The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem.

2017 ◽  
pp. 37-44
Author(s):  
Valery Serov
Keyword(s):  
Fourier Series ◽  
Integrable Function ◽  
Square Integrable Function

scholarly journals What is the Wigner Function Closest to a Given Square Integrable Function?

2018 ◽  
Vol 50 (5) ◽  
pp. 5161-5197 ◽  
Author(s):  
J. S. Ben-Benjamin ◽  
L. Cohen ◽  
N. C. Dias ◽  
P. Loughlin ◽  
J. N. Prata
Keyword(s):  
Wigner Function ◽  
Integrable Function ◽  
Square Integrable Function

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 LOG-TANGENT INTEGRALS AND THE RIEMANN ZETA FUNCTION

2019 ◽  
Vol 24 (3) ◽  
pp. 404-421
Author(s):  
Lahoucine Elaissaoui ◽  
Zine El-Abidine Guennoun
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Integrable Function ◽  
Harmonic Series ◽  
Algebraic Properties ◽  
Tangent Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Square Integrable Function ◽  
Positive Integers

We show that integrals involving the log-tangent function, with respect to any square-integrable function on (0,π/2), can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series ζh(s) :=∑n≥1hnn−s−8, where hn=∑nk=1(2k−1)−1.

On dichotomy of Riesz products

1979 ◽  
Vol 85 (1) ◽  
pp. 79-89 ◽  
Author(s):  
G. Ritter
Keyword(s):  
Integrable Function ◽  
Abelian Groups ◽  
Major Step ◽  
Riesz Product ◽  
Major Interest ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Riesz Products ◽  
Square Integrable Function ◽  
Continuous Measures

Background. Riesz products are very useful for the construction of singular measures on compact, Abelian groups. Under some circumstances, two Riesz products are either equivalent or singular in the measure-theoretic sense. Exact knowledge of these circumstances has been of major interest ever since the 1930s, when Riesz's famous example (8) was recognized as a fertile source of examples of singular continuous measures. Zygmund(11) showed that any Riesz product over a Hadamard dissociate subset of ℕ is either a square integrable function or singular with respect to Lebesgue measure. Hewitt–Zuckerman(4) generalized these products to all compact, Abelian groups, introducing the notion of a dissociate subset. They extended Zygmund's result in certain cases. The next major step was taken by Brown–Moran(3) and Peyrière(6), (7), who showed that two Riesz productsare mutually singular ifThe author (9) has improved another result of Brown–Moran (3) by showing that µa and µb are equivalent if

scholarly journals Bounded andL2-solutions to a second order nonlinear differential equation with a square integrable forcing term

1999 ◽  
Vol 22 (3) ◽  
pp. 569-571 ◽  
Author(s):  
Allan Kroopnick
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Second Order ◽  
Forcing Term ◽  
Order Nonlinear Differential Equation ◽  
Square Integrable Function

This paper presents two theorems concerning the nonlinear differential equationx″+c(t)f(x)x′+a(t,x)=e(t), wheree(t)is a continuous square-integrable function. The first theorem gives sufficient conditions when all the solutions of this equation are bounded while the second theorem discusses when all the solutions are inL2[0,∞).

scholarly journals Divergence of Fourier series

1973 ◽  
Vol 8 (2) ◽  
pp. 289-304 ◽  
Author(s):  
Masako Izuml ◽  
Shin-Ichi Izumi
Keyword(s):  
Fourier Series ◽  
Integrable Function ◽  
The Other ◽  
Other Hand ◽  
Almost Everywhere ◽  
Series Of Functions

Carleson has proved that the Fourier series of functions belonging to the class L2 converge almost everywhere.Improving his method, Hunt proved that the Fourier series of functions belonging to the class Lp (p > 1) converge almost everywhere. On the other hand, Kolmogoroff proved that there is an integrable function whose Fourier series diverges almost everywhere. We shall generalise Kolmogoroff's Theorem as follows: There is a function belonging to the class L(logL)p (p > 0) whose Fourier series diverges almost everywhere. The following problem is still open: whether “almost everywhere” in the last theorem can be replaced by “everywhere” or not. This problem is affirmatively answered for the class L by Kolmogoroff and for the class L(log logL)p (0 < p < 1) by Tandori.

On the Nörlund summability of Fourier series

1963 ◽  
Vol 59 (1) ◽  
pp. 47-53 ◽  
Author(s):  
C. T. Rajagopal
Keyword(s):  
Fourier Series ◽  
Integrable Function ◽  
Prove Theorem ◽  
Lebesgue Integrable Function ◽  
Summability Of Fourier Series ◽  
Lebesgue Set ◽  
Recent Extension

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).

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