Mathematical Analysis

Download Analytic Functionals on the Sphere by Mitsuo Morimoto PDF

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By Mitsuo Morimoto

This ebook treats round harmonic enlargement of genuine analytic features and hyperfunctions at the sphere. simply because a one-dimensional sphere is a circle, the best instance of the idea is that of Fourier sequence of periodic capabilities. the writer first introduces a method of advanced neighborhoods of the field through the Lie norm. He then experiences holomorphic features and analytic functionals at the complicated sphere. within the one-dimensional case, this corresponds to the learn of holomorphic features and analytic functionals at the annular set within the advanced aircraft, hoping on the Laurent sequence growth. during this quantity, it really is proven that an identical notion nonetheless works in a higher-dimensional sphere. The Fourier-Borel transformation of analytic functionals at the sphere is additionally tested; the eigenfunction of the Laplacian should be studied during this manner.

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X/ for all x 2 X. So far, we have proved the statements (i) and (ii) of the ergodic theorem. f; x/ ! f in L1 , R R (b) X f D X f . We begin by proving a special case. We assume that f is a bounded function assuming that jf j Ä K for a constant K > 0. x/ˇ Ä K; n ˇ ˇ n and the statement (a) follows by means of the dominated convergence theorem of Lebesgue. 1 n X 44 Chapter I. fN ; x/ ! 1 almost everywhere and also the convergence in L1 . Let " > 0. fN / fN ˇˇ d m C jfN f j d m; n and we are going to estimate each term on the right-hand side by "=3 if N and n are sufficiently large.

S 1 /; and, in addition, the set RC is dense in S 1 and of second Baire category. Exercise. Every real number 0 < x Ä 1 can be represented by a unique decimal expansion X xj x D 0:x1 x2 x3 D 10j j 1 containing infinitely many non zero digits xj 2 f0; 1; 2; : : : ; 9g. Demonstrate that on the average, the number of zeros in the decimal expansion is equal to 1=10 for 40 Chapter I. Introduction almost all 0 < x Ä 1. Hint: Consider the mapping ' W S 1 ! z/ D z 10 in the fundamental domain of the covering space.

X/j dx < 1: 0 Fourier coefficients. x/e 2 i nx dx; n 2 Z: 0 Two classical statements about Fourier coefficients are the following. n/ ! 0; jnj ! n/ O for all n 2 Z H) f D g almost everywhere: In other words an element of L1 is uniquely determined by its Fourier coefficients so that the map f 7! fO is injective. S 1 /. en ; ek / D ıij . f; en /; n 2 Z: 36 Chapter I. Introduction P Fourier series. k/ek . k/e 2 ikx : jkjÄn A classical result in Hilbert space is as follows. S 1 / ! 0 for n ! f; f /1=2 is the norm in a Hilbert space.

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