Volume 33, Issue 1 -

Recently it has been shown, that if a weight has the doubling property on its support [− 1,1], then the zeros of the associated orthogonal polynomials are uniformly spaced: if θ m,j and θ m,j +1 are the places in [0,π], for which cosθ m,j and cosθ m,j +1 is the j -th and the j +1-th zero of the m -th orthogonal polynomial, then θ m,j  − θ m,j +1  ∼ 1/ m . In this paper it is shown, that this result is also true in a local sense: if a weight has the doubling property in an interval of its support, then uniform spacing of the zeros is true inside that interval. The result contains as special cases some theorems of Last and Simon on local zero spacing of orthogonal polynomials.

In the paper we present a uniqueness polynomial for a class of meromorphic functions having the same set of poles.

In this paper the space G 2 (ℝ) is introduced and the boundedness, normality and spectral properties of the Hausdorff operator H ϕ; acting on the space G 2 (ℝ) is investigated by using the theory of Fourier transform and Hilbert transform. Existence of spectrum of the Hausdorff operator on G 2 (ℝ) is obtained.

In this paper the q -absolute Cesàro summability of order one is introduced in order to make an advanced study in the special topic of absolute summability of orthogonal series. Moreover, we give some sufficient conditions in terms of the coefficients of an orthogonal series under which such series is q -absolute Cesàro summable almost everywhere.

In the note, the authors review the past and long proofs and supply two new and short proofs for a monotonicity result that the function ψ ( x )+ln( e 1/ x  − 1) is strictly increasing on (0,∞), where ψ ( x ) is the psi function.

In this paper entries of sequences, series and infinite matrices are real or complex numbers. We introduce the ( M , λ n ) method of summability and study some of its nice properties.

This paper deals with six major developments of the theory of generalized functions or distributions as a whole new branch of modern analysis. These include (i) The Dirac delta function as the limit of sequences of ordinary functions, (ii) Schwartz´s new theory of distributions based on test functions, (iii) Temple–Lighthill´s simpler analytical approach to generalized functions based on good functions, (iv) Mikusinski´s algebraic approach to generalized functions, (v) Cauchy´s representation of distributions by analytic functions, and (vi) Sato´s less abstract and computationally more effective approach to generalized functions (or hyperfunctions) based on complex variables. Included are basic features, properties and examples of generalized functions including the Dirac delta function and the Heaviside function. Some additional properties of convolution are discussed in the end.