Geodesic
On the Continuous Linear Right Inverse for Convolution Operators in Spaces of Ultradifferentiable Functions
Matematičeskie zametki, Tome 96 (2014) no. 4, pp. 548-566

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A surjective convolution operator is considered in spaces of complex-valued Beurling ultradifferentiable functions of normal type on a finite interval. A complete description of characteristic functions for which this operator has a continuous linear right inverse is obtained. The finite- and infinite-order differential operators with constant coefficients are studied as a special case.
Keywords: convolution operator, continuous linear right inverse (CLRI), Beurling ultradifferentiable function. 
D. A. Polyakova. On the Continuous Linear Right Inverse for Convolution Operators in Spaces of Ultradifferentiable Functions. Matematičeskie zametki, Tome 96 (2014) no. 4, pp. 548-566. http://geodesic.mathdoc.fr/item/MZM_2014_96_4_a6/
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[1] B. A. Taylor, R. Meise, D. Vogt, “Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse”, Ann. Inst. Fourier (Grenoble), 40:3 (1990), 619–655 | DOI | MR | Zbl

[2] R. Meise, D. Vogt, “Characterization of convolution operators on spaces of $C^\infty$-functions admitting a continuous linear right inverse”, Math. Ann., 279:1 (1987), 141–155 | DOI | MR | Zbl

[3] S. Momm, “On the dependence of analytic solutions of partial differential equations from the right-hand side”, Trans. Amer. Math. Soc., 345:2 (1994), 729–752 | DOI | MR | Zbl

[4] Yu. F. Korobeinik, “O pravom obratnom dlya operatora svertki”, Ukr. matem. zhurn., 43:9 (1991), 1167–1176 | MR | Zbl

[5] M. Langenbruch, “Continuous linear right inverses for convolution operators in spaces of real analytic functions”, Studia Math., 110:1 (1994), 65–82 | MR | Zbl

[6] S. N. Melikhov, Z. Momm, “O lineinom nepreryvnom pravom obratnom dlya operatora svertki na prostranstvakh rostkov analiticheskikh funktsii na vypuklykh kompaktakh v $\mathbb C$”, Izv. vuzov. Matem., 1997, no. 5, 38–48 | MR | Zbl

[7] M. A. Solovev, “Lorents-kovariantnye ultraraspredeleniya, giperfunktsii i analiticheskie funktsionaly”, TMF, 128:3 (2001), 492–514 | DOI | MR | Zbl

[8] C. Roumieu, “Sur quelques extensions de la notion de distribution”, Ann. Sci. École Norm. Sup. (3), 77:1 (1960), 41–121 | MR | Zbl

[9] C. Roumieu, “Ultra-distributions définies sur $\mathbb{R}^n$ et sur certaines classes de variétés différentiables”, J. Anal. Math., 10 (1962/1963), 153–192 | DOI | MR | Zbl

[10] G. Björck, “Linear partial differential operators and generalized distributions”, Ark. Mat., 6:4-5 (1966), 351–407 | DOI | MR | Zbl

[11] R. W. Brawn, R. Meise, B. A. Taylor, “Ultradifferentiable functions and Fourier analysis”, Results Math., 17:3-4 (1990), 206–237 | DOI | MR | Zbl

[12] R. Meise, B. Taylor, D. Vogt, “Equivalence of slowly decreasing conditions and local Fourier expansions”, Indiana Univ. Math. J., 36:4 (1987), 729–756 | DOI | MR | Zbl

[13] R. Meise, B. A. Taylor, “Splitting of closed ideals in (DFN)-algebras of entire functions and the property (DN)”, Trans. Amer. Math. Soc., 302:1 (1987), 341–370 | MR | Zbl

[14] M. Langenbruch, S. Momm, “Complemented submodules in weighted spaces of analytic functions”, Math. Nachr., 157:1 (1992), 263–276 | DOI | MR | Zbl

[15] D. A. Abanina, “Razreshimost uravnenii svertki v prostranstvakh ultradifferentsiruemykh funktsii Berlinga normalnogo tipa na intervale”, Sib. matem. zhurn., 53:3 (2012), 477–494 | MR | Zbl

[16] D. A. Abanina, “Eksponentsialno-polinomialnyi bazis v prostranstve reshenii odnorodnogo uravneniya svertki na klassakh ultradifferentsiruemykh funktsii”, Vladikavk. matem. zhurn., 13:4 (2011), 3–17

[17] D. A. Abanina, “On Borel's theorem for spaces of ultradifferentiable functions of mean type”, Results Math., 44:3-4 (2003), 195–213 | DOI | MR | Zbl

[18] A. V. Abanin, Pham Trong Tien, “Almost subadditive weight functions form Braun–Meise–Taylor theory of ultradistributions”, J. Math. Anal. Appl., 363 (2010), 296–301 | DOI | MR | Zbl

[19] V. V. Zharinov, “Kompaktnye semeistva LVP i prostranstva FS i DFS”, UMN, 34:4 (1979), 97–131 | MR | Zbl

[20] A. V. Abanin, I. A. Filipev, “Analiticheskaya realizatsiya prostranstv, sopryazhennykh k prostranstvam beskonechno differentsiruemykh funktsii”, Sib. matem. zhurn., 47:3 (2006), 485–500 | MR | Zbl

[21] E. Michael, “Continuous selections. I”, Ann. of Math. (2), 63 (1956), 361–382 | DOI | MR | Zbl

[22] R. Meise, D. Vogt, Introduction to Functional Analysis, Oxf. Grad. Texts Math., 2, The Clarendon Press, New York, 1997 | MR | Zbl

[23] L. Khermander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR | Zbl

[24] R. Boas, Entire Functions, Pure Appl. Math., 5, Academic Press, New York, 1954 | MR | Zbl

[25] A. V. Abanin, Ultradifferentsiruemye funktsii i ultraraspredeleniya, Nauka, M., 2007

[26] P. Z. Agranovich, Indikatory golomorfnykh funktsii mnogikh peremennykh, Dis. kand. fiz.-matem. nauk, Kharkov, 1978

[27] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956 | MR | Zbl

[28] B. Ya. Levin, Dopolneniya i ispravleniya k knige «Raspredelenie kornei tselykh funktsii», Preprint FTINT AN USSR, Kharkov, 1978