B. Malcolm Brown

Cardiff University | CU · School of Computer Science and Informatics

In 1971 W.N. Everitt published his seminal work on the family of HELP inequalities. This paper, written in the twenty-fifth year of the HELP paper, is respectfully dedicated to him.

This paper is concerned with the numerical computation of the Titchmarsh--Weyl M matrix. We show how an algorithm may be developed which relates solutions of an initial-value problem to an approximation of the M matrix. In some special cases we compare the results from our algorithm with results obtained using special functions.

We find the adjoint of the Askey–Wilson divided difference operator with respect to the inner product on L2(–1, 1, (1– x2)½dx) defined as a Cauchy principal value and show that the Askey-Wilson polynomials are solutions of a q-Sturm–Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward w...

We find the adjoint of the Askey-Wilson divided difference operator with respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy principle value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforwa...

The inequalities we investigate are associated with M[f]=-f '' +1 2x 2 f on (0,1] and [1,∞). On [1,∞) Mf=λf is in the strong-limit-point case and the inequality is of the type attributed to Everitt (HELP). The equation Mf=λf is limit-circle and non-oscillatory (for λ∈ℝ) on (0,1] and the inequality is of the type studied by Evans and Everitt. Estima...

The repeated diagonalization techniques used by Eastham to obtain asymptotic results for solutions to linear differential systems are further developed to produce a numerical algorithm for estimating the Titchmarsh-Weyl m-coefficient in the second-order case. These analytic results have been exploited to produce a computer code (RDML1) to generate...

Recent work on the Hardy Everitt Littlewood and Pölya (HELP) inequality using numerical techniques is presented and analysed further. New techniques are used to integrate the highly oscillatory solutions that restricted the range of problems covered in earlier publications.

This is a survey of recent results on a class of series inequalities involving second-order difference operators, which includes a well-known inequality of Copson's. A connection has been established between these inequalities and the properties of the Hellinger-Nevanlinna m-function for an associated recurrence relation Mxn = λwnxn, : the validity...

We establish an integral representation of a right inverse of the Askey-Wilson finite difference operator on $L^2$ with weight $(1-x^2)^{-1/2}$. The kernel of this integral operator is $\vartheta'_4/\vartheta_4$ and is the Riemann mapping function that maps the open unit disc conformally onto the interior of an ellipse.

In this paper we discuss an integro-differential inequality formed from the square of a second-order differential expression. A connection between the existence of the inequality and the Titchmarsh-Weyl m-function is established and it is shown that the best constant in the inequality is determined by the behaviour of the m-function. Analytic resul...

Recent efforts have been focused on using numerical methods to estimate the Titchmarsh–Weyl m-coefficient. In this paper we look at interval analytic methods to provide provable bounds for these values.

In a recent paper the first two authors studied a class of series inequalities associated with a three-term recurrence relation which includes a well-known inequality of Copson's. It was shown that the validity of the inequality and the value of the best constant are determined in terms of the so-called Hellinger-Nevanlinna m-function. The theory i...

Synopsis In 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δa n = a n+1 – a n and Δ ² a n = Δ(Δa n ), then is convergent and the constant 4 being best possible. Equality occurs if and only if a n = 0 for all n. In this paper we give a result that extends...

A survey of recent work on the Hardy Everitt Littlewood and Pólya (HELP) inequality and analogous series inequalities investigated by Brown and Evans is presented. Included is a description of the numerical techniques devised by Brown, Kirby and Pryce to determine the best constants in the inequalities.

Let Δ0 denote the linear manifold of the space L 2 [0,π] defined by

A numerical method for determining the Titchmarsh-Weyl m(λ ) function for the singular eigenvalue equation -(py')' + qy = λ wy on [a,∞ ), where a is finite, is presented. The algorithm, based on Weyl's theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case...

This paper is concerned with numerical methods for finding m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation -y '' + qy = λ y on [0, ∞) and the results are applied to the problem of finding best constants for Everitt's extension to the Hardy-Little-wood-Polya (HELP) integral inequality.

David Edmunds has influenced and made major contributions to numerous branches of mathematics. These include spectral theory, functional analysis, approximation theory, the theory of function spaces, operator theory, ordinary and partial differential equations. The breadth of his impact is demonstrated by his publication record, which consists of 5...

Desmond Evans has made major influential contributions to numerous branches of mathematical analysis which include the spectral theory of both ordinary and partial differential equations, mathematical physics and functional analysis. The breadth and depth of his achievements are recorded in his 142 published papers and 3 books. Following a B.Sc (Wa...