The maximum modulus of a trigonometric trinomial

Neuwirth, Stefan

doi:10.1007/s11854-008-0028-2

Cited by 27 publications

(21 citation statements)

References 14 publications

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“…See also [8,9,10,11,12,13] for related questions concerning real or complex homogeneous polynomials of degree 2 or 3. Trigonometric trinomials (real or complex) have also been studied by Aron and Klimek [1], Neuwirth [23] and the second author [24]. This paper introduces a novelty, never studied before, by presenting a full description of the extreme points of a space of 2-homogeneous polynomials defined on a non symmetric convex body.…”

Section: Introduction and Notationmentioning

confidence: 99%

Geometry of homogeneous polynomials on non symmetric convex bodies

Muñoz-Fernández¹,

Révész²,

Seoane‐Sepúlveda³

2009

MATH. SCAND.

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If ∆ stands for the region enclosed by the triangle in R 2 of vertices (0, 0), (0, 1) and (1, 0) (or simplex for short), we consider the space P( 2 ∆) of the 2-homogeneous polynomials on R 2 endowed with the norm given by ax 2 + bxy + cy 2 ∆ := sup{|ax 2 + bxy + cy 2 | : (x, y) ∈ ∆} for every a, b, c ∈ R. We investigate some geometrical properties of this norm. We provide an explicit formula for · ∆ , a full description of the extreme points of the corresponding unit ball and a parametrization and a plot of its unit sphere. Using this geometrical information we also find sharp Bernstein and Markov inequalities for P( 2 ∆) and show that a classical inequality of Martin does not remain true for homogeneous polynomials on non symmetric convex bodies.

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Section: Introduction and Notationmentioning

confidence: 99%

Geometry of homogeneous polynomials on non symmetric convex bodies

Muñoz-Fernández¹,

Révész²,

Seoane‐Sepúlveda³

2009

MATH. SCAND.

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“…See also [5][6][7][8][9][10] for related questions concerning real or complex homogeneous polynomials of degree 2 or 3. Trigonometric trinomials (real or complex) have also been studied by Aron and Klimek [1], Neuwirth [14] and Révész [15].…”

Section: Introduction and Notationmentioning

confidence: 99%

“…Let P (x) = ax m + bx n + c. If a = 0 or b = 0 then(14) follows immediately. Suppose now that a = 0 and b = 0.…”

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Geometry of Banach spaces of trinomials

Muñoz-Fernández

Seoane‐Sepúlveda

2008

Journal of Mathematical Analysis and Applications

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“…Most of the results that are known are for real spaces, although some description of extreme and smooth points in complex spaces also exist [4,5,8,10]. The geometry of other spaces of polynomials (not necessarily homogeneous) has also been studied by several authors [3,[14][15][16].…”

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confidence: 99%

The unit ball of the complex $${{\mathcal P}(^3H)}$$

2008

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Let H be a two-dimensional complex Hilbert space and P( 3 H ) the space of 3-homogeneous polynomials on H . We give a characterization of the extreme points of its unit ball, B P( 3 H ) , from which we deduce that the unit sphere of P( 3 H ) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B P( 3 H ) remains extreme as considered as an element of B L( 3 H ) . Finally we make a few remarks about the geometry of the unit ball of the predual of P( 3 H ) and give a characterization of its smooth points. Notation, terminology and preliminary resultsGiven n (real or complex) Banach spaces X 1 , . . . , X n we denote by X 1 ⊗· · ·⊗ X n their tensor product and by π the projective norm. If X 1⊗π · · ·⊗ π X n is the completion of X 1 ⊗· · · ⊗ X n under the projective norm, then we have (X 1⊗π · · ·⊗ π X n ) * = L( n X 1 , . . . , X n ), the space of continuous n-linear forms on X 1 × · · · × X n endowed with the supremum norm. However, as remarked in [6], most of the times the multilinear theory is far from being just a simple

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The maximum modulus of a trigonometric trinomial

Cited by 27 publications

References 14 publications

Geometry of homogeneous polynomials on non symmetric convex bodies

Geometry of homogeneous polynomials on non symmetric convex bodies

Geometry of Banach spaces of trinomials

The unit ball of the complex $${{\mathcal P}(^3H)}$$

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