scholarly journals Generalized functions for applications

1985 ◽  
Vol 26 (3) ◽  
pp. 362-374 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Real Line ◽  
Generalized Functions ◽  
Generalized Function ◽  
The Real ◽  
Square Roots ◽  
Rigorous Approach ◽  
Schwartz Distributions ◽  
Delta Functions

AbstractA simple rigorous approach is given to generalized functions, suitable for applications. Here, a generalized function is defined as a genuine function on a superset of the real line, so that multiplication is unrestricted and associative, and various manipulations retain their classical meanings. The superset is simply constructed, and does not require Robinson's nonstandard real line. The generalized functions go beyond the Schwartz distributions, enabling products and square roots of delta functions to be discussed.

The theory of generalized functions

1955 ◽  
Vol 228 (1173) ◽  
pp. 175-190 ◽  
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Convergent Sequence ◽  
Generalized Functions ◽  
Generalized Function ◽  
Partial Derivatives ◽  
Full Complement ◽  
Delta Functions ◽  
Theory Of Generalized Functions

This paper gives an introductory account of the construction and properties of generalized functions f( x ) of real variables x 1 , x 2 , ..., x n . These are defined so as to ensure that (i) any generalized function f(x ) possesses its full complement of generalized partial derivatives D p f(x ) of all orders; (ii) any convergent sequence of generalized functions {f n (x)} has a generalized limit, f(x) , which is also a generalized function; (iii) the derived sequence {D p f n ( x )} converges to D p f ( x ). The construction of these generalized functions ensures that any continuous function possesses derivatives which are generalized functions, so that the delta functions of Dirac are included in the theory. The representation of generalized functions by Fourier series and integrals is considered as an example of the simplicity and generality of the theory.

GENERALIZED MEHLER SEMIGROUPS AND ORNSTEIN–UHLENBECK PROCESSES ARISING FROM SUPERPROCESSES OVER THE REAL LINE

2004 ◽  
Vol 07 (04) ◽  
pp. 591-605 ◽  
Author(s):  
ZENGHU LI ◽  
ZIKUN WANG
Keyword(s):  
Real Line ◽  
Spatial Motion ◽  
The Real ◽  
Schwartz Distributions ◽  
Dependent Spatial Motion

We study the fluctuation limits of a class of superprocesses with dependent spatial motion on the real line, which give rise to some new Ornstein–Uhlenbeck processes with values of Schwartz distributions.

Hilbert boundary value problems—a distributional approach

1977 ◽  
Vol 77 (3-4) ◽  
pp. 193-208 ◽  
Author(s):  
Marion Orton
Keyword(s):  
Boundary Value Problems ◽  
Half Space ◽  
Open Subset ◽  
Real Line ◽  
Boundary Value ◽  
Growth Conditions ◽  
The Real ◽  
Singular Support ◽  
Schwartz Distributions ◽  
Finite Set

SynopsisHilbert boundary value problems for a half-space are considered for analytic representations of Schwartz distributions: given data g ∈D'(ℛ) and a coefficient x we seek functions F(z) analytic for Jmz≠0 whose limits exist in D'(ℛ) and satisfy F+—XF– = g on an open subset U of the real line R. U is the complement of a finite set which contains the singular support and the zeros of X·X and its reciprocal satisfy certain growth conditions near the boundary points of U. Solutions F(z) are shown to exist, and their general form is determined by obtaining a suitable factorisation of x.

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

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

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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