and spectral analysis

We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods. If fEL', the Fourier transform is 7(t) --fexp (itx)f(x)dx and Z(f) denotes the set of zeros of 7. Given 4 ELI* we may form the convolution fo4 (x) ff(y)o (x-y)dy. The spectrum of 4 is defined by A())= nZ(f) where the intersection is taken over all fELL such that fo4) O. The underlying heuristic principle of this investigation is that t*A(0) if and only if the trigonometric integral fexp (-isx)0(x)dx is summable, in some suitable sense, to 0 in a neighborhood of t. Since we do not assume that fx+l I 4(y) I dy = o(l) as I x| oo , ordinary convergence is not suitable. Beurling(') has treated Abel summability and Pollard, [4], the (R, 2) method. However their results may be extended to a large class of summability methods which are quite easy to describe. (Some of our theorems are new even for the Abel and (R, 2) cases.) DEFINITION. A function kGL' is a spectral kernel if k(O) = 1 and k(x) =f-Zjk'(y)dy where k' L'. (We shall consider k' extended to negative arguments as an odd function.) Set kh(t) =fexp (-itx)k(hx)q5(x)dx. Taking the limit as h-*O gives a regular summability method.