Shifting theorem

Figure 2 Flow diagram of the DHT with N=8, P=3. Broken lines represent transfer factors -1 while full lines represent unity transfer factor. The crossover boxes perform the sign reversal called for by the shift theorem which also requires the sine and cosine factors Sn, Cn.

To derive the decomposition formula we require two theorems, the shift theorem and the similarity theorem. The shift theorem states that if/(r) has DHT H(v) then/(r+ a) has DHT... 

Selected entries from Methods in Enzymology [vol, page(s)] Application in fluorescence, 240, 734, 736, 757 convolution, 240, 490-491 in NMR [discrete transform, 239, 319-322 inverse transform, 239, 208, 259 multinuclear multidimensional NMR, 239, 71-73 shift theorem, 239, 210 time-domain shape functions, 239, 208-209] FT infrared spectroscopy [iron-coordinated CO, in difference spectrum of photolyzed carbonmonoxymyo-globin, 232, 186-187 for fatty acyl ester determination in small cell samples, 233, 311-313 myoglobin conformational substrates, 232, 186-187]. 

A proof of this relation may be found in Bracewell (1978). Note that the spectral variable used in this and the next chapter is the same as that defined in Eqs. (7) and (8). Now consider a spatial distribution /(x) and its Fourier spectrum F(w) that come close to satisfying the equality in Eq. (4). We may take Ax and Aw as measures of the width, and hence the resolution, of the respective functions. To see how this relates to more realistic data, such as infrared spectral lines, consider shifting the peak function /(x) by various amounts and then superimposing all these shifted functions. This will give a reasonable approximation to a set of infrared lines. To discuss quantitatively what is occurring in the frequency domain, note that the Fourier spectrum of each shifted function by the shift theorem is given simply by the spectrum of the unshifted function multiplied by a constant phase factor. The superimposed spectrum would then be... 

The second-shifting theorem, which permits the mathematical treatment of lag or dead time, is defined in the following manner ... 

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

Now in view of the shift theorem for Fourier transforms as applied to one-sided Fourier transforms [38], namely. 

The complex susceptibility of a ferrofluid in a weak applied field may be written directly from Eqs. (109) and (110) and the Langevin equations (98) and (99) [taking note of Eq. (102)] using the shift theorem for one-sided Eourier transforms, Eq. (30). Thus... 

Hence using the shift theorem for Fourier transforms... 

Using the shift theorem [to -> (to — ajls)] in the latter equation, we have... 

There are several theorems that are useful in obtaining Laplace transforms of various functions. The first is the shifting theorem-. 

Consider a solution where only O is present initially at a concentration Cq, semi-infinite linear diffusion conditions prevail, and a constant cathodic current i is applied for a time t (where t with ti being the forward transition time). At ti the current is reversed, that is, the direction of the current is changed from cathodic to anodic, so that R formed during the forward step is oxidized to O, and the time T2 (measured from ti) at which Cr at the electrode surface drops to zero is noted. At T2, the reverse transition time, the potential shows a rapid change toward positive values. We desire an expression for T2- This is most easily accomplished using the zero shift theorem of Section A. 1.7. Since for 0 < / < i(t) = i, and for ti < t t + T2, i(t) = —i, the expression for the current, using step function notation, is... 

The treatment involving current reversal employs the same equations and utilizes the zero-shift-theorem method, as in Section 8.4.2. Thus, for reversal of current at time t (where ti < ti). 

Once the boundary conditions are written out, they must usually be transformed. The zero-shift theorem provides the necessary basis. It is summarized by... 

Equation A. 1.63 is called the zero-shift theorem because it shows that multiplication by in transform space corresponds to a shift in the real time axis by an amount k. This effect is shown in Figure A. 1.2 for the simple function F(t) = 2t... 

The Fourier transform of (7.50) is obtained with the help of the shift theorem (see Appendix B), which shows that the Fourier transform of 8(—z — t) is equal to exp(z ) times the Fourier transform of < (—z). We thus obtain... 

The shift theorem states that if the function f(t) has the Fourier transform F(w), then the function f(t-a) has the transform F(uj) exp(-iu)a). Its derivation is also quite simple. We go through it here for illustration. 

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