Fourier Transforms
Fourier Transforms
Fourier Transforms
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FT of Higher Derivatives. The higher derivatives can be obtained in a similar manner andit is straightforward to show thatFf ′′ x −ik 2 Ffx − k 2 f̂kFf n x −ik n Ffx −ik n f̂k (3.2.4)Derivative of FT. AstheFT f k of fx is a function of k, it can be differentiated in theusual way to gived f kdk d dk −−fx e ikx dx ix fx e ikx dx−ddk fx eikx dx d f kdk iFxfx or Fxfx − i d f kdk. (3.2.5)Thus if the FT of fx is known, then the FT of xfxcan be determined by differentiation.This can easily be extended to giveF x n fx −i n d n f kdk n .Both the result in (3.2.5) and the more general result require that x n fx is absolutelyintegrable for whatever value of n is being employed. This comes with a warning: if fx isabsolutely integrable, it should not be assumed that the same property holds for x n fx.Shift Operator. Consider the function fx e icx ,wherec is a real constant and the FT offx is known.F fx e icx −fx e icx e ikx dx −fx e ikcx dx f k c . (3.2.6)The appearance of c "shifts" the value of k at which the transform is evaluated.3.2.3 Convolution of two functionsAll of the previous discussion is solely concerned with a single function and sometimes it isnecessary to consider two or more functions. This can also arise when we considerfunctions that themselves can be considered to be the products of two or more functions.Suppose that we have two functions fx and gx, both of which are bounded andabsolutely integrable. The Convolution of fx and gx is a function, written f ∗ g, definedby7