Showing papers on "Kernel (image processing) published in 1969"
69 citations
TL;DR: It is concluded that the optimum procedure is to use gaussian quadrature numerical integration to convert to a matrix equation and diagonalize the matrix with the computer program ALLMAT.
Abstract: In this paper we show that numerical and kernel expansion procedures for solving the laser mode problem do not differ in essence; both convert the integral equation into a matrix equation. Furthermore, the Fox and Li iterative method is shown to be a matrix diagonalization technique. A particular kernel expansion using Gaussian-Hermite functions is discussed, as are matrix diagonalization techniques. Numerical results are compared with other published values. We conclude that the optimum procedure is to use gaussian quadrature numerical integration to convert to a matrix equation and diagonalize the matrix with the computer program ALLMAT. This method is computationally simple and simultaneously determines many modes. Also, it is applicable to unstable and/or tilted mirror resonators with selectively coated reflectors.
58 citations
TL;DR: In this paper, a study of the values of the determining function φ as well as its derivatives and gaps in terms of the generating function f and its derivatives where f is the convolution transform of φ with variation diminishing kernel is presented.
Abstract: This is a study of the values of the determining function φ as well as its derivatives and gaps in terms of the generating function f and its derivatives where f is the convolution transform of φ with variation diminishing kernel.
6 citations
TL;DR: The Hirschman-Widder real inversion theory still holds in the sense of weak limits as discussed by the authors, which is, iff(t) belongs to a:,, where c and d depend on the kernel (or even more general spaces for those transforms whose kernels belong to class II or III).
2 citations
TL;DR: In this article, the authors derived a solution of the initial problem of linearized BOLTZMANN-VLASOV and POISSON'S equations of an isotropic, one-dimensional and nonrelativistic plasma.
Abstract: The plasma convolution equation equivalent to the initial problem of linearized BOLTZMANN-VLASOV and POISSON'S equations of an isotropic, one-dimensional and nonrelativistic plasma is derived. The integral equation obtained involves both the time t and space coordinate x.
The solution of this equation is exhibited in terms of a forcing function and a resolvent kernel. The forcing function is an exciting electric field caused by the initial disturbance of plasma equilibrium. The resolvent kernel obeys an integral kernel equation which also involves the kernel of the convolution plasma equation and the resolvent is interpreted as a plasma response to the unit impulse disturbance. The plasma kernel is directly expressed by the equilibrium velocity distribution. The plasma response and the equilibrium distribution are related in the same way as the response of a transmission-line is related to its transfer function and the integral-kernel equation plays a key role in the formulation of problems of plasma analysis and synthesis. To analyse plasma, is to determine its electric response for a given equilibrium velocity distribution of plasma components and to synthesize plasma is to design an equilibrium distribution of plasma components for a required plasma response.
In this paper the plasma analysis is carried out and the example of Lorentzian and Maxwellian plasma equilibrium distributions are considered. The plasma convolution equation equivalent to the one-point boundary problem, (x = 0), of linearized BOLTZMANN-VLASOV and POISSON'S equations is presented. The absence of discrete spectrum solutions is evident from the plasma convolution equations but the oscillation properties of plasma are preserved for the examples considered. The complete analytical results are obtained for a Lorentzian plasma. The electric response of the plasma is shown to be a temporal oscillation travelling with the velocity U = x/t and the amplitude of the travelling oscillation depending on U and decaying as t−1. For a Maxwellian plasma the asymptotic expansion of the electric response is obtained for short and long times and a fixed x. The result for the short-time limit is the same as in the Lorentzian plasma case and for the long time approximation the plasma response is shown to be an oscillation decaying as t−1.
2 citations
01 Jan 1969
1 citations
TL;DR: In this paper, the boundedness of certain operators on L(R) was studied and it was shown that these operators are too singular to be handled by the ordinary Calderón-Zygmund techniques of [ l ].
Abstract: In this note, we state some results on the boundedness of certain operators on L(R). The operators which we study are too singular to be handled by the ordinary Calderón-Zygmund techniques of [ l ] . Our first theorem concerns a sublinear operator g\\* which arises in Littlewood-Paley theory. If ƒ is a real-valued function on R, set u(x, t) equal to the Poisson integral of/, defined on i£+ = R X (0, «> ). Then for X> 1, the gx*-function on R is defined by the equation