scholarly journals On classes of summable functions and their Fourier Series

1912 ◽  
Vol 87 (594) ◽  
pp. 225-229 ◽  
Keyword(s):  
Fourier Series ◽  
Integral Equations ◽  
Bounded Variation ◽  
Summable Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Suitable Constant ◽  
Necessary And Sufficient ◽  
Class Of Functions ◽  
Formal Statement

1. Functions which are summable may be such that certain functions of them are themselves summable. When this is the case they will possess certain special properties additional to those which the mere summability involves. A remarkable instance where this has been recognised is in the case of summable functions whose squares also are summable. The—in its formal statement almost self-evident—Theorem of Parseval which asserts that the sum of the squares of the coefficients of a Fourier series of a function f ( x ) is equal to the integral of the square of f ( x ), taken between suitable limits and multiplied by a suitable constant, has been recognised as true for all functions whose squares are summable. Moreover, not only has the converse of this been shown to be true, but writers have been led to develop a whole theory of this class of functions, in connection more especially with what are known as integral equations. That functions whose (1 + p )th power is summable, where p >0, but is not necessarily unity, should next be considered, was, of course, inevitable. As was to be expected, it was rather the integrals of such functions than the functions themselves whose properties were required. Lebesgue had already given the necessary and sufficient condition that a function should be an integral of a summable function. F. Riesz then showed that the necessary and sufficient condition that a function should be the integral of a function whose (1 + p )th power is summable had a form which constituted rather the generalisation of tire expression of the fact that such a function has bounded variation, than one which included the condition of Lebesgue as a particular case.

scholarly journals Certain subclasses of analytic functions involving complex order

Filomat ◽  
2008 ◽  
Vol 22 (2) ◽  
pp. 115-122 ◽  
Author(s):  
T.N. Shanmugam ◽  
S. Sivasubramanian ◽  
B.A. Frasin
Keyword(s):  
Analytic Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Order ◽  
Necessary And Sufficient ◽  
Class Of Functions

In the present investigation, we consider an unified class of functions of complex order. Necessary and sufficient condition for functions to be in this class is obtained. The results obtained in this paper generalizes the results obtained by Srivastava and Lashin [10], and Ravichandran et al. [4]. .

On the Nørlund Summability of a Class of Fourier Series

1970 ◽  
Vol 22 (1) ◽  
pp. 86-91 ◽  
Author(s):  
Badri N. Sahney
Keyword(s):  
Fourier Series ◽  
Integrable Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lebesgue Integrable Function ◽  
Summability Of Fourier Series ◽  
Lebesgue Set ◽  
Necessary And Sufficient

1. Our aim in this paper is to determine a necessary and sufficient condition for N∅rlund summability of Fourier series and to include a wider class of classical results. A Fourier series, of a Lebesgue-integrable function, is said to be summable at a point by N∅rlund method (N, pn), as defined by Hardy [1], if pn → Σpn → ∞, and the point is in a certain subset of the Lebesgue set. The following main results are known.

scholarly journals On the asymptotic boundary behavior of functions analytic in the unit disk

1977 ◽  
Vol 67 ◽  
pp. 1-13
Author(s):  
James R. Choike
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Unit Disk ◽  
Boundary Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Asymptotic Boundary ◽  
Necessary And Sufficient ◽  
Class Of Functions

In [8] a necessary and sufficient condition was given for determining the equivalence of two asymptotic boundary paths for an analytic function w = f(p) on a Riemann surface F. In this paper we give a necessary and sufficient condition for determining the nonequivalence of two asymptotic boundary paths for f(z) analytic in |z| < R, 0 < R ≤ + ∞. We shall, also, illustrate some applications of the main result and examine a class of functions introduced by Valiron.

A note on the age-dependent branching process with immigration

1975 ◽  
Vol 12 (01) ◽  
pp. 130-134 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
General Class ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Generalization ◽  
Age Dependent ◽  
General Class Of Functions ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Class Of Functions

Let {Z(t)}t0be an age-dependent branching process with immigration. For a general class of functions Φ(x), a necessary and sufficient condition is given for whenE{Φ (Z(t))} &lt;∞. This result is a direct generalization of a theorem proven for the branching process without immigration.

Matrix Transformations Based on Dirichlet Convolution

1997 ◽  
Vol 40 (4) ◽  
pp. 498-508
Author(s):  
Chikkanna Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Bounded Variation ◽  
Matrix Transformations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractThis paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

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 On the formation of usually convergent Fourier series

1913 ◽  
Vol 88 (602) ◽  
pp. 178-188
Keyword(s):  
Fourier Series ◽  
First Principles ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Theoretical Interest ◽  
The Subject ◽  
Necessary And Sufficient ◽  
Sufficient Test ◽  
Suitable Modification

1. Series which converge expect at a set of content zero, or, using the expression very commonly adopted, series which converge usually, posses many of the properties which appertain to series which converge every-where. It becomes, therefore, of importance to device circumstances under which we can assert the consequence that a series converge in this manner. The subject has recently received considerable attention. so far as Fourier Series are concerned no result of even an approximately final character has been obtained. It may be supposed, indeed, that the result* of Jerosch and Weyl were at first so regarded, but, if we examine them closely in the light of the Riesz-Fischer theorems, which was known previously to the result of these authors, it becomes evident that they are merely equivalent to the statement that the Fourier Series of a function, whose square is summable, is changed into one which converges usually, if the typical coefficient a n and b n are divided by the sixth root of the integer n denoting their place in the series. Now it is difficult to believe that the question of the usual convergences of a Fourier Series can depended on the degree of the summability of the function with which it is associated and it is still more difficult to see how precisely the sixth root of n can have anything to do it. On the other hand Weyl's method, which itself marks an advance on that of Jerosch, does not obviously lend itself to any suitable modification which would secure a greater degree of generality in the result. The mistake is frequently made of confusing theoretical interest with pracitcal importance in the matter of a necessary and sufficient test. Tests which are only sufficient, but not necessary, are often much more convenient. Still more frequent it is convenient to work from first principles, and not to use any test at all. Instead of employing Weyl's necessary and sufficient condition that a series should converge usually, I have attacked the problem directly. The principles I have employed do not differ essentially from those already exposed in previous communication to this Society, but the generality and interest of the result obtained in the matter in hand seem to justify a further communication.

scholarly journals On the ordinary convergence of restricted fourier series

1917 ◽  
Vol 93 (651) ◽  
pp. 276-292 ◽  
Keyword(s):  
Fourier Series ◽  
General Type ◽  
Closed Interval ◽  
Single Point ◽  
Interval Of Periodicity ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Trigonometrical Series ◽  
Derived Series ◽  
Necessary And Sufficient

1. The necessary and sufficient condition that a trigonometrical series should be a Fourier series is that the integrated series should converge to an integral throughout the closed interval of periodicity, and should be the Courier series, accordingly, of an integral. Conversely, starting with the Courier series of an integral and differentiating it term by term, we obtain the Courier series of the most general type, namely, one associated with any function possessing an absolutely convergent integral. If the Courier series which is differentiated is not the Courier series of an integral, but of a function which fails to be an integral, at even a single point, the derived series will not lie a Courier series.

scholarly journals Good decomposition in the class of convex functions of higher order

2006 ◽  
Vol 80 (94) ◽  
pp. 157-169
Author(s):  
Slobodanka Jankovic ◽  
Tatjana Ostrogorski
Keyword(s):  
Positive Integer ◽  
Convex Functions ◽  
Higher Order ◽  
Sufficient Condition ◽  
Slowly Varying Function ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Class Of Functions ◽  
Good Decomposition

The problems investigated in this article are connected to the fact that the class of slowly varying functions is not closed with respect to the operation of subtraction. We study the class of functions Fk?1, which are nonnegative and i-convex for 0<_ i < k, where k is a positive integer. We present necessary and sufficient condition that guarantee that, no matter how we decompose an additively slowly varying function L ? Fk?1 into a sum L = F + G, F,G ? Fk?1, then necessarily F and G are additively slowly varying.

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