Inner functions and zero sets for ℓ^{𝑝}_{𝐴}

Cheng, Raymond; Mashreghi, Javad; Ross, William

doi:10.1090/tran/7675

Inner functions and zero sets for $\ell ^{p}_{A}$
HTML articles powered by AMS MathViewer

by Raymond Cheng, Javad Mashreghi and William T. Ross PDF

Trans. Amer. Math. Soc. 372 (2019), 2045-2072 Request permission

Abstract:

In this paper we characterize the zero sets of functions from $\ell ^{p}_{A}$ (the analytic functions on the open unit disk $\mathbb {D}$ whose Taylor coefficients form an $\ell ^p$ sequence) by developing a concept of an “inner function” modeled by Beurling’s discussion of the Hilbert space $\ell ^{2}_{A}$, the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [Dokl. Akad. Nauk SSSR 160 (1965), pp. 263–266] which says that when $p > 2$, there are zero sets for $\ell ^{p}_{A}$ which are not Blaschke sequences.

References

Evgeny Abakumov and Alexander Borichev, Shift invariant subspaces with arbitrary indices in $l^p$ spaces, J. Funct. Anal. 188 (2002), no. 1, 1–26. MR 1878629, DOI 10.1006/jfan.2001.3850
A. Aleman, S. Richter, and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996), no. 2, 275–310. MR 1440934, DOI 10.1007/BF02392623
Javier Alonso, Horst Martini, and Senlin Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012), no. 1-2, 153–189. MR 2885507, DOI 10.1007/s00010-011-0092-z
Peter J. Brockwell and Alexander Lindner, Strictly stationary solutions of autoregressive moving average equations, Biometrika, 97(3):765–772, 2010.
W. L. Bynum, Weak parallelogram laws for Banach spaces, Canad. Math. Bull. 19 (1976), no. 3, 269–275. MR 442647, DOI 10.4153/CMB-1976-042-4
W. L. Bynum and J. H. Drew, A weak parallelogram law for $l_{p}$, Amer. Math. Monthly 79 (1972), 1012–1015. MR 317008, DOI 10.2307/2318073
Stamatis Cambanis, Clyde D. Hardin Jr., and Aleksander Weron, Innovations and Wold decompositions of stable sequences, Probab. Theory Related Fields 79 (1988), no. 1, 1–27. MR 952989, DOI 10.1007/BF00319099
N. L. Carothers, A short course on Banach space theory, London Mathematical Society Student Texts, vol. 64, Cambridge University Press, Cambridge, 2005. MR 2124948
R. Cheng and J. Dragas, A Blaschke sequence that is not a zero set for $\ell ^p_A$, Proc. Amer. Math. Soc., to appear.
Raymond Cheng and Charles B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl. 404 (2013), no. 1, 64–70. MR 3061381, DOI 10.1016/j.jmaa.2013.02.064
Raymond Cheng and Charles B. Harris, Mixed-norm spaces and prediction of $\rm S\alpha S$ moving averages, J. Time Series Anal. 36 (2015), no. 6, 853–875. MR 3419671, DOI 10.1111/jtsa.12134
R. Cheng, A. G. Miamee, and M. Pourahmadi, Regularity and minimality of infinite variance processes, J. Theoret. Probab. 13 (2000), no. 4, 1115–1122. MR 1820505, DOI 10.1023/A:1007874226636
R. Cheng, A. G. Miamee, and M. Pourahmadi, On the geometry of $L^p(\mu )$ with applications to infinite variance processes, J. Aust. Math. Soc. 74 (2003), no. 1, 35–42. MR 1948256, DOI 10.1017/S1446788700003104
R. Cheng and W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hungar. 71 (2015), no. 1, 45–58. MR 3374700, DOI 10.1007/s10998-014-0078-4
Raymond Cheng and William T. Ross, An inner-outer factorization in $\ell ^p$ with applications to ARMA processes, J. Math. Anal. Appl. 437 (2016), no. 1, 396–418. MR 3451972, DOI 10.1016/j.jmaa.2016.01.009
Raymond Cheng, Javad Mashreghi, and William T. Ross, Birkhoff-James orthogonality and the zeros of an analytic function, Comput. Methods Funct. Theory 17 (2017), no. 3, 499–523. MR 3686895, DOI 10.1007/s40315-017-0191-5
Raymond Cheng, Javad Mashreghi, and William T. Ross, Multipliers of sequence spaces, Concr. Oper. 4 (2017), no. 1, 76–108. MR 3714456, DOI 10.1515/conop-2017-0007
James A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396–414. MR 1501880, DOI 10.1090/S0002-9947-1936-1501880-4
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg, Invariant subspaces in Bergman spaces and the biharmonic equation, Michigan Math. J. 41 (1994), no. 2, 247–259. MR 1278431, DOI 10.1307/mmj/1029004992
Peter Duren, Dmitry Khavinson, and Harold S. Shapiro, Extremal functions in invariant subspaces of Bergman spaces, Illinois J. Math. 40 (1996), no. 2, 202–210. MR 1398090
Peter Duren, Dmitry Khavinson, Harold S. Shapiro, and Carl Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 157 (1993), no. 1, 37–56. MR 1197044
Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
Omar El-Fallah, Karim Kellay, Javad Mashreghi, and Thomas Ransford, A primer on the Dirichlet space, Cambridge Tracts in Mathematics, vol. 203, Cambridge University Press, Cambridge, 2014. MR 3185375
Saber Elaydi, An introduction to difference equations, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2005. MR 2128146
Robert C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292. MR 21241, DOI 10.1090/S0002-9947-1947-0021241-4
Javad Mashreghi, Representation theorems in Hardy spaces, London Mathematical Society Student Texts, vol. 74, Cambridge University Press, Cambridge, 2009. MR 2500010, DOI 10.1017/CBO9780511814525
A. G. Miamee and M. Pourahmadi, Wold decomposition, prediction and parameterization of stationary processes with infinite variance, Probab. Theory Related Fields 79 (1988), no. 1, 145–164. MR 953000, DOI 10.1007/BF00319110
Raymond Mortini and Michael von Renteln, Ideals in the Wiener algebra $W^+$, J. Austral. Math. Soc. Ser. A 46 (1989), no. 2, 220–228. MR 973315
D. J. Newman and Harold S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249–255. MR 148874
Stefan Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205–220. MR 936999, DOI 10.1515/crll.1988.386.205
D. Seco, A characterization of Dirichlet inner functions, Complex Analysis and Operator Theory, to appear.
H. S. Shapiro and A. L. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217–229. MR 145082, DOI 10.1007/BF01162379
Nikolai A. Shirokov, Analytic functions smooth up to the boundary, Lecture Notes in Mathematics, vol. 1312, Springer-Verlag, Berlin, 1988. MR 947146, DOI 10.1007/BFb0082810
S. A. Vinogradov, On the interpolation and zeros of power series with a sequence of coefficients in $l^{p}$, Dokl. Akad. Nauk SSSR 160 (1965), 263–266 (Russian). MR 0176041