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 On a condition that a trigonometrical series should have a certain form

1913 ◽  
Vol 88 (606) ◽  
pp. 569-574 ◽  
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Difficult Problem ◽  
Functions Of Bounded Variation ◽  
Function Of Bounded Variation ◽  
Necessary And Sufficient Condition ◽  
Trigonometrical Series ◽  
Derived Series ◽  
Necessary And Sufficient ◽  
Series Of Functions

1. In a recent communication to the Society I have illustrated the fact that the derived series of the Fourier series of functions of bounded variation play a definite part in the theory of Fourier series. Some of the more interesting theorems in that theory can only be stated in all their generality when the coefficients of such derived series take the place of the Fourier constants of a function. I have also recently shown that Lebesgue’s theorem, whether in its original or in its extended form, with regard to the usual convergence of a Fourier series when summed in the Cesàro manner is equally true for the derived series of Fourier series of functions of bounded variation. I have also pointed out that, in considering the effect of all known convergence factors in producing usual convergence, it is immaterial whether the series considered be a Fourier series, or such a derived series. We are thus led to regard the derived series of the Fourier series of functions of bounded variation as a kind of pseudo-Fourier series, possessing properly so-called. In particular we are led to ask ourselves what is the necessary and sufficient condition that a trigonometrical series should have the form in question. One answer is of course immediate. The integrated series must converge to a function of bounded variation. This is merely a statement in slightly different language of the property in question. We require a condition of a simpler formal character, one which does not require us to solve the difficult problem as to whether an assigned trigonometrical series not only converges but also has for sum a function of bounded variation.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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 the Nemytskii Operator in the Space of Functions of Bounded (p, 2, α)-Variation with Respect to the Weight Function

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Wadie Aziz
Keyword(s):  
Weight Function ◽  
Bounded Variation ◽  
The Other ◽  
Affine Function ◽  
Nemytskii Operator ◽  
Other Hand ◽  
Generator Function ◽  
Uniformly Continuous

AbstractIn this paper, we consider the Nemytskii operator (Hf)(t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p,2,α)-variation (with respect to a weight function α) into the space of functions of bounded (q,2,α)-variation (with respect to α) 1<q<p, then H is of the form (Hf)(t) = A(t)f(t)+B(t). On the other hand, if 1<p<q then H is constant. It generalize several earlier results of this type due to Matkowski-Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

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.

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