scholarly journals On the series of Legendre

1918 ◽  
Vol 94 (660) ◽  
pp. 292-295
Keyword(s):  
Fourier Series ◽  
General Type ◽  
General Term ◽  
Present Communication ◽  
Legendre Series ◽  
Normal Functions ◽  
Sturm Liouville ◽  
To Come ◽  
In Virtue Of

In recent communications to the Society, I have confined myself largely to the Theory of Fourier Series, partly because much seemed to me still to require doing in this subject, partly because I believed its thorough investigation to be the natural preparation for the study of other series of normal functions. It has, indeed, been known for some time that the behaviour of, for instance, series of Sturm-Liouville functions exactly corresponds to that of Fourier series. The introduction that I have recently made into Analysis of what I have called restricted Fourier series enables us to notably extend the range of such analogies. I propose in the present communication to illustrate this remark with reference to series of Legendre coefficients. Whereas Fourier series may be said to be “naturally unrestricted,” in virtue of the fact that the convergence of the integrated series to an integral necessarily involves the tendency towards zero of its own general term, so that the consideration of the more general type of series does not at once suggest itself, Legendre series may be said to come into being “restricted,” even when the coefficients are expressible in what may be called the Fourier form by means of integrals involving Legendre’s coefficients. In other words, such series correspond precisely to restricted Fourier series, instead of to ordinary Fourier series like the analogous series of Sturm-Liouville functions.

scholarly journals On Fourier series and functions of bounded variation

1913 ◽  
Vol 88 (606) ◽  
pp. 561-568 ◽  
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
General Theorem ◽  
General Term ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Trivial Case ◽  
Present Communication ◽  
Other Hand ◽  
Derived Series

1. In a previous communication to the Society I have pointed out that the succession of constants obtained by multiplying together two successions of Fourier constants in the manner which naturally suggests itself is a succession of Fourier constants, and I have discussed the summability of the function with new constants are associated. We may express the matter in another way by saying that I have shown that the use of the Fourier constants of an even function g(x) as convergence factors in the Fourier series of a function f(x) changes the latter series into a series which is associated with the new series is increased. The use of the Fourier constants of an odd function as convergence factors, on the other hand, has the effect of changing the allied series of the Fourier series of f (x) into a Fourier series, even when the allied series is not itself a Fourier series. It at once suggests itself that the former of the two statements in this form of the result is not the most that can be said. Indeed, the series, whose general term is cos nx , and whose coefficients are accordingly unity, may clearly take the place of the Fourier series of g(x) , although it is not a Fourier series. On the other hand, it is the derived series of the Fourier serious of a function of bounded variation, which is, moreover, odd. We are thus led to ask ourselves whether this is not the trivial case of a general theorem. In the present communication I propose to show, among other things, that the answer to this questions is in affirmative. The following theorems are, in fact, true:—

scholarly journals The double Fourier series of a discontinuous function

1924 ◽  
Vol 106 (737) ◽  
pp. 299-314 ◽  
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Double Fourier Series ◽  
General Term ◽  
Discontinuous Function ◽  
Lebesgue Integral ◽  
Function Of Two Variables

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function f ( x, y ) possesses a double Lebesgue integral over the square (– π < π ; – π < y < π ). Then the general term of the double Fourier series of this function is given by cos = є mn { a mn cos mx cos ny + b mn sin mx sin ny + c mn cos mx sin ny + d mn sin mx cos ny } There є 00 = ¼, є m0 = ½ ( m > 0), є 0n = ½ ( n > 0), є ms = 1 ( m > 0, n >0). the coefficients are given by the formulæ a mn = 1/ π 2 ∫ π -π ∫ π -π f ( x, y ) cos mx cos ny dx dy , obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫ ms = Σ m μ = 0 Σ n v = 0 A μ v = 1/π 2 ∫ π -π ∫ π -π f (s, t) sin( m +½) ( s - x ) sin ( n + ½) ( t - y )/2 sin ½ ( s - x ) 2 sin ½ ( t - y ) if f ( s , t ) is defined outside the original square by double periodicity, we have sub S ms = 1/π 2 ∫ π 0 ∫ π 0 f ( x + s , y + t ) + f ( x + s , y - t ) + f ( x - s , y + t ) + f ( x - s , y - t ) sin ( m + ½) s / 2 sin ½ s sin ( n + ½) t / 2 sin ½ t ds dt .

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Basic Fourier series

1979 ◽  
Vol 369 (1736) ◽  
pp. 115-136 ◽  
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Stochastic Processes ◽  
Numerical Approximation ◽  
Basic Number ◽  
Second Order ◽  
Orthogonal Functions ◽  
Sturm Liouville

The concept of basic number is applied to the development of a simple analogue of the Sturm–Liouville system of the second order. This is then employed to deduce a family of q -orthogonal functions, which leads to a generalization of the Fourier and Fourier–Bessel expansions. The numerical approximation of basic integrals is discussed and some aspects of the evaluation of C a (q; x) are mentioned. A few of the zeros of this function are listed, and, in conclusion, an indication is given of the possibility of applying the analysis presented in this paper to thé study of stochastic processes and time-series.

Ecology and the Ethics of Environmental Restoration

1994 ◽  
Vol 36 ◽  
pp. 31-43
Author(s):  
Robert Elliot
Keyword(s):  
Intrinsic Value ◽  
Environmental Restoration ◽  
Initial Value ◽  
Value Judgment ◽  
Human Interference ◽  
Rational Debate ◽  
To Come ◽  
In Virtue Of

Some people think that nature has intrinsic value, that it has value in itself quite apart from its present and future economic, intellectual, recreational and aesthetic uses. Some people think that nature's intrinsic value grounds an obligation to preserve it and to minimise human interference with it. I agree. It is important, however, to try to say exactly why nature has intrinsic value, to go beyond merely stating some idiosyncratic attitude and to provide some justification of that attitude with which others might engage. Presumably there are properties that wild nature exemplifies in virtue of which it is intrinsically valued. Only when these are indicated is rational debate as to whether wild nature has intrinsic value possible. Only when these are indicated is it possible to begin to persuade dissenters to change their views. Indeed, unless one can at least begin to say what these properties are it is not clear that the attitude could have any meaningful content. While it is perhaps possible to value something without immediately understanding what it is about the thing that makes it valuable, the failure to come up with any candidate value-adding property after some reflection suggests that the initial value-judgment is vacuous.

scholarly journals On the mode of approach to zero of the coefficients of a Fourier series

1917 ◽  
Vol 93 (654) ◽  
pp. 455-467
Keyword(s):  
Fourier Series ◽  
General Theory ◽  
Practical Interest ◽  
Present Communication ◽  
Single Instance ◽  
Order Of Magnitude ◽  
Real Importance ◽  
Derived Series ◽  
Insight Into ◽  
De La Vallée Poussin

1. In the present communication I give a number of results on the mode of approach to zero of the coefficients of a Fourier series, to which I have already made allusion in my paper on “The Order of Magnitude of the Coefficients of a Fourier Series.” These results are not merely curious, they have a real importance, and give one an insight into the nature of these series, which cannot easily be gained without them. Indeed, while the earlier paper leads, as I showed in a subsequent communication, to the discovery of classes of derived series of Fourier series, which, although not themselves Fourier series, none the less converge, and are utilisable in a similar manner, and is therefore in a certain sense of practical interest, the present paper does something towards the elucidation of the general theory of the convergence of Fourier series themselves, as well as of their derived series. It will be sufficient to give a single instance. I have recently shown that the well-known test of Dirichlet for the convergence of a Fourier series admits of a remarkable generalisation. It follows from Theorem. 1, given below (7), that the convergence secured by that test and by its generalisation alike possess what may be called greater strength than the rival tests of Dini and de la Vallée Poussin.

Judging by Refraining from Judgment

2019 ◽  
pp. 95-114
Author(s):  
Gerhard Richter
Keyword(s):  
A Priori ◽  
Aesthetic Theory ◽  
Critical Practice ◽  
Work Of Art ◽  
Critical Engagement ◽  
Singular Form ◽  
To Come ◽  
In Virtue Of

This chapter focuses on Adorno’s understanding of the category of judgment. Proceeding from Adorno’s apodictic interpretation of a poem by the German Biedermeier writer Eduard Mörike, it reconstructs what it might mean for Adorno to argue for the critical practice of judging by refraining from judgment. Mörike’s children’s poem “Mousetrap Rhyme” is the only poem that Adorno chooses to quote in its entirety in his Aesthetic Theory. His surprising choice reveals how the uncoercive gaze can never be reduced to a set of ideological operations or a priori correspondences but rather must confront, in the space of the work of art, the question of its judgment—and the typically unspoken premises and presuppositions of any judgment—always one more time. Here, the uncoercive gaze fastens upon the artwork in a way that allows art to become world without reducing the art to the condition of being merely that which already is the case or that which already claims to be world. The artwork keeps alive the singular form of judgment as judgment without judging, in which the ultimate arrest of judgment remains deferred in virtue of another judgment, based on a future critical engagement, that is always still to come.

Detection Protocol of Possible Crime Scenes Using Internet of Things (IoT)

2020 ◽  
pp. 783-799
Author(s):  
Bashar Alohali
Keyword(s):  
Internet Of Things ◽  
Scientific Knowledge ◽  
Data Classification ◽  
General Term ◽  
Scientific Principles ◽  
To Come ◽  
Investigation Process ◽  
Crime Scenes

Forensics is a science that deals with using scientific principles in order to aid an investigation of a civil or criminal crime. It is a system of procedures that allow an investigator to use as much resources as possible in order to come up with a conclusion for an investigation. Since forensics is a very general term that encompasses an investigation process using scientific knowledge, one can separate a system of investigation based on how it is conducted. This chapter introduces of internet of things (IoT) forensics, IoT application in forensics field. Art-of-states for IoT forensics are provided. The issues for IoT forensics are identified. Also, we have introduced the proposed data classification in Iot forensics protocol. At the end of this chapter, we point out a brief summary and conclusion.

scholarly journals Sturm-liouville series of normal functions in the theory of integral equations

1911 ◽  
Vol 84 (574) ◽  
pp. 573-575
Keyword(s):  
Integral Equations ◽  
Complete System ◽  
Singular Values ◽  
Positive Type ◽  
Normal Functions ◽  
Sturm Liouville

In a memoir read before the Society on May 13 last I proved a theorem* which may be stated as follows:— Let ψ 1 ( s ), ψ 2 ( s ),..., ψ n ( s ),... be a complete system of normal functions relating to a function K ( s,t ) which is of positive type in the square a ≤ s ≤ b , a ≤ t ≤ b ; and let λ 1 , λ 2 ,.., λ n ,... be the corresponding singular values. Then the series ψ 1 ( s ) ψ 1 ( t )/λ 1 -λ + ψ 2 ( s ) ψ 2 ( t )/λ 2 -λ + ... + ψ n ( s ) ψ n ( t )/λ n -λ +... converges absolutely and uniformly, and has for its sum function K λ ( s, t ), the solving function of k ( s, t ).

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