Talk:Sine and cosine transforms

Cosine Transform (CT) has been underestimated. Since virtually all physical quantities do not actually require description by negative values but each of them has a natural zero, CT is the tailor made transform if one asks for primary physical reletionships, not the complex-valued Fourier transform (FT). There is no negative distance. Radius, elapsed time, energy, frequency, temperature, charge of an electron, mass, wave length, etc. are also always positive. Cartesian coordinates and the ordinary time demand arbitrary reference points. This redundant information on linear phase is the only one that makes the FT of a function of elapsed time more 'complete' as compared to CT by various immediate and subsequent notorius worries: required double redundancy, non-causality, ambiguity, misinterpretation of apparent symmetry, the need for arbitrary windows in signal processing, etc. Our ears have no knownledge of the zero of time exactly related to Christ's birth midnight New Year in Greenwich. They just relate to the very moment. Performing CT they are able to add a one-way rectification to the motion of basilar membrane according to CT. This would be impossible with FT. Therefore we could not distiguish by ear between rarefaction and condensation clicks. Fortunately, hearing is still much better than the FT and Ohm's law of acoustics allow. What about the outcome of the second expensive in history physical experiment, I am curious. If Higg's boson is an artifact of improper use of FT, then it will never be found.

For more details see: http://iesk.et.uni-magdeburg.de/~blumsche/M283.html The paper "Adaptation of spectral analysis to reality" has been amended following suggestions by R. Fritzius. The old version is available via IEEE.

Blumschein 13:25, 10 September 2007 (UTC)[reply]

Yes, I think you are right (apart from what you speculate about the Higgs). However, unless you can produce (other) mainstream publications beside your own that support your view, the statement constitutes original research and cannot be included in the article.--149.217.1.6 (talk) 22:32, 14 January 2009 (UTC)[reply]

Symmetry around the origin?

Integrating from -inf to inf can't be considered symmetric, because the origin can be arbitrarily chosen. So unless I misunderstand something, the ground of the discussion is not sound. —Preceding unsigned comment added by Michael Litvin (talk • contribs) 21:26, 26 November 2010 (UTC)[reply]

Restricting the Integration Range to the Positive Real Axis

It appears to me that the integration range of the cosine transform should run from $-\infty$ to $\infty$ for general functions. In the post above from November 2010 Michael Litvin already pointed out that the symmetry applies only to particular functions. To verify this simply assume the function
$f(x)=\left\{{\begin{array}{cc}0&x>0\\\exp(x)&x<0\end{array}}\right.$ .
Obviously the fourier integral over the positive half axis is zero, whilst over the negative half axis it is nonzero.

A short research on google books shows that many authors are rather sloppy regarding the integration range. A reference where it is stated correctly is Wolfram Mathworld: http://mathworld.wolfram.com/FourierCosineTransform.html

--Pia novice (talk) 13:48, 18 July 2012 (UTC)[reply]