By Elias M. Stein, Guido Weiss

The authors current a unified remedy of uncomplicated subject matters that come up in Fourier research. Their goal is to demonstrate the function performed by way of the constitution of Euclidean areas, fairly the motion of translations, dilatations, and rotations, and to encourage the examine of harmonic research on extra normal areas having the same constitution, e.g., symmetric areas.

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23(b)) #X = #(Ao U X\Ao) < #(Aox(X\Ao)) < #(A0 xA0) = #Ao. 22). 7. 0 Note that #A0 < #X (for A0 c X). Suppose #Ao < #X. In this case #A0 < #(X\Ao) by Claim 2. Thus there exists a proper subset of X\Ao, say At, such that #A0 = #A 1. 23(b)) #(A; x Aj) < #(A0 x AO) = #A0 for all combinations of i, j in (0, 1), and hence #[(Ao x A 1) U (A I X AO) U (A i x A 1)] < #Ao according to Claim 0. 6 that #[(Ao x A 1) U (A 1 x AO) U (A 1 x A 1)] = #A0. Set A = AOUAi and observe that (AxA)\(AoxA0) _ (AoxA1) U (A1 xAo) U (A1 xA1) because Ao and At are disjoint.

A) First verify that the above assertion holds whenever X and Y are both (nonempty) finite sets. Consider the relations ^-x and -y on the Cartesian product X x Y defined as follows. y (x2, y2) yl = ),-2. (b) Show that -x and -y are both equivalence relations on X x Y. Now take x E X and y E Y arbitrary and consider the equivalence classes, [x j c X x Y and [y] c X x Y, of the point (x, y) E X x Y with respect to ^-x and --y, respectively. (c) Show that [x] H Y and [y] H X. Hint: Y [x] y [y] X x Next suppose one of the sets X or Y, say X, is infinite and consider the singleton {(x, y)) on (x, y) E X X Y.

The previous proposition ensured the converse for countably infinite sets. The next theorem (which is another application of Zorn's Lemma) ensures the converse for every infinite set. Therefore, the identity #X = #(X x X) actually characterizes the infinite sets (of any cardinality) among the nonempty sets. 9. If X is an infinite set, then #X = #(X x X). Proof. First we verify the following auxiliary result. Claim 0. Let C, D and E be nonempty sets. If #(E x E) = #E, then #(C U D) < #E whenever #C < #E and #D < #E.

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Introduction to Fourier Analysis on Euclidean Spaces by Elias M. Stein, Guido Weiss
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