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

By Yitzhak Katznelson

ISBN-10: 0486633314

ISBN-13: 9780486633312

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Extra info for An Introduction to Harmonic Analysis (second corrected edition)

Example text

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. Hence, if ](11)= O(jllr k ) where k > 2. and if 1- '{k-2 [k]-l k integer otherwise then fis [-times continuously differentiable.

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J) = lim - j sgn (j)l. -+00 1-+ P = tl(O) + ·HI + if) '" L j(j) eiJt o = lU) . II. 'the Convergence of Fourier Series 49 is a well-defined. bounded linear operator on B. Conversely. if the mapping f -TP is well-defined in a space B. then B admits conjuga. tion since 1 = - i(2P - f - 1(0». Theorem: Let B be a homogeneous Banach space on T and assume that for feB and for all n. 7) Then B admits conjugation if, and only if. B admits convergence in norm. 1 and the foregoing remarks, it is clearly sufficient to prove that the mappingf -T is well defined in B if, and only if.

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An Introduction to Harmonic Analysis (second corrected edition) by Yitzhak Katznelson


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