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In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit.

We offer sharp inequalities for trigonometric sums of the classical Fejér–Jackson–Young-type, and we present sharp bounds for alternating trigonometric sums involving binomial coefficients.

We say that a simply connected domain in the plane has an o-minimal corner at the origin if its boundary at the origin is given by two curves forming a corner of nonzero angle which are definable in a given o-minimal structure. We investigate the mapping function at o-minimal corners. We describe the asymptotic behaviour if the boundary curves are definable in an arbitrary polynomially bounded o-minimal structure. If they are definable in the o-minimal structure ℝ an ℝ we establish an asymptotic expansion in terms of a generalized power series with logarithmic perturbation.

This paper deals with derivation of the Fourier Integral Theorem, convolution theorem, dual convolution theorem, many classical and new algebraic and analytic properties of convolution, Poisson´s summation formula, integral representation of a band-limited signal, and Shannon´s sampling theorem. It is shown that the convolution originates in the context of partial sums of the Fourier series of a function f and the n th partial sum is expressed in terms of convolution of f and the Dirichlet kernel. The Cessaro and Abel means of a Fourier series of f are also expressed in terms of convolution of f and the Fejer and the Poisson kernels. It is shown that the Fourier inverse integral is Gauss, Abel and Cesaro summable to f ( x ) in the L 1 -norm, and the pointwise summability theorems can be used to find the Fourier inversion formula almost everywhere in ℝ. Applications of convolution to the Fourier diffusion equation and the classical wave equation are presented.

The main result of Nopper´s paper [8] in this same journal reads as follows. If a transcendental entire function f has small growth order, then there is at most one rational point w such that all f ( n ) ( w ), n  = 0,1,..., are rational with denominators increasing sufficiently slowly with n . The principal purpose of our work is to considerably relaxe this arithmetic growth condition, to generalize Nopper´s result to meromorphic functions, and to improve on similar statements of Nikishin [6], and Nopper and Wallisser [10]. From all these assertions, some well-known but not obvious irrationality results can be deduced in a systematic and simple way. In each case, the method of Newton´s interpolation series is essentially used, and it is also explained how this one leads to new and transparent proofs of Straus´ [16] and Schneider´s [13] achievements on integer-valued meromorphic functions.

Analysis Vol. 31(2011) Issue 4, pp. 355–379 Please note the correct version available online at: http://www.oldenbourg-link.com/toc/anly/31/4/1122 DOI: 10.1524/anly.2011.1122