On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

Complete monotonicity of two functions involving the tri-and tetra-gamma functions

The psi function ??(x) is defined by ??(x) = ????(x)/??(x) and ?? ⁽ⁱ⁾ (x), for i ?? ?, denote the polygamma functions, where ??(x) is the gamma function. In this paper, we prove that the functions $ [\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }} $ and $ \frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\} $ are completely monotonic on (0,??).     

The parabolic analytic functions and the derivative of real functions

Among the bidimensional hypercomplex-number systems defined as the parabolic (dual) numbers are introduced with the rule α = 0. As well as the functions of a complex variable, the analytic functions of a parabolic variable can be introduced as analytic continuation of the real functions of a real variable. These functions hold the property that the “imaginary” part is linked to the derivative of the “real” part. In this paper we will show how this property allows one to demonstrate in an algebraic way some rules of the differential calculus for the real functions of a real variable.     

Identification of survival functions through hazard functions in the Clayton-family

In a multivariate nonparametric setup the survival function is not identifiable from the hazard function. Things may change, however, if we restrict ourselves to semiparametric submodels. In this note we show that for the Clayton survival model, the answer is affirmative.     

Monotonicity of some functions involving the gamma and psi functions

Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.

     

Cubic analogues of the Jacobian theta functions and the Jacobian elliptic functions

In this paper, the Jacobian theta functions, the Jacobian elliptic functions, and their cubic analogues are introduced. Then, the properties satisfied the cubic analogues are discussed.     

On the rate of approximation of singular functions by step functions

We consider approximations of a monotone function on a closed interval by step functions having a bounded number of values: the dependence on the number of values of the rate of approximation in the norm of the spaces L _p is studied. A criterion for the singularity of the function in terms of the rate of approximation is obtained. For self-similar functions, we obtain sharp estimates of the rate of approximation in terms of the self-similarity parameters. Functions with arbitrarily fast and arbitrarily slow (down to the theoretic limit) rate of approximation are constructed.     

An analog of the system of reference functions corresponding to the walsh functions

An orthonormal system of entire functions of exponential type in L₂(–; ) is constructed with the aid of the system of the Walsh functions and its approximation properties are considered.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 485–493, April, 1977.The author thanks A. V. Efimov for useful advice and suggestions.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

Limiting functions of series of functions

A series with a given set of limit functions is constructed. A theorem proved by Besicovich concerning conditionally convergent series is strengthened.Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 25–32, July. 1970.     

On the boundedness and continuity of convex functions and additive functions

Aequationes mathematicae -     

On theta type functions associated with the zeros of the Selberg zeta functions

In this paper, we consider a kind of theta type function concerning the zeros of the Selberg zeta function. This is obtained from an application of Cartier-Voros type Selberg trace formula for non co-compact but co-finite volume discrete subgroups ofPSL(2, R).