Necessary Conditions for Integrability of the Fourier Transform

2009 ◽  
Vol 16 (3) ◽  
pp. 553-559
Author(s):  
Elijah Liflyand
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Necessary Conditions ◽  
The Other ◽  
Necessary Condition ◽  
Convergent Fourier Series ◽  
Other Hand ◽  
The Fourier Transform

Abstract We prove the necessary conditions for the integrability of the Fourier transform. The result is a generalization, on one hand, of the well known necessary condition for absolutely convergent Fourier series and, on the other hand, of an earlier multidimensional result of Trigub.

UN-Sector and Compositeness Conditions in the Bronzan-Lee Model

1974 ◽  
Vol 29 (11) ◽  
pp. 1531-1542
Author(s):  
A. L. Choudhury
Keyword(s):  
Fourier Transform ◽  
Composite Particle ◽  
Composite Model ◽  
The Other ◽  
Final States ◽  
Lee Model ◽  
Other Hand ◽  
Initial States ◽  
The Fourier Transform ◽  
Limiting Values

The Lehmann-Symanzik-Zimmermann (LSZ) technique has been used to calculate all τ -functions of the UN-sector of the Bronzan-Lee model. Using the prescription of Liossatos, the ZV → 0 limit has been carried out for the fourier transform of the τ-functions in the sector. These limiting functions τ̂α,LUN are then compared with the τ̂C,αUN functions derived from a composite model, proposed by the foregoing author, where V is considered to be a composite particle. It has been found that when the so called composite V-particle does not appear in the initial and the final states, these τ-functions coincide. On the other hand, the limiting values of some τ-functions differ from those of the composite model, when such particles appear in the final or initial states.

Elements and Knowledge in the Theaetetus

Elenchos ◽  
2013 ◽  
Vol 34 (2) ◽  
pp. 299-326
Author(s):  
Jie Tian
Keyword(s):  
Present Article ◽  
Necessary Conditions ◽  
The Other ◽  
Necessary Condition ◽  
The Third ◽  
Other Hand ◽  
Adequate Definition ◽  
Dream Theory ◽  
The One ◽  
Definition Of

Abstract Plato's Theaetetus develops an inquiry concerning the definition of knowledge. Famously, after Socrates and Theaetetus have discussed the three candidates for the definition of knowledge, the end of the dialogue seems to leave us in a situation of aporia. The present article focuses on the last hypothesis raised in the dialogue and tries to determine whether this hypothesis can be seen, under appropriate qualification, as acceptable within a Platonic framework. This hypothesis is connected with a dream theory that unfolds two crucial factors for understanding the definition of knowledge, i.e. elements and logos. So the aim of this paper is twofold: on the one hand, to make clear what elements properly are; on the other hand, to find an account of logos suitable to make it a necessary condition for the definition of knowledge. As will emerge from this paper, the first two candidates for the definition of knowledge are indeed not sufficient for gaining an adequate definition, but they nonetheless foreshadow the third hypothesis and are necessary conditions for understanding the third one.

The infrared spectrum of CS

1984 ◽  
Vol 62 (12) ◽  
pp. 1414-1419 ◽  
Author(s):  
R. J. Winkel Jr. ◽  
Sumner P. Davis ◽  
Rubén Pecyner ◽  
James W. Brault
Keyword(s):  
Fourier Transform ◽  
Infrared Emission ◽  
Solar Observatory ◽  
Band Head ◽  
The Other ◽  
Fourier Transform Spectrometer ◽  
National Solar Observatory ◽  
Carbon Monosulfide ◽  
Vibration Rotation ◽  
The Fourier Transform

The infrared emission spectrum of carbon monosulfide was observed as a sequence of vibration–rotation bands in the X1Σ+ state, with strong heads of the Δν = 2 sequence degraded to the red. Eight bands of 12C32S were identified, and bands corresponding to the isotope 12C34S were also observed. The most prominent band head, that of the (2–0) band, is at 2585 cm−1, with the other heads spaced approximately 26 cm−1 to smaller wavenumbers. Our data, taken with the Fourier transform spectrometer at the National Solar Observatory (Kitt Peak) include the first reported laboratory observations of the band heads and as many as 200 lines in each band. These observations allowed the calculation of vibrational and rotational constants to higher order than previously reported.

scholarly journals Nietzsche’s Dionysos

2017 ◽  
Vol 3 (3) ◽  
pp. 604
Author(s):  
Dieter Mersch
Keyword(s):  
Necessary Conditions ◽  
Central Point ◽  
The Other ◽  
Other Hand ◽  
The Absolute ◽  
The Aesthetic

Nietzsche’s Dionysus, admittedly, represents a direct provocation and an attack on the classical interpretation accepted since Winckelmann, an interpretation that elevates the Apollonian to its central point of focus; Nietzsche’s introduction of another principle to oppose it, rather than representing a genuine invention, in actuality bridges the small gap between Hegel and Hölderlin. If, namely, the Hegelian aesthetic from the very beginning points to Schein and Erscheinung – as necessary conditions of truth, for the truth would not exist if it were not to “superficially appear” (scheinen) and “make its appearance” (erscheinen), writes Hegel – Schein and Erscheinung would still nonetheless be bound up everywhere with the criterium of the absolute; after all, the untruth of the aesthetic rests squarely in the fact that it cannot do other than to draw upon the language of Erscheinung. For Hölderlin, on the other hand, the Dionysian advances to become a metapoetic symbol combining itself – the enigmatic and continually transforming – with the practice of art. Nietzsche follows those very same lines even while giving the metaphor a thoroughly different twist.

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

scholarly journals Approximation byq-Bernstein Polynomials in the Caseq→1+

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Xuezhi Wu
Keyword(s):  
Bernstein Polynomials ◽  
The Other ◽  
Necessary Condition ◽  
Standard Case ◽  
Other Hand ◽  
Approximating Sequence ◽  
Sufficient And Necessary Condition

LetBn,q(f;x),q∈(0,∞)be theq-Bernstein polynomials of a functionf∈C[0,1]. It has been known that, in general, the sequenceBn,qn(f)withqn→1+is not an approximating sequence forf∈C[0,1], in contrast to the standard caseqn→1-. In this paper, we give the sufficient and necessary condition under which the sequenceBn,qn(f)approximatesffor anyf∈C[0,1]in the caseqn>1. Based on this condition, we get that if1<qn<1+ln⁡2/nfor sufficiently largen, thenBn,qn(f)approximatesffor anyf∈C[0,1]. On the other hand, ifBn,qn(f)can approximateffor anyf∈C[0,1]in the caseqn>1, then the sequence(qn)satisfieslim¯n→∞n(qn-1)≤ln2.

Leibniz on Apperception and Animal Souls

Dialogue ◽  
1994 ◽  
Vol 33 (4) ◽  
pp. 701-724 ◽  
Author(s):  
Murray Miles
Keyword(s):  
The Other ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Human Beings ◽  
Mental Processes ◽  
Other Hand ◽  
Forms Of Life ◽  
The One

InLeibniz: Perception, Apperception, and Thought, Robert McRae alleges a flat “contradiction” (McRae 1976, p. 30) at the heart of Leibniz's doctrine of three grades of monads: bare entelechies characterized by perception; animal souls capable both of perception and of sensation; and rational souls, minds or spirits endowed not only with capacities for perception and sensation but also with consciousness of self or what Leibniz calls (introducing a new term of art into the vocabulary of philosophy) “apperception.” Apperception is a necessary condition of those distinctively human mental processes associated with understanding and with reason. Insofar as it is also a sufficient condition of rationality, it is not ascribable to animals. But apperception is a necessary condition of sensation or feeling as well; and animals are capable of sensation, according to Leibniz, who decisively rejected the Cartesian doctrine that beasts are nothing but material automata. “On the one hand,” writes McRae, “what distinguishes animals from lower forms of life is sensation or feeling, but on the other hand apperception is a necessary condition of sensation, and apperception distinguishes human beings from animals” (McRae 1976, p. 30). “We are thus left with an unresolved inconsistency in Leibniz's account of sensation, so far as sensation is attributable both to men and animals” (ibid., p. 34).

ORIGINATIONS AND EXTINCTIONS: ANALYSES OF FOSSIL DATA AND SIMULATIONS

2000 ◽  
Vol 11 (02) ◽  
pp. 277-285 ◽  
Author(s):  
TANE RAY ◽  
LEO MOSELEY ◽  
NAEEM JAN
Keyword(s):  
Fourier Transform ◽  
Frequency Dependence ◽  
Frequency Spectrum ◽  
Fossil Record ◽  
Trophic Levels ◽  
The Other ◽  
Simulation Data ◽  
The Past ◽  
Other Hand ◽  
Fossil Data

We analyse the fossil data of Benton1 with and without interpolation schemes. By Fourier transform analysis, we find a frequency dependence of the amplitude of 1/f for the various interpolation schemes used in the past. We illustrate that shuffling the interpolated data changes the spectra only slightly. On the other hand, an identical analysis performed on the raw (uninterpolated) fossil data gives a flat frequency spectrum. We conclude that the 1/f behavior is an artifact of the interpolation schemes. We next introduce a simulation of extinctions driven only by interactions between two trophic levels. Fourier transform analysis of the simulation data shows a frequency dependence of 1/f. When the data are grouped into a form resembling the fossil record the frequency dependence vanishes, giving a flat spectrum. Our simulation produces a frequency spectrum that agrees with the observed fossil record.

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