# An Introduction to Harmonic Analysis (second corrected by Yitzhak Katznelson

By Yitzhak Katznelson

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**Example text**

The case p = 2 will be proved in the following section. The case 1 < p < 2 will be proved in chapter IV. 7 cannot be extended to p > 2. '(T) with p> 2, thenfEe(T) and consequently I < co; this is all that we can assert. There exist continuous functions f such 2 that I -. = OCJ for all 8> O. IJ(nW 1/(,,)1 An Introduction to Harmonic Analysis 26 EXERCISES FOR SECTION 4 l. } of positive numbers such that ca. - 0 as In 1_ 00. 1 and for all 11. L 2. Show that if IJ(n) I I nl 1< 00, then f is [-times continuously differentiable.

1 and the foregoing remarks, it is clearly sufficient to prove that the mappingf -T is well defined in B if, and only if. the operators S" are uniformly bounded on B. IIB ~ K. ~B ~ K. Let feB and I: > 0; let P e B be a trigonometric polynomial satisfying ~f-pIIB ~ t/2K. 9) I S~W - S~(P) liB = I S~U - P} II. ~ i· If nand m are both greater than the degree of p. (f) ~. ,(P) and 1:. W E B. t Assume conversely that f ..... f~ is well defined, hence bounded, in 8. W = f~ - i(2"+1)t(e-1I2.. +1)'l)~ S!

6) is bounded by B. 8). (f(t -I- 0) -I- /(t- 0» and in particular to J(t) at every point oj continuity. The convergence is uniform on closed intervals oj continuity oj J. 5) is valid (cf. 5). 3. Lemma: Let JELI(T) and assume LII I I dt < co. Then J(;) lim 8nCf, 0) = 0 . 9) = 1 21t f J(t) . )tdt . t sm = 2 2~f /(t)cosntdt -I- 2~ f t f(t)cos 2 -----sinntdt. t sm 2 54 An Introduction to Harmonic Analysis . fcost/2 By our assumptIon '--;--/2 SlOt E 1 T . 9) tend to zero. 4 Theorem (Principle of localization): Let feLI(T) alld assume that f vanishes ill an open interval I.