Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

Analyticity on families of circles

It is known that iff is a continuous function on the complex plane which extends holomorphically from each circle surrounding the origin, thef is not necessarily holomorphic. In the paper we prove that if, in addition,f extends holomorphically from each circle belonging to an open family of circles which do not surround the origin, thef is holomorphic.     

On derivatives of Bernstein type rational functions of two variables

In this paper the author defines Bernstein type rational functions of two variables and prove the approximation theorems for the derivatives of them.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are

(x²+y)uxx+2xyuxy+y²uyy+gxux+g(y−1)uy=λu,     

Defining continuity of real functions of real variables

Continuity of a real function of a real variable has been defined in various ways over almost 200 years. Contrary to popular belief, the definitions are not all equivalent, because their consequences for four somewhat pathological functions reveal five essentially different cases. The four defensible ones imply just two cases for continuity on an interval if that is defined by using pointwise continuity at each point. Some authors had trouble: two different textbooks each gave two arguably inconsistent definitions, three more changed their definitions in their second editions, two more claimed continuity at a point for functions not defined there, and one gave a definition implying it for a function with no limit there.     

Reducibility of rational functions in several variables

We prove an analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.     

Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Positive definite rational functions of two variables which are not the sum of three squares

Cassels, Ellison, and Pfister have shown that there is a positive semidefinite function of R(x, y) that is not the sum of three squares. In this paper positive definite functions of R(x, y) are found having the same property. The proof involves showing the nonexistence of points on some elliptic curves defined over C(x), and extends the methods of [1].     

Hypergeometric functions of two variables

Summary The systematic investigation of contour integrals satisfying the system of partial differential equations associated with Appell's hypergeometric functionF ₁ leads to new solutions of that system. Fundamental sets of solutions are given for the vicinity of all singular points of the system of partial differential equations. The transformation theory of the solutions reveals connections between the system under consideration and other hypergeometric systems of partial differential equations. Presently it is discovered that any hypergeometric system of partial differential equations of the second order (with two independent variables) which has only three linearly independent solutions can be transformed into the system ofF ₁ or into a particular or limiting case of this system. There are also other hypergeometric systems (with four linearly independent solutions) the integration of which can be reduced to the integration of the system ofF ₁.     

A method of approximation by rational functions on the real line

For a given system of numbers {z _k } _k=1 ⁿ , IMz _k > 0, rational functions of order 4n — 2 are constructed which effect for a functionf(x∈C _∞) an approximation of the same order as the best approximation by proper rational functions having poles at the points {z _{k k=1} ⁿ and . Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 375–380, September, 1977. In conclusion the author thanks E. P. Dolzhenko and S. B. Stechkin, whose discussions contributed to improvements in this work.     

Baire one, Gibson and weakly Gibson real functions of several real variables

We investigate Darboux-like properties for real-valued, Baire one functions defined on Euclidean n-space which belong to the classes of Gibson and weakly Gibson functions recently introduced and investigated by Kenneth Kellum.     

Jensen's operator inequality for functions of two variables

The operator convex functions of two variables are characterized in terms of a non-commutative generalization of Jensen's inequality.