Recovery of band limited functions via cardinal splines

The main result of this paper asserts that if a function f is in the class B_π,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its derivatives f^(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric ofL _q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞.     

Multiplication formulas for Kubert functions

The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of （generalized） Kubert functions. Second, we apply the special case of Carlitz＇s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz＇s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz （for sums） and Mordell （for integrals）, both of which are generalizations of preceding results by Frasnel, Landau, Mikolas, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck＇s lamma is the same as Carlitz＇s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.     

Structure formulas for univalent functions

With the aid of special integrating factors for the Loewner-Kufarev equation, one determines the general form of the structure formulas for functions of the class S and one obtains a series of new concrete structure formulas for certain subclasses of functions from S.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 128–134, 1983.     

Bosonic formulas for affine branching functions

In this paper, we derive two bosonic (alternating sign) formulas for branching functions of affine Kac-Moody Lie algebras \(\mathfrak{g}\). Both formulas are expressed in terms of the Weyl group and string functions of \(\mathfrak{g}\).     

Asymptotic formulas for certain arithmetic functions

This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere. The last named author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12.     

On product formulas for exponential functions

A direct and inverse theorem on the generalized Chernoff formula in Banach algebras is proved by using only elementary properties of exponential and logarithm functions. The main result in this paper forms a part of the first author’s Master thesis (1981) at National Central University.     

Weighted cubature formulas for periodic functions

This article contains a study of weighted cubature formulas for periodic functions in n-dimensional Euclidean space with matrix period H. The principal error terms are considered.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 319–326, March, 1968.The author wishes to express his deep appreciation of S. L. Sobolev's guidance in this work.     

Cauchy and Pompey formulas for generalized analytic functions

We use the apparatus of the theory of generalized functions by which we can define strictly the fundamental functional solution of an equation, thereby making it possible to obtain the Cauchy-Pompey formulas without any additional conditions on the behavior of solutions at infinity when the coefficients of the equation are constant. The kernels of these formulas are obtained in explicit form.Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 263–268, September, 1972.In conclusion the author wishes to express his gratitude to V. S. Vinogradov for formulating the problem and for help in writing the paper.     

On the explicit formulas for zeta functions

In this paper, we study the properties of a large class of zeta functions that arises in geometric analysis and mathematical physics. They are attached to some elliptic operators. This method can be used to evaluate explicitly the special values of zeta functions of elliptic operators defined on some symmetric spaces.     

Asymptotic formulas for some retricted partition functions

We use Rademacher’s method to obtain asymptotic expressions for some restricted partition functions. For fixed positive integers m > 1 and l, we consider the number p _m (n) of partitions of n into summands not divisible by m, and the number p _m ^l (n) of partitons of n with the further restriction that any integer occurs at most l times as a summand. 2000 Mathematics Subject Classification Primary—11P82; Secondary—11P83     

Cauchy and Poisson formulas for polyanalytic functions and applications

We obtain new Cauchy and Poisson integral formulas for polyanalytic functions. As an application, we establish mean value theorems for polyanalytic and real polyharmonic functions in a disk. We also give applications to sharp estimates of generalized maximum modulus principle type for associated functions, and, in particular, to estimates for rational functions (components) in the problem of singularity separation for polyrational functions.     

Decomposition formulas for some triple hypergeometric functions

With the help of some techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Srivastava's hypergeometric functions HA, HB and HC in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 15 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly.     

Optimal stochastic quadrature formulas for convex functions

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods, we prove that adaptive Monte Carlo methods are much better.Supported by a Heisenberg scholarship of the DFG.     

Complexity and depth of formulas for symmetric Boolean functions

A new approach for implementation of the counting function for a Boolean set is proposed. The approach is based on approximate calculation of sums. Using this approach, new upper bounds for the size and depth of symmetric functions over the basis B₂ of all dyadic functions and over the standard basis B₀ = {∧, ∨,^- } were non-constructively obtained. In particular, the depth of multiplication of n-bit binary numbers is asymptotically estimated from above by 4.02 log₂n relative to the basis B₂ and by 5.14log₂n relative to the basis B₀.     

Symbolic operational images and decomposition formulas for hypergeometric functions

Based upon the classical derivative and integral operators we introduce a new operator which allows the derivation of new symbolic operational images for hypergeometric functions. By means of these symbolic operational images a number of decomposition formulas involving quadruple series are then found. Other closely-related results are also considered.