On series representation of second order random elements and stochastic processes

1980 ◽  
pp. 318-324
Author(s):  
M. R. Żeberski
Keyword(s):  
Stochastic Processes ◽  
Series Representation ◽  
Second Order ◽  
Random Elements

Basic Fourier series

1979 ◽  
Vol 369 (1736) ◽  
pp. 115-136 ◽  
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Stochastic Processes ◽  
Numerical Approximation ◽  
Basic Number ◽  
Second Order ◽  
Orthogonal Functions ◽  
Sturm Liouville

The concept of basic number is applied to the development of a simple analogue of the Sturm–Liouville system of the second order. This is then employed to deduce a family of q -orthogonal functions, which leads to a generalization of the Fourier and Fourier–Bessel expansions. The numerical approximation of basic integrals is discussed and some aspects of the evaluation of C a (q; x) are mentioned. A few of the zeros of this function are listed, and, in conclusion, an indication is given of the possibility of applying the analysis presented in this paper to thé study of stochastic processes and time-series.

On Two Alternative Field Solutions for Second-Order Wave Diffraction

1991 ◽  
Vol 113 (3) ◽  
pp. 185-192 ◽  
Author(s):  
K. F. Cheung ◽  
M. Isaacson
Keyword(s):  
Free Surface ◽  
Nonlinear Boundary ◽  
Series Representation ◽  
Taylor Series Expansion ◽  
Perturbation Series ◽  
Two Dimensions ◽  
Second Order ◽  
Alternative Methods ◽  
Time Stepping ◽  
Alternative Field

The present paper set out a nonlinear boundary value problem involving the interaction of surface waves with large submerged structures in two dimensions. The problem is solved to second order by a time-stepping procedure on the basis of two alternative methods. Both of these involve the application of Green’s theorem, a Taylor series expansion of variables at the free surface and a perturbation series representation of the pertinent variables. Differences between the two solutions are associated with alternative integral equations applicable to second-order terms. The two second-order solutions are compared with each other and with previous theoretical and experimental results, and their applicability is assessed.

Approximate series representations of linear operations on second-order stochastic processes: application to Simulation

2006 ◽  
Vol 52 (4) ◽  
pp. 1789-1794 ◽  
Author(s):  
J. Navarro-Moreno ◽  
J.C. Ruiz-Molina ◽  
R.M. Fernandez-Alcala
Keyword(s):  
Stochastic Processes ◽  
Second Order ◽  
Series Representations

Banach space valued weak second order stochastic processes

2021 ◽  
pp. 71-96
Author(s):  
Yûichirô Kakihara
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Abelian Group ◽  
Stochastic Processes ◽  
Compact Abelian Group ◽  
Covariance Function ◽  
Second Order ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Content Type

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

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