COMMUTATIVE NEUTRIX CONVOLUTION PRODUCT OF GENERALIZED FRESNEL COSINE INTEGRALS AND APPLICATIONS

2021 ◽  
Vol 3 (2) ◽  
pp. 83-83–91
Keyword(s):  
Real Line ◽  
Convolution Product ◽  
The Real ◽  
Associated Functions

The generalized Fresnel cosine integral $C_k(x)$ and its associated functions $C_{k+}(x)$ and $C_{k-}(x)$ are defined as locally summable functions on the real line. The generalized Fresnel cosine integrals have huge applications in physics, specially in optics and electromaghetics. In many diffraction problems the generalized Fresnel integrals plays an important role. In this paper are calculated the commutative neutrix convolutions of the generalized Fresnel cosine integral and its associated functions with $x^r, r=0,1,2,\dots$.

scholarly journals On the Fresnel sine integral and the convolution

2003 ◽  
Vol 2003 (37) ◽  
pp. 2327-2333 ◽  
Author(s):  
Adem Kılıçman
Keyword(s):  
Real Line ◽  
The Real ◽  
Associated Functions

The Fresnel sine integralS(x), the Fresnel cosine integralC(x), and the associated functionsS+(x), S−(x), C+(x), andC−(x)are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel sine integral and its associated functions withx+r, xrare evaluated.

scholarly journals On the sine integral and the convolution

2002 ◽  
Vol 30 (6) ◽  
pp. 365-375
Author(s):  
Brian Fisher ◽  
Fatma Al-Sirehy
Keyword(s):  
Real Line ◽  
The Real ◽  
Associated Functions

The sine integralSi(λx)and the cosine integralCi(λx)and their associated functionsSi+(λx),Si−(λx),Ci+(λx),Ci−(λx)are defined as locally summable functions on the real line. Some convolutions of these functions andsin(μx),sin+(μx), andsin−(μx)are found.

scholarly journals Further Results on the Dilogarithm Integral

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Biljana Jolevska-Tuneska ◽  
Brian Fisher
Keyword(s):  
Real Line ◽  
The Real ◽  
Associated Functions

The dilogarithm integral Li(xs) and its associated functionsLi+(xs)andLi-(xs)are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.

scholarly journals Dual Convolution for the Affine Group of the Real Line

2021 ◽  
Vol 15 (4) ◽  
Author(s):  
Yemon Choi ◽  
Mahya Ghandehari
Keyword(s):  
Real Line ◽  
Banach Algebras ◽  
Trace Class ◽  
Affine Group ◽  
Convolution Product ◽  
Fourier Algebra ◽  
Algebra Structure ◽  
Connected Component ◽  
The Real ◽  
Trace Class Operators

AbstractThe Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $$L^2({{\mathbb {R}}}^\times , dt/ |t|)$$ L 2 ( R × , d t / | t | ) . In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $$L^p({{\mathbb {R}}}^\times , dt/ |t|)$$ L p ( R × , d t / | t | ) for $$p\in (1,2)\cup (2,\infty )$$ p ∈ ( 1 , 2 ) ∪ ( 2 , ∞ ) .

scholarly journals On the Fresnel integrals and the convolution

2003 ◽  
Vol 2003 (41) ◽  
pp. 2635-2643 ◽  
Author(s):  
Adem Kiliçman ◽  
Brian Fisher
Keyword(s):  
Real Line ◽  
The Real ◽  
Associated Functions

The Fresnel cosine integralC(x), the Fresnel sine integralS(x), and the associated functionsC+(x),C−(x),S+(x), andS−(x)are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel cosine integral and its associated functions withx+randxrare evaluated.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

AN OSCILLATION FUNCTION ON THE REAL LINE

2000 ◽  
Vol 26 (1) ◽  
pp. 237
Author(s):  
Duszyński
Keyword(s):  
Real Line ◽  
The Real

DERIVATION BASES ON THE REAL LINE (I)

1982 ◽  
Vol 8 (1) ◽  
pp. 67 ◽  
Author(s):  
Thomson
Keyword(s):  
Real Line ◽  
The Real

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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