An Inversion Formula for the Weierstrass Transform

1961 ◽  
Vol 13 ◽  
pp. 593-601 ◽  
Author(s):  
G. G. Bilodeau
Keyword(s):  
Inversion Formula ◽  
Basic Properties ◽  
Weierstrass Transform ◽  
Gauss Transform ◽  
Image Position

The Weierstrass transform f(x) of a function ϕ(y) is defined by1.1wherewhenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operatorwill invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definitionfor f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

Two-dimensional (P,Q)-Mellin transform and applications

2021 ◽  
pp. 2250081
Author(s):  
Pankaj Jain ◽  
Chandrani Basu ◽  
Vivek Panwar
Keyword(s):  
Integral Equations ◽  
Inversion Formula ◽  
Mellin Transform ◽  
Type Theorem ◽  
Two Dimensional ◽  
Basic Properties

In this paper, we have introduced and studied two-dimensional [Formula: see text]-Mellin transform which extends the known results of two-dimensional [Formula: see text]-Mellin transform. We provide its several basic properties, the appropriate convolution, the inversion formula and the Parseval-type relations. Some applications of [Formula: see text]-Mellin transform have been pointed out in solving integral equations. Finally, a Titchmarsh-type theorem has been proved.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

scholarly journals An Inversion Formula for the Generalised Laplace Transform

1949 ◽  
Vol 8 (3) ◽  
pp. 126-127 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Whittaker Functions ◽  
Laplace Integral ◽  
Image Position

Recently I have given a generalisation of the Laplace integralin the formwhere Wk, m (x) stands for Whittaker Functions.

Sets of Uniqueness for the Group of Integers of a p-Series Field

1979 ◽  
Vol 31 (4) ◽  
pp. 858-866 ◽  
Author(s):  
William R. Wade
Keyword(s):  
Dual Group ◽  
Negative Integer ◽  
Detailed Account ◽  
Basic Properties ◽  
Dyadic Group ◽  
Image Position ◽  
Sets Of Uniqueness ◽  
Prime 2

Let G denote the group of integers of a p-series field, where p is a prime ≦ 2. Thus, any element can be represented as a sequence {xi }i = 0∞ with 0 ≦ xi < p for each i ≦ 0. Moreover, the dual group {Ψm}m = 0∞ of G can be described by the following process. If m is a non-negative integer with for each k , and if then(1)where for each integer k ≧ 0 and for each x = {xi} ∈ G the functions Φk are defined by(2)In the case that p = 2, the group G is the dyadic group introduced by Fine [1] and the functions are the Walsh-Paley functions. A detailed account of these groups and basic properties can be found in [4].

scholarly journals Approximation by a composition of Chlodowsky operators and Százs–Durrmeyer operators on weighted spaces

2013 ◽  
Vol 16 ◽  
pp. 388-397 ◽  
Author(s):  
Aydın İzgi
Keyword(s):  
Modulus Of Continuity ◽  
Weighted Spaces ◽  
Order Of Approximation ◽  
Weighted Modulus Of Continuity ◽  
Durrmeyer Operators ◽  
Basic Properties ◽  
Weighted Modulus ◽  
Image Position

AbstractIn this paper we deal with the operators $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$. Also, we establish the order of approximation by using weighted modulus of continuity.

On the Modular Value and Fractional Part of a Random Variable

1995 ◽  
Vol 9 (4) ◽  
pp. 551-562
Author(s):  
Stephen J. Herschkorn
Keyword(s):  
Fourier Series ◽  
Inversion Formula ◽  
Distribution Density ◽  
Probabilistic Method ◽  
Fractional Part ◽  
Renewal Theory ◽  
Random Variable ◽  
General Distribution ◽  
Complex Numbers ◽  
Basic Properties

Let X be a random variable with characteristic function ϕ. In the case where X is integer-valued and n is a positive integer, a formula (in terms of ϕ) for the probability that n divides X is presented. The derivation of this formula is quite simple and uses only the basic properties of expectation and complex numbers. The formula easily generalizes to one for the distribution of X mod n. Computational simplifications and the relation to the inversion formula are also discussed; the latter topic includes a new inversion formula when the range of X is finite.When X may take on a more general distribution, limiting considerations of the previous formulas suggest others for the distribution, density, and moments of the fractional part X — [X]. These are easily derived using basic properties of Fourier series. These formulas also yield an alternative inversion formula for ϕ when the range of X is bounded.Applications to renewal theory and random walks are suggested. A by-product of the approach is a probabilistic method for the evaluation of infinite series.

GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550047 ◽  
Author(s):  
JOSÉ L. RAMÍREZ ◽  
GUSTAVO N. RUBIANO ◽  
BORUT JURČIČ ZLOBEC
Keyword(s):  
Inversion Formula ◽  
Koch Curve ◽  
Basic Properties ◽  
Sierpinski Triangle ◽  
Fractal Patterns

In this paper, we introduce the [Formula: see text]-circle inversion which generalizes the classical inversion with respect to a circle ([Formula: see text]) and the taxicab inversion [Formula: see text]. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the [Formula: see text]-circle inversion.

The fractional Hankel wavelet transformation

2015 ◽  
Vol 08 (02) ◽  
pp. 1550030 ◽  
Author(s):  
Akhilesh Prasad ◽  
K. L. Mahato
Keyword(s):  
Inversion Formula ◽  
Wavelet Transformation ◽  
Basic Properties

The main objective of this paper is to investigate the fractional Hankel wavelet transformation and to study some basic properties. An inversion formula for this fractional Hankel wavelet transformation is also obtained. Some examples of fractional Hankel wavelet transformation are given.

scholarly journals Simultaneous dual integral equations

1973 ◽  
Vol 14 (1) ◽  
pp. 73-76
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Inversion Formula ◽  
Hankel Transform ◽  
Dual Integral Equations ◽  
Image Position

Lowengrub [l] has considered simultaneous dual integral equations of the formwhere i = 1,2 …n, I1= {x:0 ≦ x x <1}, I2 {= x:0 ≦ x >1}, the cIJ are constants, the f1(x) are known functions and the functions φ(x) are to be determined.denotes the modified operator of the Hankel transform with the inversion formula

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