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On the infimum convolution inequality

R. Latała, J. O. Wojtaszczyk (2008)

Studia Mathematica

We study the infimum convolution inequalities. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure μ. In particular, we prove an optimal IC inequality for product log-concave measures and for uniform measures on the ℓ ⁿ p balls. Such an optimal inequality implies, for a given measure,...

On the order of magnitude of Walsh-Fourier transform

Bhikha Lila Ghodadra, Vanda Fülöp (2020)

Mathematica Bohemica

For a Lebesgue integrable complex-valued function f defined on ℝ + : = [ 0 , ∞ ) let f ^ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f ^ ( y ) → 0 as y → ∞ . But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L 1 ( ℝ + ) there is a definite rate at which the Walsh-Fourier transform tends to zero. We...

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality ∥ p ' ∥ [ - 1 , 1 ] ≤ 1 2 ∥ p ∥ [ - 1 , 1 ] for a constrained polynomial p of degree at most n , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval ( - 1 , 1 ) and establish a new asymptotically sharp inequality.

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