The measure transition problem for meromorphic polar functions
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The measure transition problem for meromorphic polar functions J. Buescu1
· A. C. Paixão2
Received: 8 July 2019 / Revised: 31 August 2020 / Accepted: 15 October 2020 © Springer Nature Switzerland AG 2020
Abstract In very general conditions, meromorphic polar functions (i.e. functions exhibiting some kind of positive or co-positive definiteness) separate the complex plane into horizontal or vertical strips of holomophy and polarity, in each of which they are characterized as integral transforms of exponentially finite measures. These measures characterize both the function and the strip. We study the problem of transition between different holomorphy strips, proving a transition formula which relates the measures on neighbouring strips of polarity. The general transition problem is further complicated by the fact that a function may lose polarity upon strip crossing and in general we cannot expect polarity, or even some specific related form of integral representation, to exist. We show that, even in these cases, a relevant analytical role will be played by exponentially finite signed measures, which we construct and study. Applications to especially significant examples like the , ζ or Bessel functions are performed. Keywords Meromorphic functions · Positive definite functions · Measure theory · Fourier transform · Laplace transform · Gamma function · Zeta function Mathematics Subject Classification Primary 30D30 · 42A38; Secondary 42A82 · 28A25 · 44A10 · 30E20 · 33B15
The first author acknowledges partial support by Fundação para a Ciência e Tecnologia, UID/MAT/04561/2019.
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J. Buescu [email protected] A. C. Paixão [email protected]
1
Dep. Matemática, FCUL and CMAFCIO, Lisboa, Portugal
2
Área Departamental de Matemática, ISEL, Lisboa, Portugal 0123456789().: V,-vol
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J. Buescu, A. C. Paixão
1 Introduction A wide range of meromorphic functions split the complex plane into disjoint horizontal or vertical strips separated respectively by poles on the imaginary or real axis, in each of which the function exhibits some kind of polarity, i.e. positive/negative definiteness or co-positive/co-negative definiteness. The most immediate class of such functions are holomorphic extensions of analytic positive definite functions, such as characteristic functions from probability theory. In each strip of holomorphy the function has a fixed polarity and there exists a unique global exponentially finite measure which determines the function by a Fourier-Laplace or Laplace-Fourier integral representation. This measure does not extend beyond the corresponding holomorphy strip, and so both the strip and the function are effectively characterized by the measure. Now, as the consideration of a significant collection of examples shows (see [18]), the meromorphic (co-)positive/(co-)negative definite functions defined on maximal horizontal (vertical) strips of holomorphy may have a rather subtle behaviour with respect to definiteness upon crossing of these strips. Indeed, they may enter a new maximal ho
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