Norm Control for Inverses of Convolutions and Large Matrices - - PowerPoint PPT Presentation

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Norm Control for Inverses of Convolutions and Large Matrices - - PowerPoint PPT Presentation

Jan 16, 2023 26 likes •248 views

Norm Control for Inverses of Convolutions and Large Matrices Nikolai Nikolski University Bordeaux 1 Steklov Institute / Chebyshev Lab St.Petersburg Motivation: Effective Inversions (constructive, algorithmic, norm controlled)

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Norm Control for Inverses of
 Convolutions and Large Matrices

Nikolai Nikolski University Bordeaux 1 Steklov Institute / Chebyshev Lab St.Petersburg

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  • Motivation: Effective Inversions

(constructive, algorithmic, norm controlled)

  • Example 1: Convolution T: f↦ f⋆S on a

group G as a map on a Banach function space X

  • The visible spectrum of T: the range of the

Fourier transform Sˆ(Gˆ). A necessary condition for inversion: δ=: inf| Sˆ| > 0 «Well posed inversion»: ||T⁻¹||≤ c(δ), δ>0.

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  • Motivation: Effective Inversions

(constructive, algorithmic, norm controlled)

  • Example 2: Condition Numbers of

Matrices T, n⨯n: CN(T)= ||T||·||T⁻¹|| .

  • The visible spectrum of T: eigenvalues

λᵢ(T), i= 1,…,n; an invertibility condition: δ=: inf|λᵢ(T)| > 0 «Well posed inversion»: CN(T)= ||T||·||T⁻¹||≤ c(δ/||T||), δ>0.

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Enough motivations?..

Finally, let him who has never used a convolution or a large matrix cast the first stone…

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My goal in effective inversions is to understand:


  • Relations «Full Spectrum»/

«Visible Spectrum» (the Wiener- Pitt phenomenon)

  • «Invisible» but Numerically

Detectable Spectrum (c(δ)= ∞)

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Plan for today:

1.Convolutions/Fourier multipliers 2.Large Matrices 3.Some Integration Operators

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  • II. Large Matrices
  • A is an n⨉n matrix
  • ||A||≤ 1 ⇒ ||A⁻¹||≤ 1/|det(A)|≤ 1/δⁿ where

δ= min|σ(A)|

  • For a Banach normed ℂⁿ, |·| it is √n times

worth: ||A⁻¹||≤ √en/|det(A)| - J.Schäffer (1970); sharpness – E.Gluskin, M.Meyer, A.Pajor; J.Bourgain; H.Queffelec (1993)

  • My subject below: Matrices commuting

with a given A (or, just functions of A)

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Summary

  • The nature of «invisible spectrum» is different

(characters measurable wrt «thin σ-algebras»,

  • r forced holomorphic extensions, or complex

homomorphisms in fibers over the boundary).

  • The «invisible» but numerically detectable

spectrum comes from a «true invisible spectrum» and its discontinuity wrt to a weak approximation.

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The End
 ***

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Thank you!

And Happy Birthday

to N.G.M.!!