We obtain analogs of the Parseval equality and convolution theorem and establish some other properties of the convolution of functions from the Hardy–Smirnov spaces in an arbitrary convex unbounded polygon.
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References
A. Beurling, “On two problems concerning linear transformations in Hilbert space,” Acta Math., 81, No. 1, 239–255 (1949).
P. Lax, “Translation invariant subspaces,” Acta Math., 101, 163–178 (1959).
N. K. Nikol’skii, “Invariant subspaces in the theory of operators and theory of functions,” in: VINITI Series in Mathematical Analysis [in Russian], Vol. 12, VINITI, Moscow (1974), pp. 199–412.
N. K. Nikolski, Operators, Functions and Systems: an Easy Reading, Vol. 1, American Mathematical Society, Providence (2002).
P. D. Lax and R. S. Phillips, Scattering Theory for Automorphic Functions, Princeton University, Princeton (1976).
P. Lax, Functional Analysis, Wiley, New York (2002).
V. P. Gurarii, “Group methods of commutative harmonic analysis,” in: VINITI Series in Mathematics [in Russian], Vol. 25, VINITI, Moscow (1988), pp. 1–312.
B. Korenblum, “An extension of a Nevanlinna theory,” Acta Math., 135, No. 1, 187–219 (1974).
B. Korenblum, “A Beurling-type theorem,” Acta Math., 138, No. 1, 265–293 (1977).
H. S. Shapiro, “Weakly invertible elements in certain function spaces and generators in l 2;” Mich. Math. J., 11, 161–165 (1964).
H. S. Shapiro, “Some observations concerning the weighted polynomial approximation of holomorphic functions,” Mat. Sb., 73, 320– 330 (1967).
M. M. Dzhrbashyan, Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moscow (1966).
B. V. Vinnitskii, “On zeros of functions analytic in a half-plane and completeness of systems of exponents,” Ukr. Mat. Zh., 46, No. 5, 484–500 (1994); English translation: Ukr. Math. J., 46, No. 5, 514–532 (1994).
V. M. Dil’nyi, “On the representation of a class of analytic functions in an angular domain,” Visn. Nats. Univ. “L’viv. Politekhn.,” 718, No. 718, 15–18 (2012).
B. V. Vinnitskii, “Approximation properties of systems of exponentials in one space of analytic functions,” Ukr. Mat. Zh., 48, No. 2, 168–183 (1996); English translation: Ukr. Math. J., 48, No. 2, 189–206 (1996).
B. V. Vynnyts’kyi, “On solutions of a homogeneous convolution equation in a class of functions analytic in a half-layer,” Mat. Stud., 7, No. 1, 41–52 (1997).
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals [Russian translation], Tekhnicheskaya Literatura, Moscow (1948).
J. Garnett, Bounded Analytic Functions, Academic Press, New York (1981).
N. Wiener and R. E. A. C. Paley, Fourier Transforms in the Complex Domain [Russian translation], Nauka, Moscow (1964).
I. I. Privalov, Boundary Properties of Analytic Functions [in Russian], Gostekhteorizdat, Moscow (1950).
B. V. Vynnyts’kyi, “Convolution equation and angular boundary values of analytic functions,” Dopov. Nats. Akad. Nauk Ukr., Ser. A, No. 10, 13–17 (1995).
B. Vinnitskii and V. Dil’nyi, “On extension of Beurling–Lax theorem,” Math. Notes, 79, 362–368 92006).
V. M. Dil’nyi, “On the equivalence of some conditions for weighted Hardy spaces,” Ukr. Mat. Zh., 58, No. 9, 1257–1263 (2006); English translation: Ukr. Math. J., 58, No. 9, 1425–1432 (2006).
V. M. Dil’nyi, “On the existence of solutions of a convolution-type equation,” Dopov. Nats. Akad. Nauk Ukr., No. 11, 7–10 (2008).
V. Dilnyi, “On cyclic functions in weighted Hardy spaces,” Zh. Mat. Fiz., Anal., Geometr., 7, 19–33 (2011).
M. M. Dzhrbashyan and V. M. Martirosyan, “Theorems of Wiener–Paley and Müntz–Szász types,” Izv. Akad. Nauk SSSR, Ser. Mat., 41, 868–894 (1974).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1155–1164, September, 2012.
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Dil’nyi, V.M. On convolution of functions in angular domains. Ukr Math J 64, 1315–1325 (2013). https://doi.org/10.1007/s11253-013-0719-4
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DOI: https://doi.org/10.1007/s11253-013-0719-4