Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions

deLyra, Jorge L.

doi:10.48550/arxiv.1409.2582

Cited by 6 publications

(59 citation statements)

References 2 publications

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“…In this paper we will discuss the question of the convergence of DP Fourier series, in the light of the correspondence between FC pairs of DP Fourier series and Taylor series of inner analytic functions, which was established in a previous paper [1]. This discussion was started in that paper, and will be continued here in order to include the more complex and difficult cases.…”

Section: Introductionmentioning

confidence: 88%

“…In the previous paper [1] we discussed the issue of the convergence of FC pairs of DP Fourier series such as…”

Section: Weak Convergencementioning

confidence: 99%

“…Note that since w(0) = 0 the integrand is in fact analytic on the open unit disk, if we define it at z = 0 by continuity, and therefore the path of integration from 0 to z is irrelevant, so long as it is contained within that disk. It is easier to see this using the series representation of w(z), which converges in the open unit disk, since w(z) is analytic there, 1) .…”

Section: Classification Of Singularitiesmentioning

confidence: 99%

“…The concepts of the degrees of hardness and of the degrees of softness of singularities in the complex plane, which were introduced in the previous paper [1], will be discussed and defined in more precise terms in this paper. The basic concept is that a soft singularity is one where the complex function is still well defined, although not analytic, when one takes the limit to the singular point.…”

Section: Introductionmentioning

confidence: 99%

“…For organizational reasons this discussion must be separated into several parts. In the previous paper [1] we tackled the extreme cases which we qualify as those of strong divergence and strong convergence, for which results can be established in full generality. In this paper we will deal with the remaining cases, which we will qualify as those of weak convergence and very weak convergence.…”

Section: Introductionmentioning

confidence: 99%

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Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities

deLyra

2014

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The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of convergence of the Taylor series of the inner analytic function is the open unit disk. An essentially complete classification, in terms of the singularity structure of the corresponding inner analytic functions, of the modes of convergence of a large class of DP Fourier series, is established. Given a weakly convergent Fourier series of a DP real function, it is shown how to generate from it other expressions involving trigonometric series, that converge to that same function, but with much better convergence characteristics. This is done by a procedure of factoring out the singularities of the corresponding inner analytic function, and works even for divergent Fourier series. This can be interpreted as a resummation technique, which is firmly anchored by the underlying analytic structure.

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Section: Introductionmentioning

confidence: 88%

“…In the previous paper [1] we discussed the issue of the convergence of FC pairs of DP Fourier series such as…”

Section: Weak Convergencementioning

confidence: 99%

Section: Classification Of Singularitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities

deLyra

2014

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Low-Pass Filters, Fourier Series and Partial Differential Equations

deLyra

2014

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When Fourier series are used for applications in physics, involving partial differential equations, sometimes the process of resolution results in divergent series for some quantities. In this paper we argue that the use of linear low-pass filters is a valid way to regularize such divergent series. In particular, we show that these divergences are always the result of oversimplification in the proposition of the problems, and do not have any fundamental physical significance. We define the first-order linear low-pass filter in precise mathematical terms, establish some of its properties, and then use it to construct higher-order filters. We also show that the first-order linear low-pass filter, understood as a linear integral operator in the space of real functions, commutes with the second-derivative operator. This can greatly simplify the use of these filters in physics applications, and we give a few simple examples to illustrate this fact.

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Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters

deLyra¹

2014

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When Fourier series are employed to solve partial differential equations, low-pass filters can be used to regularize divergent series that may appear. In this paper we show that the linear low-pass filters defined in a previous paper can be interpreted in terms of the correspondence between Fourier Conjugate (FC) pairs of Definite Parity (DP) Fourier series and inner analytic functions, which was established in earlier papers. The action of the first-order linear low-pass filter corresponds to an operation in the complex plane that we refer to as "singularity splitting", in which any given singularity of an inner analytic function on the unit circle is replaced by two softer singularities on that same circle, thus leading to corresponding DP Fourier series with better convergence characteristics. Higher-order linear low-pass filters can be easily defined within the unit disk of the complex plane, in terms of the first-order one. The construction of infinite-order filters, which always result in C ∞ real functions over the unit circle, and in corresponding DP Fourier series which are absolutely and uniformly convergent to these functions, is presented and discussed.

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Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions

Cited by 6 publications

References 2 publications

Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities

Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities

Low-Pass Filters, Fourier Series and Partial Differential Equations

Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters

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