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 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 A Note about Functions in Lip α

1954 ◽  
Vol 10 (2) ◽  
pp. 100-100 ◽  
Author(s):  
N. Du Plessis
Keyword(s):  
Fourier Series ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Trigonometrical Series ◽  
Necessary And Sufficient

This note gives a proof of the result:A necessary and sufficient condition that a trigonometrical series T (x) be the Fourier series of a function is that σn – σm = O(n-n) uniformly in [0, 2π] for all m≤n, where σn is the nth (C, l) mean of T (x).

EQUIVALENT SEQUENTIAL AND PARALLEL REDUCTIONS IN ARBITRARY BINARY PICTURES

2014 ◽  
Vol 28 (07) ◽  
pp. 1460009 ◽  
Author(s):  
KÁLMÁN PALÁGYI
Keyword(s):  
Single Point ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sufficient Criterion ◽  
Translation Invariant ◽  
Necessary And Sufficient

A reduction transforms a binary picture only by changing some black points to white ones, which is referred to as deletion. Sequential reductions traverse the black points of a picture, and consider a single point for possible deletion, while parallel reductions can delete a set of black points simultaneously. Two reductions are called equivalent if they produce the same result for each input picture. A deletion rule is said to be equivalent if it yields a pair of equivalent parallel and sequential reductions. This paper introduces a class of equivalent deletion rules that allows us to establish a new sufficient condition for topology-preserving parallel reductions in arbitrary binary pictures. In addition we present a method of verifying that a deletion rule given by matching templates is equivalent, a necessary and sufficient condition for order-independent deletion rules, and a sufficient criterion for order-independent and translation-invariant parallel subfield-based algorithms.

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 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 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.

Travelling waves for a nonlocal double-obstacle problem

1997 ◽  
Vol 8 (6) ◽  
pp. 581-594 ◽  
Author(s):  
PAUL C. FIFE
Keyword(s):  
Obstacle Problem ◽  
Travelling Waves ◽  
Single Point ◽  
Travelling Wave ◽  
Wave Profile ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Piecewise Constant ◽  
Regularity Properties ◽  
Necessary And Sufficient

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problemut =J*u−u−f (u),the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.

scholarly journals On the Continuity of Stationary Gaussian Processes

1969 ◽  
Vol 34 ◽  
pp. 89-104 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Fourier Series ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Covariance Function ◽  
Stationary Gaussian Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stationary Gaussian Processes ◽  
Necessary And Sufficient ◽  
Special Case

Let us consider a stochastically continuous, separable and measurable stationary Gaussian process X = {X(t), − ∞ < t < ∞} with mean zero and with the covariance function p(t) = EX(t + s)X(s). The conditions for continuity of paths have been studied by many authors from various viewpoints. For example, Dudley [3] studied from the viewpoint of ε-entropy and Kahane [5] showed the necessary and sufficient condition in some special case, using the rather neat method of Fourier series.

scholarly journals Functions of positive and negative type, and their connection with the theory of integral equations

1909 ◽  
Vol 83 (559) ◽  
pp. 69-70 ◽  
Keyword(s):  
Integral Equations ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Closed Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Double Integral ◽  
Positive Type ◽  
Negative Type ◽  
Necessary And Sufficient

This memoir is concerned with continuous symmetric functions k ( s, t ) for which the double integral ∫ a b ∫ a b k ( s, t ) θ ( s ) θ ( t ) dsdt is either not negative, or not positive, for each function θ( s ) which is continuous in the interval ( a, b ) ; in the former case the function k ( s, t ) is said to be of positive type, while in the latter it is said to be of negative type. The importance of these classes of functions in the theory of integral equations will be gathered from Part I. The greater portion of the second part is devoted to a proof of the theorem that the necessary and sufficient condition, under which a continuous symmetric function, k ( s, t ), is of positive type, is that the functions k ( s 1 , s 1 ), k ( s 1 , s 2 s 1 , s 2 ),....., k ( s 1 , s 2 ...., s n s 1 , s 2 , ......, s n ), ... should never be negative, when the variable s 1 , s 2 , ...., s n , .... are each confined to the closed interval ( a, b ). This leads to several interesting properties of such a function ; for instance, if k ( a 1 , a 1 ) = 0, the functions k ( s , a 1 ), k ( a 1 , t ) are identically zero.

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