On Density of Fourier Coefficients

1973 ◽  
Vol 16 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Rafat N. Siddiqi
Keyword(s):  
Fourier Series ◽  
Total Variation ◽  
Bounded Variation ◽  
Fourier Coefficients ◽  
Function Of Bounded Variation ◽  
Fourier Sine ◽  
Image Position

Letfbe anLintegrable real valued function of period 2π and let(1)be its Fourier series. It is known that iffis of bounded variation then allnanandnbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] whereV(f) is the total variation off. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.

scholarly journals On the Fourier constants of a function

1911 ◽  
Vol 85 (575) ◽  
pp. 14-24
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Fourier Coefficients ◽  
General Term ◽  
Summable Function ◽  
Function Of Bounded Variation ◽  
Internal Point ◽  
General Sense ◽  
Definition Of ◽  
Unique Limit

1. Considerable progress has been made lately in the study of the properties of the constants in a Fourier series, using this term in the most general sense possible consistent with the extended definition of integration due to Lebesgue. Thus we now know that these coefficients necessarily under all circumstances have the unique limit zero, as the integer denoting their place in the series increases indefinitely, and that the same is true if we substitute for that integer any other quantity which increases without limit. Further, we know that the series whose general term is b n / n , where b n is the typical coefficient of the sine terms, always converges, and we are able to write down its sum. That the series whose general term is a n , where a n is the typical coefficient of the cosine terms, converges when the origin is an internal point of an interval throughout which the function has bounded variation, and that accordingly the series whose general term is a n / n q , (0< q ), converges, is an immediate consequence of known results. Should the function have its square summable, we know that the series whose general term is ( a n 2 + b n 2 ) converges, and we can write down its sum. We can also sum the series of the products of the Fourier coefficients of two such functions. From the property that Ʃ ( a n 2 + b n 2 ) converges, we can deduce that the series Ʃ a n / n q and Ʃ b n / n q , (½< q ) necessarily converge absolutely. Again, making use of a theorem recently proved, we may integrate the Fourier series of any summable function, after multiplying it term by term by any function of bounded variation, with a certainty that we shall obtain the same result as if the Fourier series converged to the function to which it corresponds, and such term-by-term integration were allowable.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals Fourier series with small gaps

1989 ◽  
Vol 46 (2) ◽  
pp. 212-219
Author(s):  
P. Isaza ◽  
D. Waterman
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Bounded Variation ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Order Condition ◽  
Circle Group ◽  
The Difference ◽  
Bounded Below

AbstractA trigonometric series has “small gaps” if the difference of the orders of successive terms is bounded below by a number exceeding one. Wiener, Ingham and others have shown that if a function represented by such a series exhibits a certain behavior on a large enough subinterval I, this will have consequences for the behavior of the function on the whole circle group. Here we show that the assumption that f is in any one of various classes of functions of generalized bounded variation on I implies that the appropriate order condition holds for the magnitude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmundtype condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

On the Fourier coefficients of a discontinuous function

1941 ◽  
Vol 6 (4) ◽  
pp. 231-256 ◽  
Author(s):  
S. P. Bhatnagar
Keyword(s):  
Fourier Series ◽  
Fourier Coefficients ◽  
Discontinuous Function ◽  
Image Position

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).We writeand suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).

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.

On absolute summability factors of infinite series and their application to Fourier series

1967 ◽  
Vol 63 (1) ◽  
pp. 107-118 ◽  
Author(s):  
R. N. Mohapatra ◽  
G. Das ◽  
V. P. Srivastava
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
Summability Factors ◽  
Partial Sum ◽  
Absolute Summability Factors ◽  
Image Position

Definition. Let {sn} be the n-th partial sum of a given infinite series. If the transformationwhereis a sequence of bounded variation, we say that εanis summable |C, α|.

On the absolute Nörlund summability of a Fourier series

1967 ◽  
Vol 7 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let Σn−0∞an, be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us writeIfas n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

On Measures Determined by Continuous Functions that are not of Bounded Variation

1970 ◽  
Vol 13 (1) ◽  
pp. 121-124 ◽  
Author(s):  
J. H. W. Burry ◽  
H. W. Ellis
Keyword(s):  
Continuous Function ◽  
Total Variation ◽  
Bounded Variation ◽  
Real Line ◽  
Open Interval ◽  
Continuous Functions ◽  
Outer Measure ◽  
Function Of Bounded Variation ◽  
Borel Sets ◽  
Nondifferentiable Function

In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.

Curvature and Radius of Curvature for Functions with Bounded Boundary Rotation

1973 ◽  
Vol 25 (5) ◽  
pp. 1015-1023 ◽  
Author(s):  
J. W. Noonan
Keyword(s):  
Bounded Variation ◽  
Radius Of Curvature ◽  
Function Of Bounded Variation ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position ◽  
Class Of Functions

For k ≧ 2 denote by Vk the class of functions f regular in and having the representation(1.1)where μ is a real-valued function of bounded variation on [0, 2π] with(1.2)Vk is the class of functions with boundary rotation at most kπ.

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