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Bernstein's Analytic Continuation of Complex Powers of Polynomials Bernsteins analytic continuation of complex p owers c Paul Garrett garrettmathumnedu version January Analytic continuation of distributions Statement of the theorems on analytic continuation Bernsteins pro ofs Pro of of the Lemma the Bernstein p olynomial Pro of of the Prop osition estimates on zeros Garrett Bernsteins analytic continuation of complex p owers Let f b e a p olynomial in x x with real co ecients For complex s let n s f b e the function dened by s s f x f x if f x s f x if f x s Certainly for s the function f is lo cally integrable For s in this range s we can dened a distribution denoted by the same symb ol f by Z s s f x x dx f n R n R the space of compactlysupp orted smo oth realvalued where is in C c n functions on R s The ob ject is to analytically continue the distribution f as a meromorphic distributionvalued function of s This typ e of question was considered in several provo cative examples in IM Gelfand and GE Shilovs Generalized Functions volume I One should also ask ab out analytic continuation as a temp ered distribution In a lecture at the Amsterdam Congress IM Gelfand rened this question to require further that one show that the p oles lie in a nite numb er of arithmetic progressions Bernstein proved the result in under a certain regularity hyp othesis on the zeroset of the p olynomial f Published in Journal of Functional Analysis and Its Applications translated from Russian The present discussion includes some background material from complex function theory and from the theory of distributions Garrett Bernsteins analytic continuation of complex p owers 1 Analytic continuation of distributions First we recall the nature of the top ologies on test functions and on distributions Let C U b e the collection of compactlysupp orted smo oth functions with c n supp ort inside a set U R As usual for U compact we have a countable family of seminorms on C U c f sup jD f j x where for we write as usual n 1 n D x x n It is elementary to show that for U compact C U is a complete lo cally c convex top ological space a Frechet space To treat U not necessarily compact n eg R itself let U U b e compact subsets of U so that their union is all of U Then C U is the c union of the spaces C U and we give it with the lo cally convex direct limit i c top ology n n The spaces D U and D R of distributions on U and on R resp ec n n R For U and D R C tively are the continuous duals of D U C c c present purp oses the top ology we put on the continuous dual space V of a top ological vectorspace V is the weak top ology a subbasis near in V is given by sets U f V jv j g v In this context a V valued function f on an op en subset of C is holo morphic on if for every v V the Cvalued function z f z v on is holomorphic in the usual sense This notion of holomorphy might b e more p edantically termed weak holomorphy since reference to the top ology might b e required If z for an op en subset of C and f is a holomorphic V valued o function on z say that f is weakly meromorphic at z if for every o o v V the Cvalued function z f z v has a p ole as opp osed to essential singularity at z Say that f is strongly meromorphic at z if the orders of o o these p oles are b ounded indep endently of v That is f is strongly meromorphic Garrett Bernsteins analytic continuation of complex p owers at z if there is an integer n and an op en set containing z so that for all o o v V the Cvalued function n z z z f z v o n is holomorphic on If n is the least integer f so that z z f is holomorphic o at z then f is of order n at z etc o o To say that f is strongly meromorphic on an op en set is to require that there b e a set S of p oints of with no accumulation p oint in so that f is holomorphic on S and so that f is strongly meromorphic at each p oint of S For brevity but risking some confusion we will often say meromorphic instead of strongly meromorphic 2 Statement of the theorems on analytic continua tion n Let O b e the p olynomial ring Rx x For z R let O b e the lo cal n z ring at z ie the ring of ratios P Q of p olynomials where the denominator do es not vanish at z Let m b e the maximal ideal of O consisting of elements z z of O whose numerator vanishes at z Let I dep ending up on f b e the ideal z z in O generated by z f f x x n n A p oint z R is simple with resp ect to the p olynomial f if f z N for some N we have I m z z P f There are m so that f i i z i x i Remarks The second condition is equivalent to the assertion that O I is z z nitedimensional The simplest situation in which the second condition holds is when I O ie some partial derivative of f is nonzero at z The third z z condition do es not follow from the rst two For example Bernstein p oints out that with f x y x y x y the rst two conditions hold but the third do es not Theorem lo cal version If z is a simple p oint with resp ect to f then there is a neighb orho o d U of x so that the distribution Z s s f f x x dx U Garrett Bernsteins analytic continuation of complex p owers on test functions C U on U has an analytic continuation to a meromor c phic element in the continuous dual of C U c Theorem global version If all real zeros of f x are simple with resp ect s to f then f is a meromorphic distributionvalued function of s C 3 Bernsteins pro of Let R b e the ring of linear dierential op erators with co ecients in O Note z z that R is b oth a left and a right O mo dule for D R for f g O and a z z smo oth function near z the denition is f D g f D g Lemma There is a dierential op erator D R and a nonzero Bernstein z p olynomial H in a single variable so that n n D f H nf for any natural numb er n Pro of b elow Pro of of Lo cal Theorem from Lemma Let U b e a smallenough neighb or ho o d of z so that on it all co ecients of D are holomorphic on U For suciently n is continuously dierentiable so we large natural numb ers n the function f have n n D f H nf For each xed C U consider the function c s s g s D f H sf The hyp otheses of the prop osition b elow are satised so the equality for all largeenough natural numb ers implies equality everywhere s s D f H sf This gives us s D f s f H s s Now we claim that for any n Z the distribution f on C U is c meromorphic for s n For n this is certain The formula just derived then gives the induction step Further this argument makes clear that s the p oles of f restricted to C U are concentrated on the nite collection c of arithmetic progressions i i i Garrett Bernsteins analytic continuation of complex p owers s where the are the ro ots of H s In particular the order of the p ole of f at i a p oint s is equal to the numb er of ro ots so that s lies among o i o i i i s In particular the distribution f really is strongly meromorphic This proves the Theorem granting the Lemma and granting the Prop osition Prop osition attributed by Bernstein to Carlson If g is an analytic cs function for s and jg sj be and if g n for all suciently large natural numb ers n then g Pro of of Global Theorem Invoking the Lo cal Theorem and its pro of ab ove n s for each z R we cho ose a neighb orho o d U of z in which f is meromorphic z so that U is Zariskiop en ie is the complement of a nite union of zero sets z of p olynomials Indeed writing X f f x i x i with O as in the pro of of the Lo cal Theorem let g h with i z i i i p olynomials g and h and take U to b e the complement of the union of the i i z zerosets of the denominators h i n Then Hilb erts Basis Theorem implies that the whole R is covered by of these Zariskiop ens Then make a partition of U nitelymany U z z n 1 unity sub ordinate to this nite cover ie take so that n i P Then and spt U i i z i X s s f f i i the righthand side is a nite sum of mero By choice of the neighb orho o ds U z i morphic distributionvalued functions 4 Pro of of the Lemma the Bernstein p olynomial Now we prove existence of the dierential op erator D and the Bernstein p oly nomial H This is the most serious part of this pro of The complex function theory prop osition is not entirely trivial but is approximately standard Pro of of Lemma Let X R P z i x i where the m are so that i z X f R f z i x i Garrett Bernsteins analytic continuation of complex p owers Also put f S P f i x x i i f P Q x x i i Then we have X f f P f i x i i f f S f f f i x x i i Thus by Leibniz formula n n n P f nf S f i Sublemma There is a nonzero p olynomial M in one variable so that M P can b e written in the form X f M P J i x i i for some J R i z Pro of of Sublemma Write as usual j j n For a natural numb er m write X m D P m j jm where O That is we move all the co ecients to the right of the m z dierential op erators That this is p ossible is easy to see for example x x i i x x j j is or as i j or not Further the co ecients are p olynomials in the Thus m i j j m m z Then taking M P of the form X X m M P b P D b m m m m mq Garrett Bernsteins analytic continuation of complex p owers with b R the condition of the sublemma will b e met if m X b I m m z m N for all indices If j j N where I m then this condition is automatically fullled Thus there are nitelymany conditions X b m m m N where is the image of in O m Since the latter quotient
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