Transform kernel

K v,t) is called the transform kernel. For the Fourier transform the kernel is e j ". Other transforms (see Section 40.8) are the Hadamard, wavelet and the Laplace transforms [4]. 

The transform kernel is a DCT with a time-shift component added ... 

It is remarkable that the integral equation (3.693) proved to be formally exact [51], so that all theories differ only in their definition of the kernel, given as E(f) or V(.v). Sometimes the kernel is explicitly defined in the original works, but more often the reduction to the integral form and extraction of the kernel is a separate problem solved in Ref. 46. In a few cases this procedure was nontrivial and required rather long and sophisticated calculations that are of no interest except for the final results represented by the Laplace transformed kernels S in Table V. 

Laplace Transforms. When Eqs. (2.16.2) and (2.16.3) are modified by using a real transform kernel exp(—kx) instead of exp(zfcx), then we have the Laplace transform ... 

It is noteworthy, that the transform kernel is represented by direct product ( ) of one-dimensional complex functions. The kernel is thus a 2 -dimensional vector. One can represent both signal s T) and spectrum S f) in a similar fashion ... 

Bieniasz LK (2011) A highly accurate, inexpensive procedure for computing integral transformation kernel and its moment integrals for cylindrical wire electrodes. J Electroanal Chem 661 280-286... 

Kinetic response of surfaces defined by finite fractals has been addressed in the context of interaction of finite time independent fractals with a time-dependent diffusion field by a novel approach of Cantor Transform that provides simple closed form solutions and smooth transitions to asymptotic limits (Nair Alam, 2010). In order to enable automatic simulation of electrochemical transient experiments performed under conditions of anomalous diffusion in the framework of the formalism of integral equations, the adaptive Huber method has been extended onto integral transformation kernel representing fractional diffusion (Bieniasz, 2011). 

Most integral transforms are special cases of the equation g(.s) = f t)K s, t) dt in which g(s) is said to be the transform ofjlt) and K(s, t) is called the kernel of the transform. A tabulation of the more important kernels and the interval (a, b) of apphcability follows. The first three transforms are considered here. 

To analyse the equations represented by (236), Gal-Or (G3) has introduced an integral transformation with a suitable kernel K fa, t) defined by... 

The integral transformation (241) with kernel (250) is seen to be accomplished by taking the Laplace transforms of Eqs. (236) with e ", where s= 1/f, and dividing the transformed quantity by t. Hence, the expected value of is simply given by... 

Thus, using the modified kernel, the set of transport equations given by (236) can be solved for any real system, and again the inverse Laplace transformation becomes an unnecessary procedure. This removes a considerable... 

Thus, a single-valued connection is established between the kernel of this equation R(t) (a memory function ) and KM(t). Their Laplace transforms R and Km are related by... 

In order to describe quantitatively the Q-branch transformation at this stage, one should use the secular simplification of the problem [133, 257]. It neglects completely the off-diagonal elements of the kernel of integral operator (6.9), i.e. terms taking into account transfer from the other branches in equation (6.4) ... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

Resorting to the definition of the Fourier transform, Eq. (2.9), we notice that for the redundant coordinates the harmonic kernel degenerates and becomes exp (0) = 1. Thus for the redundant coordinates the Fourier transform turns into a simple integration with respect to the respective reciprocal coordinate20 - a projection . 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

Support Vector Machines (SVMs) generate either linear or nonlinear classifiers depending on the so-called kernel [149]. The kernel is a matrix that performs a transformation of the data into an arbitrarily high-dimensional feature-space, where linear classification relates to nonlinear classifiers in the original space the input data lives in. SVMs are quite a recent Machine Learning method that received a lot of attention because of their superiority on a number of hard problems [150]. 

This we interpret as the kernel of an integral operator, D, which transforms an arbitrary function / on 2-space into a function D f on 2-space defined by... 

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