VB
V. Bogouslavskii
info
Please Note
<p>This page displays the records of the person named above and is not linked to a unique person identifier. This record may need to be merged to a profile.</p>
1 records found
1
This thesis studies sharp inequalities for sums of two functions in Lp. The classical triangle inequality gives a general upper bound for ||f + g||p, but it does not use information about how much the two functions overlap. The aim of this thesis is to obtain sharper bounds by also taking into account the quantity ||fg||p/2p/2 . Together with ||f||pp and ||g||pp, this overlap term gives an associated triple (x, y, z) = ( |f||pp, |g||pp, ||fg||p/2p/2 which lies in a three-dimensional admissible cone.
The main part of this thesis is based on the envelope method developed by Ivanisvili and Mooney, together with related work on sharpened triangle inequalities in Lp. The problem of estimating ||f + g||pp is translated into a geometric problem on the admissible cone. Two envelope functions, denoted by Fp and Gp, are constructed from the boundary data of the cone. Depending on the value of p > 0, these envelopes give the sharp upper and lower bounds for ||f + g||pp The thesis explains how these envelopes are constructed, how their concavity and convexity are checked, and why they give optimal inequalities. It also discusses equality and extremal cases by studying the affine regions of the envelopes.
In the final part of the thesis, the p-power problem is extended to q-powers. Instead of estimating only ||f + g||pp, we study inequalities for ||f + g||qp where q > 2. Using the sharp p-power envelope, the infinite-dimensional problem is reduced to a finitedimensional optimization problem. The behavior of this optimization depends on the exponent q/p. For 2 < q ≤p, the problem reduces to a proportional family of functions. For q≥p, the optimization is controlled by the balanced case x = y. This shows that the envelope method remains useful for q-powers, but that an additional optimization step is needed once the final power is changed. Beyond the obtained inequalities, the results illustrate how envelope methods can reduce functional optimization problems to finite-dimensional optimization problems. This provides additional insight into the structure of sharp inequalities and the role of overlap between functions. ...
The main part of this thesis is based on the envelope method developed by Ivanisvili and Mooney, together with related work on sharpened triangle inequalities in Lp. The problem of estimating ||f + g||pp is translated into a geometric problem on the admissible cone. Two envelope functions, denoted by Fp and Gp, are constructed from the boundary data of the cone. Depending on the value of p > 0, these envelopes give the sharp upper and lower bounds for ||f + g||pp The thesis explains how these envelopes are constructed, how their concavity and convexity are checked, and why they give optimal inequalities. It also discusses equality and extremal cases by studying the affine regions of the envelopes.
In the final part of the thesis, the p-power problem is extended to q-powers. Instead of estimating only ||f + g||pp, we study inequalities for ||f + g||qp where q > 2. Using the sharp p-power envelope, the infinite-dimensional problem is reduced to a finitedimensional optimization problem. The behavior of this optimization depends on the exponent q/p. For 2 < q ≤p, the problem reduces to a proportional family of functions. For q≥p, the optimization is controlled by the balanced case x = y. This shows that the envelope method remains useful for q-powers, but that an additional optimization step is needed once the final power is changed. Beyond the obtained inequalities, the results illustrate how envelope methods can reduce functional optimization problems to finite-dimensional optimization problems. This provides additional insight into the structure of sharp inequalities and the role of overlap between functions. ...
This thesis studies sharp inequalities for sums of two functions in Lp. The classical triangle inequality gives a general upper bound for ||f + g||p, but it does not use information about how much the two functions overlap. The aim of this thesis is to obtain sharper bounds by also taking into account the quantity ||fg||p/2p/2 . Together with ||f||pp and ||g||pp, this overlap term gives an associated triple (x, y, z) = ( |f||pp, |g||pp, ||fg||p/2p/2 which lies in a three-dimensional admissible cone.
The main part of this thesis is based on the envelope method developed by Ivanisvili and Mooney, together with related work on sharpened triangle inequalities in Lp. The problem of estimating ||f + g||pp is translated into a geometric problem on the admissible cone. Two envelope functions, denoted by Fp and Gp, are constructed from the boundary data of the cone. Depending on the value of p > 0, these envelopes give the sharp upper and lower bounds for ||f + g||pp The thesis explains how these envelopes are constructed, how their concavity and convexity are checked, and why they give optimal inequalities. It also discusses equality and extremal cases by studying the affine regions of the envelopes.
In the final part of the thesis, the p-power problem is extended to q-powers. Instead of estimating only ||f + g||pp, we study inequalities for ||f + g||qp where q > 2. Using the sharp p-power envelope, the infinite-dimensional problem is reduced to a finitedimensional optimization problem. The behavior of this optimization depends on the exponent q/p. For 2 < q ≤p, the problem reduces to a proportional family of functions. For q≥p, the optimization is controlled by the balanced case x = y. This shows that the envelope method remains useful for q-powers, but that an additional optimization step is needed once the final power is changed. Beyond the obtained inequalities, the results illustrate how envelope methods can reduce functional optimization problems to finite-dimensional optimization problems. This provides additional insight into the structure of sharp inequalities and the role of overlap between functions.
The main part of this thesis is based on the envelope method developed by Ivanisvili and Mooney, together with related work on sharpened triangle inequalities in Lp. The problem of estimating ||f + g||pp is translated into a geometric problem on the admissible cone. Two envelope functions, denoted by Fp and Gp, are constructed from the boundary data of the cone. Depending on the value of p > 0, these envelopes give the sharp upper and lower bounds for ||f + g||pp The thesis explains how these envelopes are constructed, how their concavity and convexity are checked, and why they give optimal inequalities. It also discusses equality and extremal cases by studying the affine regions of the envelopes.
In the final part of the thesis, the p-power problem is extended to q-powers. Instead of estimating only ||f + g||pp, we study inequalities for ||f + g||qp where q > 2. Using the sharp p-power envelope, the infinite-dimensional problem is reduced to a finitedimensional optimization problem. The behavior of this optimization depends on the exponent q/p. For 2 < q ≤p, the problem reduces to a proportional family of functions. For q≥p, the optimization is controlled by the balanced case x = y. This shows that the envelope method remains useful for q-powers, but that an additional optimization step is needed once the final power is changed. Beyond the obtained inequalities, the results illustrate how envelope methods can reduce functional optimization problems to finite-dimensional optimization problems. This provides additional insight into the structure of sharp inequalities and the role of overlap between functions.