Fourier Analysis is the Cell theory of True Calculus #43 Uni-text 8th

Alright, this really belongs in Advanced Calculus (graduate school math) but here I am just going to mention some facts and details, for I never want to frighten students, in that math requires age to learn it. We cannot learn to read a clock at too young of an age but at a prime age, we pick it up almost instantly. Same thing for most of mathematics.

Now I must not excuse myself for being rather slow in realizing what the general formula other than the Power formula with its consequence of Chain rule, was for True Calculus. The general formula for all derivatives and integrals of functions is the Fourier Analysis method. All functions, I mean all, can be made into sine and cosine summations. In True Calculus all functions are continuous because of the gap or hole between finite numbers allows all functions to be continuous and when we divide by 0 we provide the function a patch over that division so as to force the function continuous.

So why was I slow in recognizing the Cell theory had a general method, more universal than the Power formula with it's consequence of the Chain Rule? Why was I slow to realize that the dy and the dx is naturally sine and cosine and that my new method of deriving the general formula of derivative and integral would be a Fourier transform? Why was I slow? I was slow because I always feel that pure geometry of lines and angles-- Power formula of y = x, is more basic and primal than is the trigonometry functions which build upon pure geometry. Somehow I refused to rise above pure geometry of the triangle and go to the levels higher of trigonometry to find that Universal Generalized Method of deriving the derivative and integral.

But now that I am here, and happy, let me just tell what the Fourier Analysis proof involves in proving that great theorem that all functions are summations of sine and cosine wavelets. For in the Fourier Analysis starts with Fourier Series and the series is built upon a *interval* and that interval is a uniform interval such as what Wikipedia describes as [x0, x0 + P]. In other words, Fourier Analysis is built upon the Cell theory of Calculus.

The Cell theory is built from the fact that finite and infinity require a borderline and that produces a **interval** of microinfinity, the holes and gaps between successive finite numbers. So when we fetch the macroinfinity border, we cause the existence of the microinfinity border and that creates uniform small intervals. That allows the existence of the Fourier Series of its interval.

Now out of curiosity, I wanted to see if any of my authors of Stewart, Strang, Ellis & Gulick, Fisher & Ziebur discuss the Fourier Analysis. I would have guessed the answer beforehand to be absolutely no.
But to my surprise, Stewart mentions the Fourier series on page 489 in his 2003, 5th ed. book. Strang goes really strong on the Fourier Analysis and let me quote his page 291 of his 1991 text Calculus:

--- quoting Strang page 291 ---
This is not for the sake of making up new problems. I believe these are the most important definite integrals in this chapter ( p and q are 0, 1, 2, ....). They may be the most important in all of mathematics, especially because the question has such a beautiful answer. The integrals are zero. On that fact rests the success of Fourier series, and the whole industry of signal processing.
--- end quote ---

Strang is skewed in his feelings about those integrals because Strang is in Old Math with its pollution of the phony limit concept and thus, nothing is really pretty or beautiful when surrounded by pollution.

What is beautiful is that the Fourier Analysis is all coming out of the Cell theory.

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Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium