Quaternionic Hankel operators and approximation by slice regular functions

Sarfatti, Giulia

doi:10.1512/iumj.2016.65.5896

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…Slice monogenic functions with values in a Clifford algebra and their main properties were studied in [58,72,73,74,75,78,81,122,155]. Approximation of slice hyperholomorphic functions are collected in the works [77,94,95,96,97,98,99,100,145]. The case of several variables was treated in [2,79], but a lot of work has still to be done in this direction since the theory is at the beginning.…”

Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

112

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The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.

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Section: Introductionmentioning

confidence: 99%

Entire Slice Regular Functions

Colombo

Sabadini

Struppa

2016

SpringerBriefs in Mathematics

112

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“…As in the case of slice regular functions, it holds (f * g) c = g c * f c ; see [Sar16]. The reciprocal f − * of f (q) = n∈Z q n a n with respect to the * -product is then given by…”

Section: Preliminary Results and Slice Idempotent Functionsmentioning

confidence: 97%

“…We now recall the definition of slice L p spaces. These spaces were introduced by the second author in [Sar16]. We focus here only on the Hilbert case p = 2 and the case p = ∞ since it is enough for the purposes of this work.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Shift invariant subspaces of slice L^2 functions

Monguzzi¹,

Sarfatti²

2018

Ann. Acad. Sci. Fenn. Math.

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In this paper we characterize the closed invariant subspaces for the ( * -)multiplier operator of the quaternionic space of slice L 2 functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity. KEY WORDS AND PHRASES: functions of a quaternionic variable; invariant subspaces; inner-outer factorization MATHEMATICS SUBJECT CLASSIFICATION: 30G35, 30H10, 30J05

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“…Finding the exact value of Π p,p for all p's might shed some light on the geometry of slice regular functions, and we think it is an interesting open problem. The study presented in this paper began when discussing some problems related to Preprint [6].…”

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confidence: 99%

The orthogonal projection on slice functions on the quaternionic sphere

Arcozzi¹,

Sarfatti²

2015

J. Phys.: Conf. Ser.

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Abstract. We study the L p norm of the orthogonal projection from the space of quaternion valued L 2 functions to the closed subspace of slice L 2 functions.The aim of this short note is to study the orthogonal projection Π from the space of quaternion valued L 2 functions to its closed subspace of slice L 2 functions. In particular, to compute the norm of the projection operator we will first show that we can write Π in terms of a quaternionic slice Poisson kernel.Let H = R + Ri + Rj + Rk denote the non commutative 4-dimensional real algebra of quaternions and let B = {q ∈ H : |q| < 1} be the unit ball in H. Its boundary ∂B contains elements of the form q = e It , I ∈ S, t ∈ R, where S = {q ∈ H : q 2 = −1} is the two dimensional sphere of imaginary units in H. We endow ∂B with the measure dΣ e It = dσ(I)dt, which is naturally associated with the Hardy space H 2 (B) of the slice regular functions on B, see [1]. We here normalize the measures so to have Σ(∂B) = σ(S) = 1. The functions we are concerned with here satisfy algebraic conditions and size conditions. A function f : ∂B → H is a slice function if for any I, J ∈ SThat is, slice functions are affine in the S variable:where a, b : ∂B → H are functions depending on t alone. Condition (2) has an algebraic nature. Equation (1) also provides a formula to extend a function f I defined on a slice ∂B I = ∂B∩(R+RI) to a slice function f := ext(f I ) defined on the entire sphere ∂B. Slice functions were introduced in the more general setting of real alternative algebras by Ghiloni and Perotti in [5]. A notable class of slice functions is provided by the restriction to the unit sphere in H of slice regular functions. Such functions have been object of intensive research

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Quaternionic Hankel operators and approximation by slice regular functions

Cited by 6 publications

References 18 publications

Entire Slice Regular Functions

Entire Slice Regular Functions

Shift invariant subspaces of slice L^2 functions

The orthogonal projection on slice functions on the quaternionic sphere

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