Journal of Mathematical Sciences

, Volume 243, Issue 6, pp 965–980

# Interpolation Through Approximation in a Bernstein Space

Article

Let Bσ be the Bernstein space of entire functions of exponential type at most σ bounded on the real axis. Consider a sequence Λ = {zn}n∈ℤ, zn = xn + iyn, such that xn+1 − xn ≥ l > 0 and |yn| ≤ L, n ∈ ℤ. Using approximation by functions from Bσ, we prove that for any bounded sequence A = {an}n∈ℤ, |an| ≤ M, n ∈ ℤ, there exists a function f ∈ Bσ with σ ≤ σ0(l,L) such that f|Λ = A.

## References

1. 1.
A. Beurling, The Collected Works of Arne Beurling, Vol. 2, Birkhäuser, Boston (1989), pp. 351–365.
2. 2.
J. Ortega-Cerdà and K. Seip, “Multipliers for entire functions and an interpolation problem of Beurling,” J. Funct. Anal., 162, 400–415 (1999).
3. 3.
O. V. Silvanovich and N. A. Shirokov, “Approximation by entire functions on a countable union of segments of the real axis,” Vestnik St. Petersburg Univ. Ser. 1, 3, No. 4, 644–650 (2016).
4. 4.
O. V. Silvanovich and N. A. Shirokov, “Approximation by entire functions on a countable union of segments of the real axis. 2. Proof of the main theorem, Vestnik St.Petersburg Univ. Ser. 1,” 4, No. 1, 53–63 (2017).
5. 5.
V. I. Belyi, “Conformal mappings and the approximation of analytic functions in domains with a quasiconformal boundary,” Math. USSR-Sb., 31, No. 3, 289–317 (1977).
6. 6.
B. Ya. Levin, “Majorants in classes of subharmonic functions II,” Teor. Funkts., Funkts. Anal. Prilozh., No. 52, 3–33 (1989).