Integral representations of positive definite functions of one variable, associated with operator $$$\frac{d^4}{dx^4}$$$

O.V. Lopotko

doi:10.15421/241610

Integral representations of positive definite functions of one variable, associated with operator $$$\frac{d^4}{dx^4}$$$

O.V. Lopotko (Ukrainian National Forestry University)

Abstract

We obtain integral representations for positive definite functions of one variable, when kernels $$$K(x,y)$$$ are positive definite. The proof is based on the spectral theory of differential operators of fourth order.

Keywords

integral representation; positive definite functions

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References

Berezanskij Yu.M. Eigenfunction expansion of self-adjoint operators, Naukova dumka, Kyiv, 1965; 800 p. (in Russian)

Krejn M.G. "On one general method of expansion of positive definite kernels into elementary products", Doklady AN SSSR, 1946; 53(1): pp. 3-6. (in Russian)

Lopotko O.V. "Integral representation of even positive definite functions of one variable", Ukrainian Math. J., 2010; 62(2): pp. 281-284. (in Ukrainian) doi:10.1007/s11253-010-0355-1

Lopotko O.V. "Integral representation of even positive definite functions of two variables", Ukrainian Math. J., 2011; 63(6): pp. 844-853. (in Ukrainian) doi:10.1007/s11253-011-0558-0

Lopotko O.V. "Integral representation of positive definite functions of one variable, associated with operator $$$\frac{d^3}{dx^3}$$$", Visnyk ONU, 2013; 18(4(20)): pp. 31-38. (in Russian)