On the set of points with absolutely convergent trigonometric series

AbstractExact analytical expressions for the inverse Laplace transforms of the functions are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order n. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

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A remark on the norm of the Banach space of uniformly strong convergent trigonometric series

Acta Mathematica Hungarica ◽

10.1007/bf01903632 ◽

1988 ◽

Vol 51 (1-2) ◽

pp. 201-203 ◽

Cited By ~ 2

Author(s):

J. Dobi

Keyword(s):

Banach Space ◽

Trigonometric Series ◽

Convergent Trigonometric Series

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On the K-dependence of the electrical field in a crystalline slab of oscillating charges

Canadian Journal of Physics ◽

10.1139/p81-120 ◽

1981 ◽

Vol 59 (7) ◽

pp. 929-933 ◽

Cited By ~ 3

Author(s):

J. Grindlay

Keyword(s):

Electric Field ◽

Trigonometric Series ◽

Wave Vector ◽

Short Range ◽

Electrical Field ◽

Ewald Sum ◽

Convergent Trigonometric Series ◽

Short Range Part ◽

Range Part

The short range part of the electric field of a crystalline slab array of oscillating charges (a) is related to the Ewald sum and (b) can be represented by a rapidly convergent trigonometric series involving the wave vector K. Values for the coefficients of the first few terms of this series are reported for lattice sites in the sc, fcc, bcc, NaCl, CsCl structures and the symmetry directions (1,0,0), (1,1,0), (1,1,1).

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Self-Centred Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-098-3 ◽

1966 ◽

Vol 18 ◽

pp. 974-980

Author(s):

H. Kestelman

Keyword(s):

Abelian Group ◽

Trigonometric Series ◽

Absolute Convergence ◽

Rational Numbers ◽

Real Numbers ◽

Space Group ◽

Space Groups ◽

Set Of Points

A subset S of an abelian group G is said to have a centre at a if whenever x belongs to S so does 2a — x. This note is mainly concerned with self-centred sets, i.e. those S with the property that every element of S is a centre of S. Such sets occur in the study of space groups: the set of inversion centres of a space group is always self-centred. Every subgroup of G is self-centred, so is every coset in G: this is the reason why the set of points of absolute convergence of a trigonometric series is self-centred or empty (1). A self-centred set of real numbers that is either discrete or consists of rational numbers must in fact be a coset (see §3); this does not hold for an arbitrary enumerable self-centred set of real numbers (§3.3).

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