Monomial convergence for holomorphic functions on ℓr

Bayart, Frédéric; Defant, Andreas; Schlüters, Sunke

doi:10.1007/s11854-019-0022-x

Cited by 35 publications

(9 citation statements)

References 20 publications

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“…Proof Note that

y \in m o n (P (^{m} ℓ_{p}))

if and only if its decreasing rearrangement

y^{*}

belongs to

m o n (P (^{m} ℓ_{p}))

. In [, Theorem 5.3] it was proved that for any

ε > \frac{1}{p}

we have,

\frac{1}{{boldp}^{σ_{m}} \log {(p)}^{ε}} \cdot ℓ_{p} \subset m o n (P (^{m} ℓ_{p})),

where

σ_{m} = (\frac{m - 1}{m}) (1 - \frac{1}{p})

. We will show that for every

y \in d (w_{λ}, q_{m})

we have that

y^{*} \in \frac{1}{{boldp}^{σ_{m}} \log {(p)}^{ε_{0}}} \cdot ℓ_{p}

for some

ε_{0} > \frac{1}{p}

.…”

Section: Mixed Unconditional Basis Constant For Homogeneous Polynomiamentioning

confidence: 93%

“…See and the references therein for what is known about these sets. There is a strong relation between monomial convergence and mixed unconditionality.…”

Section: Mixed Unconditional Basis Constant For Homogeneous Polynomiamentioning

confidence: 99%

“…This type of comparison results have shown several applications to different problems in complex and harmonic analysis, number theory and even in modern physics. Among them we include the contributions to the study of: the asymptotic behavior of the Bohr radius or Dirichlet–Bohr radius , Sidon constants and convergence of Dirichlet series , monomial series expansions of holomorphic functions in infinitely many variables and multipliers of Dirichlet series , classical inequalities in operator theory and lower bounds on the classical bias obtainable in multiplayer XOR games .…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we provide correct estimates of the asymptotic growth of the mixed‐

(p, q)

unconditional constant as n tends to infinity. To achieve this we use some major results given in about domains of monomial convergence, our bounds on Problem and multilinear interpolation. Moreover, we give an analog of a result of Pisier and Schütt in this context.…”

Section: Introductionmentioning

confidence: 99%

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The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Galicer

Mansilla

Muro

2019

Mathematische Nachrichten

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Let, ( ) be the best constant that fulfills the following inequality: for every -homogeneous polynomial ( ) = ∑ | |= in complex variables,For every degree , and a wide range of values of , ∈ [1, ∞] (including any in the case ∈ [1, 2], and any and for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as (the number of variables) tends to infinity.Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., -homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of , . As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality. K E Y W O R D SHardy-Littlewood inequalities, multivariable von Neumann's inequality, unconditionality in spaces of polynomials, unimodular polynomials M S C ( 2 0 1 0 ) 11C08, 15A60, 15A69, 46G25, 47A30, 47H60 Problem 1.3. Let , ( ) and , ( ) be the smallest constants that fulfill the following inequalities: for every -homogeneous polynomial in complex variables,

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“…Proof Note that

y \in m o n (P (^{m} ℓ_{p}))

if and only if its decreasing rearrangement

y^{*}

belongs to

m o n (P (^{m} ℓ_{p}))

. In [, Theorem 5.3] it was proved that for any

ε > \frac{1}{p}

we have,

\frac{1}{{boldp}^{σ_{m}} \log {(p)}^{ε}} \cdot ℓ_{p} \subset m o n (P (^{m} ℓ_{p})),

where

σ_{m} = (\frac{m - 1}{m}) (1 - \frac{1}{p})

. We will show that for every

y \in d (w_{λ}, q_{m})

we have that

y^{*} \in \frac{1}{{boldp}^{σ_{m}} \log {(p)}^{ε_{0}}} \cdot ℓ_{p}

for some

ε_{0} > \frac{1}{p}

.…”

Section: Mixed Unconditional Basis Constant For Homogeneous Polynomiamentioning

confidence: 93%

“…See and the references therein for what is known about these sets. There is a strong relation between monomial convergence and mixed unconditionality.…”

Section: Mixed Unconditional Basis Constant For Homogeneous Polynomiamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we provide correct estimates of the asymptotic growth of the mixed‐

(p, q)

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Galicer

Mansilla

Muro

2019

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…To obtain the lower bounds the proof is divided in several cases. For p < 2 we have combined an appropriate way to divide and distinguish certain subsets of monomials together with the upper estimates for the unconditional basis constants of spaces of polynomials on ℓ p spanned by finite sets of monomials given in [BDS16]. The interplay between monomial convergence and mixed unconditionality for spaces of homogeneous polynomials presented in [DMP09, Theorem 5.1.]…”

Section: Introductionmentioning

confidence: 99%

Mixed Bohr radius in several variables

Galicer

Mansilla

Muro

2019

Trans. Amer. Math. Soc.

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Let K (B ℓ n p , B ℓ n q ) be the n-dimensional (p, q)-Bohr radius for holomorphic functions on C n . That is, K (B ℓ n p , B ℓ n q ) denotes the greatest constant r ≥ 0 such that for every entire function f (z) = α c α z α in n-complex variables, we have the following (mixed) Bohr-type inequalitywhere B ℓ n r denotes the closed unit ball of the n-dimensional sequence space ℓ n r . For every 1 ≤ p, q ≤ ∞, we exhibit the exact asymptotic growth of the (p, q)-Bohr radius as n (the number of variables) goes to infinity.where a n ∈ C and s = σ + i t is a complex variable. The regions of convergence, absolute convergence and uniform convergence of these series define half-planes of the form [Re(s) > σ 0 ] in the complex field. Bohr was mainly interested in controlling the region of convergence of a series. To achieve this, he related different types of convergence and focused on finding the width of the greatest strip for which a Dirichlet series can converge uniformly but not absolutely. This question is popular and known nowadays as the Bohr's absolute convergence problem.Although the solution of this problem problem appeared two decades after it was proposed (given by Bohnenblust and Hille [BH31] who showed that the maximum width of this strip is 1 2 ), Bohr made major contributions in the area (arguably, even more important than the solution of the problem 2010 Mathematics Subject Classification. 32A05,32A22 (primary),46B15,46B20,46G25,46E50 (secondary).

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