Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

Abstract. We obtain a characterization of S {Mp} {Mp} (R n ) and S(Mp) (R n ), the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier results by Gramchev et al. (Proc. Amer. Math. Soc. 139 (2011), 4361-4368) for Gevrey weight sequences. It also generalizes to R n recent results by Dasgupta and Ruzhansky which were obtained in the setting of compact manifolds. IntroductionBack in 1969 Seeley characterized [18] real analytic functions on a compact analytic manifold via the decay of their Fourier coefficients with respect to eigenfunction expansions associated to a normal analytic elliptic differential operator. In recent times, this result by Seeley has attracted much attention and has been generalized in several directions. In a recent article [6], Dasgupta and Ruzhansky extended Seeley's work and achieved the eigenfunction expansion characterization of DenjoyCarleman classes of ultradifferentiable functions, of both Roumieu and Beurling type, and the corresponding ultradistribution spaces on a compact analytic manifold. See also [5] for Gevrey classes on compact Lie groups.Such results have also a global Euclidean counterpart. In this setting, it is natural to consider differential operators of Shubin type, that is, differential operators with polynomial coefficientsIn [9] Gramchev, Pilipović, and Rodino used this type of operators to give an analogue to Seeley's result for some classes of Gelfand-Shilov spaces. The aim of this paper is to extend the results from [9] by supplying a characterization of the general Gelfand-Shilov spaces S {Mp} (R n ) = S {Mp} {Mp} (R n ) and(Mp) (R n ) of ultradifferentiable functions of Roumieu and Beurling type [3,4,7,8,15]. Our characterization is as follows. We refer to Section 2 for the notation. Note that if P is globally elliptic and normal (P P * = P * P ), then there is an orthonormal basis of L 2 (R n ) consisting of eigenfunctions of P . Properties of

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“…As a corollary, we recover a result first observed in [2]: If P is globally elliptic then all its eigenfunctions belong to…”

Section: ) In the Beurling Case We Obtainsupporting

confidence: 84%

“…In this section we exploit the iterative approach from [2,9,18] in order to obtain a structural characterization of S * (R n ) in terms of the growth of the L 2 norms of the iterates of the operator P . The regularity result Theorem 1.2 will readily follow from Theorem 3.4 below.…”

Section: Iterates Of the Operator And Regularity Of Solutionsmentioning

confidence: 99%

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $$\mathbb R^n$$ R n

Vučković

Vindas

2016

J. Pseudo-Differ. Oper. Appl.

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“…provided (1.4) holds; see [3], [4] for details and more general results. We assume, as in Seeley [16], that P is a normal operator (i.e., P * P = P P * ) satisfying the global ellipticity condition (1.4).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Eigenfunction expansions in ℝⁿ

Gramchev¹,

Pilipović

Rodino

2011

Proc. Amer. Math. Soc.

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Abstract. The main goal of this paper is to extend in R n a result of Seeley on eigenfunction expansions of real analytic functions on compact manifolds. As a counterpart of an elliptic operator in a compact manifold, we consider in R n a selfadjoint, globally elliptic Shubin type differential operator with spectrum consisting of a sequence of eigenvalues λ j , j ∈ N, and a corresponding sequence of eigenfunctions u j , j ∈ N, forming an orthonormal basis of L 2 (R n ).

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“…The other standard way is to use Theorem 25.4 (with m = m 0 = 2) in [16], which implies that 4) and using appropriate mapping properties of pseudo-differential operators with symbols satisfying (0.4). In [3] more refined estimates are established for the solutions f ∈ S ′ (R d ) of the equation Hf = g and for more general differential operators when g belongs to the GelfandShilov space S s (R d ), s ≥ 1/2 (cf. Section 1 for the definitions).…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2.1, Proposition 2.2 ′ and Remarks 2.4 and 2.5). Starting from the estimate (0.6) for b d , it might be interesting to study general symbols satisfying estimates of the same type and to establish regularity results for the related operators in the setting of Gelfand-Shilov spaces as it has been done in [3][4][5]. We will treat these applications in future papers and focus here only on the model H −1 .…”

Section: Introductionmentioning

confidence: 99%

On the Inverse to the Harmonic Oscillator

Cappiello

Rodino

Toft

2015

Communications in Partial Differential Equations

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Abstract. Let b d be the Weyl symbol of the inverse to the harmonic oscillator on R d . We prove that b d and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for b d . In the even-dimensional case we characterize b d in terms of elementary functions.In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions. IntroductionA fundamental operator in quantum physics and classical analysis is the harmonic oscillatorIn physics the operator H appears in the stationary Schrödinger equation for a particle under the action of a quadratic potential. In classical analysis, H is also known as the Hermite operator, and possesses several convenient properties. For example, the operator H is strictly positive in L 2 (R d ) with discrete spectrum, and the eigenfunctions are the Hermite functions, see for example [7,15,19].By means of the Hermite functions one can express also the kernel of the inverse H −1 . On the other hand, coherently with the point of view of the quantum physics, H −1 can be written as Weyl pseudo-differential operator(see for example the general calculus in [9,10,12,16]

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Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

Cited by 33 publications

References 22 publications

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $$\mathbb R^n$$ R n

Eigenfunction expansions of ultradifferentiable functions and ultradistributions in $$\mathbb R^n$$ R n

Eigenfunction expansions in ℝⁿ

On the Inverse to the Harmonic Oscillator

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