scholarly journals Some Extensions of the Hausdorff-Young and Paley Theorems

1961 ◽  
Vol 4 (2) ◽  
pp. 123-138
Author(s):  
P. S. Bullen
Keyword(s):  
Special Cases ◽  
Image Position

Orthonormal sequences, o.n. s., {ϕn} defined on [0,1] and satisfying1have been studied in [3] and [1]. One of the objects of this paper is to indicate that the methods used to study such o. n. s. can be used for a much wider class, and that, although there seems to be no super theorem to cover all cases, a knowledge of the results and methods of proof in some fairly broad special cases enables one to state and prove theorems for other classes of o. n. s.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

Connection formulae for spectral functions associated with singular Dirac equations

2004 ◽  
Vol 134 (1) ◽  
pp. 215-223 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Dirac Equation ◽  
Spectral Function ◽  
Limit Point ◽  
Initial Condition ◽  
Spectral Functions ◽  
Dirac Equations ◽  
Special Cases ◽  
Limit Point Case ◽  
Image Position ◽  
Connection Formulae

We consider the Dirac equation given by with initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

Some Properties of Generalized Euler Numbers

1981 ◽  
Vol 33 (3) ◽  
pp. 606-617 ◽  
Author(s):  
D. J. Leeming ◽  
R. A. Macleod
Keyword(s):  
Positive Integer ◽  
Roots Of Unity ◽  
Euler Numbers ◽  
Special Cases ◽  
Image Position

We define infinitely many sequences of integers one sequence for each positive integer k ≦ 2 by(1.1)where are the k-th roots of unity and (E(k))n is replaced by En(k) after multiplying out. An immediate consequence of (1.1) is(1.2)Therefore, we are interested in numbers of the form Esk(k) (s = 0, 1, 2, …; k = 2, 3, …).Some special cases have been considered in the literature. For k = 2, we obtain the Euler numbers (see e.g. [8]). The case k = 3 is considered briefly by D. H. Lehmer [7], and the case k = 4 by Leeming [6] and Carlitz ([1]and [2]).

A Note on Divergent Series

1952 ◽  
Vol 4 ◽  
pp. 445-454 ◽  
Author(s):  
G. M. Petersen
Keyword(s):  
Divergent Series ◽  
Matrix Methods ◽  
Special Cases ◽  
Image Position

In this note we shall discuss certain matrix methods of summation, though otherwise, §1 and §2 are not connected.In this section, we shall study some properties of the method (Bh) where we say the series ∑uv is summable (Bh) whenThe method (Bh) has been studied in special cases airsing from different values of h by Rogosinski [11; 12], Bernstein [2], and more recently by Karamata [3; 4].

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups

2018 ◽  
Vol 61 (3) ◽  
pp. 673-703 ◽  
Author(s):  
Benjamin Klopsch ◽  
Anitha Thillaisundaram
Keyword(s):  
Rooted Tree ◽  
Maximal Subgroups ◽  
Irreducible Representations ◽  
Prime Field ◽  
Infinite Dimensional ◽  
Special Cases ◽  
Profinite Completions ◽  
Spinal Group ◽  
Groups Acting On Trees ◽  
Image Position

AbstractLet p ≥ 3 be a prime. A generalized multi-edge spinal group $$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$ is a subgroup of the automorphism group of a regular p-adic rooted tree T that is generated by one rooted automorphism a and p families $b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$ of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalizes the concepts of multi-edge spinal groups, including the widely studied GGS groups (named after Grigorchuk, Gupta and Sidki), and extended Gupta–Sidki groups that were introduced by Pervova [‘Profinite completions of some groups acting on trees, J. Algebra310 (2007), 858–879’]. Extending techniques that were developed in these more special cases, we prove: generalized multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore, we use tree enveloping algebras, which were introduced by Sidki [‘A primitive ring associated to a Burnside 3-group, J. London Math. Soc.55 (1997), 55–64’] and Bartholdi [‘Branch rings, thinned rings, tree enveloping rings, Israel J. Math.154 (2006), 93–139’], to show that certain generalized multi-edge spinal groups admit faithful infinite-dimensional irreducible representations over the prime field ℤ/pℤ.

scholarly journals A Pair of Dual Integral Equations Occurring in Diffraction Theory

1962 ◽  
Vol 13 (2) ◽  
pp. 179-187 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Integral Equations ◽  
Diffraction Theory ◽  
Dual Integral Equations ◽  
Special Cases ◽  
Image Position

Dual integral equations of the formwhere f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

General series involving H-functions

1969 ◽  
Vol 65 (2) ◽  
pp. 461-465
Author(s):  
R. N. Jain
Keyword(s):  
Hypergeometric Function ◽  
Analytic Functions ◽  
General Nature ◽  
Complex Numbers ◽  
Important Class ◽  
Vast Number ◽  
Finite Series ◽  
General Series ◽  
Special Cases ◽  
Image Position

MacRobert (4–7) and Ragab(8) have summed many infinite and finite series of E-functions by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration. Verma (9) has given two general expansions involving E-functions from which, in addition to some new results, all the expansions given by MacRobert and Ragab can be deduced. Proceeding similarly, we have studied general summations involving H-functions. The H-function is the most generalized form of the hypergeometric function. It contains a vast number of well-known analytic functions as special cases and also an important class of symmetrical Fourier kernels of a very general nature. The H-function is defined as (2)where 0 ≤ n ≤ p, 1 ≤ m ≤ q, αj, βj are positive numbers and aj, bj may be complex numbers.

scholarly journals Application of the Ritz method to non-standard eigenvalue problems

1969 ◽  
Vol 10 (3-4) ◽  
pp. 367-384 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Positive Definite ◽  
Linear Operators ◽  
Extensive Literature ◽  
Maximum Eigenvalue ◽  
Special Cases ◽  
Image Position ◽  
Linear Manner

There is an extensive literature on application of the Ritz method to eigenvalue problems of the type where L1, L2 are positive definite linear operators in a Hilbert space (see for example [1]). The classical theory concerns the case in which there exists a minimum (or maximum) eigenvalue, and subsequent eigenvalues can be located by a well-known mini-max principle [2; p. 405]. This paper considers the possibility of application of the Ritz method to eigenvalue problems of the type (1) where the linear operators L1L2 are not necessarily positive definite and a minimum (or maximum) eigenvalue may not exist. The special cases considered may be written with the eigenvalue occurring in a non-linear manner.

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