# An Introduction to the Uncertainty Principle: Hardy’s by Sundaram Thangavelu

By Sundaram Thangavelu

Motivating this fascinating monograph is the improvement of a couple of analogs of Hardy's theorem in settings coming up from noncommutative harmonic research. this can be the crucial subject matter of this work.

Specifically, it really is dedicated to connections between quite a few theories bobbing up from summary harmonic research, concrete challenging research, Lie thought, precise features, and the very fascinating interaction among the noncompact teams that underlie the geometric items in query and the compact rotation teams that act as symmetries of those objects.

A educational creation is given to the required history fabric. the second one bankruptcy establishes a number of types of Hardy's theorem for the Fourier remodel at the Heisenberg team and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier remodel on rank one symmetric areas is handled. lots of the effects provided listed here are legitimate within the normal context of solvable extensions of H-type groups.

The suggestions used to turn out the most effects run the gamut of contemporary harmonic research akin to illustration thought, round capabilities, Hecke-Bochner formulation and certain functions.

Graduate scholars and researchers in harmonic research will tremendously make the most of this book.

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**Extra resources for An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups**

**Example text**

So we are left with proving the case ab = The case ab = which characterises the Gaussian, can be viewed as a theorem for entire functions of order 2. In fact, the theorem amounts to saying that if F(z) is an entire function of one complex variable of order 2 and type b that decays like e-b~2 when restricted to the real line, then F(z) = c e- bz2. 1 due to Pfannschmidt [49] which uses several properties of entire functions including the notion of proximate orders. 1 and a more refined version in the higher-dimensional case.

Sin I and therefore we have sin(e - ~) cos(e - ~)} 2 2 . a +ibl~1 . a . SIn 2 SIn 2 w(~ ,ex)=exp { -bl~1 (n Let GI = (~ + i)-mG(nw(~, ex) . We observe that Iw( I~I ,ex)1 = ebl ~1 and Iw(I~leia ,ex)1 = e-bl ~l. Therefore, on the boundary of na we have IGI(nl :::: M. By the Phragmen-Lindelof maximum principle we obtain the estimate IGI (nl :::: M for 0 :::: 61 :::: ex,O < ex < tt . This simply means that I(~ + i)-mG(nl By keeping 61 fixed and letting ex :::: C exp { bl~1 ~ Sin(e-~)} . a sm 2 2 • rr we obtain I(~ +i)-mG(nl:::: C e-bl~lcos8,O:::: 61 < n which also holds for 61 = n by continuity.

Then f = 0 whenever ab > and when ab = i, i, where the functions p and q do not grow fa ster than P and Q. The subadditivity of log P and log Q allow us to estimate the Radon transform Rf and its one-dimensional Fourier tran sform. 2. 2 is still unknown. 2 are the best possible since faster growth assumptions on P and Q will make the theorem incorrect. 1 can be proved on jRn if we replace the pointwise estimates on f and by integral conditions. 2, sn - I whe re the fun ctions P and Q are such that .