A problem on completeness of exponentials

Abstract

Let $\mu$ be a finite positive measure on the real line. For $a>0$, denote by $\mathcal{E}_a$ the family of exponential functions $$\mathcal{E}_a=\{e^{ist}| s\in[0,a]\}.$$ The exponential type of $\mu$ is the infimum of all numbers $a$ such that the finite linear combinations of the exponentials from $\mathcal{E}_a$ are dense in $L^2(\mu)$. If the set of such $a$ is empty, the exponential type of $\mu$ is defined as infinity. The well-known type problem asks to find the exponential type of $\mu$ in terms of $\mu$. In this note we present a solution to the type problem and discuss its relations with known results.

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      YEAR = {1993},
      NUMBER = {2},
      PAGES = {189--210},
      ISSN = {0234-0852},
      MRCLASS = {30D55},
      MRNUMBER = {1223178},
      MRREVIEWER = {D. Sarason},
      ZBLNUMBER = {0833.30018},
     }
  • [GAP] Go to document A. Poltoratski, "Spectral gaps for sets and measures," Acta Math., vol. 208, iss. 1, pp. 151-209, 2012.
    @article {GAP, MRKEY = {2910798},
      AUTHOR = {Poltoratski, Alexei},
      TITLE = {Spectral gaps for sets and measures},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {208},
      YEAR = {2012},
      NUMBER = {1},
      PAGES = {151--209},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {42A38 (28A12 30D15 43A10 47B35)},
      MRNUMBER = {2910798},
      MRREVIEWER = {George A. Willis},
      DOI = {10.1007/s11511-012-0076-4},
      ZBLNUMBER = {1245.42008},
      }
  • [PS] Go to document A. Poltoratski and D. Sarason, "Aleksandrov-Clark measures," in Recent Advances in Operator-Related Function Theory, Providence, RI: Amer. Math. Soc., 2006, vol. 393, pp. 1-14.
    @incollection {PS, MRKEY = {2198367},
      AUTHOR = {Poltoratski, Alexei and Sarason, Donald},
      TITLE = {Aleksandrov-{C}lark measures},
      BOOKTITLE = {Recent Advances in Operator-Related Function Theory},
      SERIES = {Contemp. Math.},
      VOLUME = {393},
      PAGES = {1--14},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2006},
      MRCLASS = {30D50 (30D55 30E05 30E20 47A45 47A55 47B15 47B33)},
      MRNUMBER = {2198367},
      MRREVIEWER = {Javad Mashreghi},
      DOI = {10.1090/conm/393/07366},
      ZBLNUMBER = {1102.30032},
      }

Authors

Alexei Poltoratski

Texas A&M University, College Station, TX