338:
If f is a mean periodic function, then it is the limit of a certain sequence of exponential polynomials which are finite linear combinations of term t^^n exp(at) where n is any non-negative integer and a is any complex number; also Df is a mean periodic function (ie mean periodic) and if h is an
171:
348:
For linear differential difference equations such as Df(t) - af(t - b) = g where a is any complex number and b is a positive real number, then f is mean periodic if and only if g is mean periodic.
342:
If f and g are mean periodic then f + g and the truncated convolution product of f and g is mean periodic. However, the pointwise product of f and g need not be mean periodic.
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330:
bounded mean-periodic function is almost periodic (in the sense of Bohr). In the other direction, there exist almost periodic functions which are not mean-periodic.
261:
228:
194:
364:
have been suggested to correspond to automorphicity of the related L-function. There is a certain class of mean-periodic functions arising from number theory.
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If L(D) is a linear differential operator with constant co-efficients, and L(D)f = g, then f is mean periodic if and only if g is mean periodic.
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602:
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326:, but they are not almost periodic as they are unbounded. Still, there is a theorem which states that any
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360:, the mean-periodicity of certain (functions related to) zeta functions associated to an
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Mean-periodic functions are a separate generalization of periodic functions from the
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exponential polynomial, then the pointwise product of f and h is mean periodic).
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is some arbitrary nonzero measure with compact (hence bounded) support.
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for which there exists a compactly supported (signed) Borel measure
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is mean-periodic if it satisfies the same equation (1), but where
310:. For instance, exponential functions are mean-periodic since
393:
Delsarte, Jean (1935). "Les fonctions moyenne-périodiques".
166:{\displaystyle \int f(x-t)\,d\mu (t)=0\qquad \qquad (1)}
445:"Fonctions moyenne-périodiques (d'après J.-P. Kahane)"
298:
There are several well-known equivalent definitions.
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182:
108:
473:"Théorie générale des fonctions moyenne-périodiques"
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429:. Tata Institute of Fundamental Research, Bombay.
237:, so that a mean-periodic function is a function
519:Journal of the Australian Mathematical Society
8:
515:"Some properties of mean periodic functions"
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395:Journal de Mathématiques Pures et Appliquées
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24:is a generalization introduced in 1935 by
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181:
130:
107:
385:
557:"Mean-periodicity and zeta functions"
302:Relation to almost periodic functions
233:Equation (1) can be interpreted as a
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423:Lectures on Mean Periodic Functions
555:; Ricotta, G.; Suzuki, M. (2012).
14:
153:
152:
32:. Further results were made by
196:is the difference between the
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143:
137:
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115:
1:
561:Annales de l'Institut Fourier
619:
70:precisely if for all real
531:10.1017/s1446788700011058
374:almost periodic functions
308:almost periodic functions
99:. This can be written as
358:Langlands correspondence
288:{\displaystyle f*\mu =0}
200:at 0 and
64:is periodic with period
58:variable. The function
356:In work related to the
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257:
224:
190:
167:
44:Consider a continuous
22:mean-periodic function
603:Mathematical analysis
513:Laird, P. G. (1972).
334:Some basic properties
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258:
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191:
168:
18:mathematical analysis
328:uniformly continuous
267:
256:{\displaystyle \mu }
247:
223:{\displaystyle \mu }
214:
189:{\displaystyle \mu }
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28:of the concept of a
20:, the concept of a
452:Séminaire Bourbaki
441:Malgrange, Bernard
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163:
469:Schwartz, Laurent
362:arithmetic scheme
48:-valued function
30:periodic function
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583:10.5802/aif.2737
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567:(5): 1819–1887.
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204:. The function
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36:and J-P Kahane.
34:Laurent Schwartz
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525:(4): 424–432.
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486:(2): 857–929.
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454:(97): 425–437.
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198:Dirac measures
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418:Kahane, J.-P.
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26:Jean Delsarte
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480:Ann. of Math
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352:Applications
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553:Fesenko, I.
235:convolution
401:: 403–453.
380:References
263:for which
76:, we have
40:Definition
574:0803.2821
539:0004-9735
277:μ
274:∗
251:μ
218:μ
184:μ
135:μ
122:−
110:∫
597:Category
471:(1947).
443:(1954).
420:(1959).
368:See also
500:1969386
46:complex
537:
498:
316:+1) −
176:where
569:arXiv
496:JSTOR
476:(PDF)
448:(PDF)
427:(PDF)
324:) = 0
320:.exp(
97:) = 0
54:of a
535:ISSN
312:exp(
85:) −
56:real
579:doi
527:doi
488:doi
16:In
599::
577:.
565:62
563:.
559:.
533:.
523:14
521:.
517:.
494:.
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460:^
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93:−
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490::
322:x
318:e
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283:0
280:=
271:f
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155:(
150:0
147:=
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141:t
138:(
132:d
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125:t
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116:(
113:f
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89:(
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81:(
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67:a
61:f
51:f
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