Related papers: On some transforms of trigonometric series
The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.
In the present paper, we give a brief review of $L^{1}$-convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.
In this paper we show an alternative way of defining Fourier Series and Transform by using the concept of convolution with exponential signals. This approach has the advantage of simplifying proofs of transforms properties and, in our view,…
The present paper proposes a new condition to replace both the ($O$-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain…
We introduce a transformation for converting a series in a parameter, \lambda, to a series in the inverse of the parameter \lambda^{-1}. By applying the transform on simple examples, it becomes apparent that there exist relations between…
In this paper we consider trigonometric series with p-bounded variation coefficients. We presented a sufficient condition for uniform convergance of such series in case p > 1. This condition is significantly weaker than these obtained in…
Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Transformations that depend not only on the sequence elements or…
The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which…
A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating…
A linear sequence transformation is defined that accelerates the convergence of the negative binomial series when the terms of the binomial have the same sign. The transformed series can be used to extend the region of applicability of the…
In the present paper we consider the trigonometric series with (b,r)-general monotone and (b,r)-rest bounded variation coefficients. Necessary and sufficien conditions of L-convergence for such series are obtained in terms of the…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
We study the convergence sets of a class of alternating series. Among other things, our results establish the convergence of the series $\sum_n (-1)^n|\sin n|/n$.
In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made…
We extend Riemann's rearrangement theorem on conditionally convergent series of real numbers to multiple instead of simple sums.
Additional remarks and questions for transseries. In particular: properties of composition for transseries; the recursive nature of the construction of R[[[ x ]]]; modes of convergence for transseries. There are, at this stage, questions…
In this note we prove a converse of Bohr's equivalence theorem for Dirichlet series under some natural assumptions.
Considering the sets of subsums of series (or achievement sets) we show that for conditionally convergent series the multidimensional case is much more complicated than that of the real line. Although we are far from the full topological…
In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial…
Methods of constructing trigonometric fundamental splines with constant sign and sign-changing convergence factors are given. An example and graphics illustrating the concepts of convergence and interpolation grids are given. Some methods…