Bounds on strong unicity for Chebyshev approximation with bounded - PowerPoint PPT Presentation

Bounds on strong unicity for Chebyshev approximation with bounded coefficients Andrei Sipos , Technische Universit¨ at Darmstadt Institute of Mathematics of the Romanian Academy August 15, 2019 Logic Colloquium 2019 Praha, ˇ Cesko

Proof mining Proof mining: an applied subfield of mathematical logic

Proof mining Proof mining: an applied subfield of mathematical logic goals: to find explicit and uniform witnesses or bounds and to remove superfluous premises from concrete mathematical statements by analyzing their proofs

Proof mining Proof mining: an applied subfield of mathematical logic goals: to find explicit and uniform witnesses or bounds and to remove superfluous premises from concrete mathematical statements by analyzing their proofs tools used: primarily proof interpretations (modified realizability, negative translation, functional interpretation)

A brief history Early efforts David Hilbert: “¨ Uber das Unendliche” (1926) Grete Hermann: “The Question of Finitely Many Steps in Polynomial Ideal Theory” (1926)

A brief history Early efforts David Hilbert: “¨ Uber das Unendliche” (1926) Grete Hermann: “The Question of Finitely Many Steps in Polynomial Ideal Theory” (1926) Georg Kreisel’s program of “unwinding of proofs” the shift of emphasis in the early 1950s Kreisel: Littlewood’s theorem, Hilbert’s 17th problem (1957) the publication of G¨ odel’s Dialectica interpretation (1958) Jean-Yves Girard: bounds on van der Waerden numbers by strategic cut elimination (1987) Horst Luckhardt: growth conditions on Herbrand terms and the number of solutions in Roth’s theorem (1989)

A brief history Early efforts David Hilbert: “¨ Uber das Unendliche” (1926) Grete Hermann: “The Question of Finitely Many Steps in Polynomial Ideal Theory” (1926) Georg Kreisel’s program of “unwinding of proofs” the shift of emphasis in the early 1950s Kreisel: Littlewood’s theorem, Hilbert’s 17th problem (1957) the publication of G¨ odel’s Dialectica interpretation (1958) Jean-Yves Girard: bounds on van der Waerden numbers by strategic cut elimination (1987) Horst Luckhardt: growth conditions on Herbrand terms and the number of solutions in Roth’s theorem (1989) Ulrich Kohlenbach: contemporary proof mining uniqueness in approximation theory (since 1990) nonlinear analysis, convex optimization et al. (since 2001) ergodic theory, commutative algebra, differential algebra: work by Avigad, Towsner, Simmons (since 2007)

A brief history Early efforts David Hilbert: “¨ Uber das Unendliche” (1926) Grete Hermann: “The Question of Finitely Many Steps in Polynomial Ideal Theory” (1926) Georg Kreisel’s program of “unwinding of proofs” the shift of emphasis in the early 1950s Kreisel: Littlewood’s theorem, Hilbert’s 17th problem (1957) the publication of G¨ odel’s Dialectica interpretation (1958) Jean-Yves Girard: bounds on van der Waerden numbers by strategic cut elimination (1987) Horst Luckhardt: growth conditions on Herbrand terms and the number of solutions in Roth’s theorem (1989) Ulrich Kohlenbach: contemporary proof mining uniqueness in approximation theory (since 1990) nonlinear analysis, convex optimization et al. (since 2001) ergodic theory, commutative algebra, differential algebra: work by Avigad, Towsner, Simmons (since 2007)

Chebyshev approximation We have the following classical Chebyshev approximation result. Theorem (de la Vall´ ee Poussin, Young – 1900s) For every n ∈ N and every continuous f : [0 , 1] → R there is an unique p ∈ P n (the set of real polynomials of degree at most n) such that � f − p � = min q ∈ P n � f − q � (where � · � denotes the supremum norm).

Chebyshev approximation We have the following classical Chebyshev approximation result. Theorem (de la Vall´ ee Poussin, Young – 1900s) For every n ∈ N and every continuous f : [0 , 1] → R there is an unique p ∈ P n (the set of real polynomials of degree at most n) such that � f − p � = min q ∈ P n � f − q � (where � · � denotes the supremum norm). Kohlenbach extracted in 1990 a modulus of uniqueness – a function Ψ with the property that if p 1 and p 2 are such that � f − p 1 � , � f − p 2 � ≤ min +Ψ( δ ), then � p 1 − p 2 � ≤ δ .

Chebyshev approximation We have the following classical Chebyshev approximation result. Theorem (de la Vall´ ee Poussin, Young – 1900s) For every n ∈ N and every continuous f : [0 , 1] → R there is an unique p ∈ P n (the set of real polynomials of degree at most n) such that � f − p � = min q ∈ P n � f − q � (where � · � denotes the supremum norm). Kohlenbach extracted in 1990 a modulus of uniqueness – a function Ψ with the property that if p 1 and p 2 are such that � f − p 1 � , � f − p 2 � ≤ min +Ψ( δ ), then � p 1 − p 2 � ≤ δ . He did this by analyzing the uniqueness proof and obtaining an approximate version of it. Let us see how the original proof flows.

A sketch of de la Vall´ ee Poussin’s proof Take p 1 and p 2 that attain the minimum distance E . Then also p 1 + p 2 attains the minimum and we denote it by p . 2

A sketch of de la Vall´ ee Poussin’s proof Take p 1 and p 2 that attain the minimum distance E . Then also p 1 + p 2 attains the minimum and we denote it by p . By a result 2 called the alternation theorem , we have that there is a j ∈ { 0 , 1 } and x 1 < . . . < x n +1 in [0 , 1] such that for every i ∈ { 1 , . . . , n + 1 } , ( p − f )( x i ) = ( − 1) i + j E .

A sketch of de la Vall´ ee Poussin’s proof Take p 1 and p 2 that attain the minimum distance E . Then also p 1 + p 2 attains the minimum and we denote it by p . By a result 2 called the alternation theorem , we have that there is a j ∈ { 0 , 1 } and x 1 < . . . < x n +1 in [0 , 1] such that for every i ∈ { 1 , . . . , n + 1 } , ( p − f )( x i ) = ( − 1) i + j E . Let i ∈ { 1 , . . . , n + 1 } and assume wlog that i + j is even. Then ( p − f )( x i ) = E , so p 1 ( x i ) − f ( x i ) + p 2 ( x i ) − f ( x i ) = E . 2 2

A sketch of de la Vall´ ee Poussin’s proof Take p 1 and p 2 that attain the minimum distance E . Then also p 1 + p 2 attains the minimum and we denote it by p . By a result 2 called the alternation theorem , we have that there is a j ∈ { 0 , 1 } and x 1 < . . . < x n +1 in [0 , 1] such that for every i ∈ { 1 , . . . , n + 1 } , ( p − f )( x i ) = ( − 1) i + j E . Let i ∈ { 1 , . . . , n + 1 } and assume wlog that i + j is even. Then ( p − f )( x i ) = E , so p 1 ( x i ) − f ( x i ) + p 2 ( x i ) − f ( x i ) = E . 2 2 Since � p 1 − f � = E , p 1 ( x i ) − f ( x i ) ≤ E . Similarly, p 2 ( x i ) − f ( x i ) ≤ E . By the above, we have that both are actually equal to E and so p 1 ( x i ) = p 2 ( x i ).

A sketch of de la Vall´ ee Poussin’s proof Take p 1 and p 2 that attain the minimum distance E . Then also p 1 + p 2 attains the minimum and we denote it by p . By a result 2 called the alternation theorem , we have that there is a j ∈ { 0 , 1 } and x 1 < . . . < x n +1 in [0 , 1] such that for every i ∈ { 1 , . . . , n + 1 } , ( p − f )( x i ) = ( − 1) i + j E . Let i ∈ { 1 , . . . , n + 1 } and assume wlog that i + j is even. Then ( p − f )( x i ) = E , so p 1 ( x i ) − f ( x i ) + p 2 ( x i ) − f ( x i ) = E . 2 2 Since � p 1 − f � = E , p 1 ( x i ) − f ( x i ) ≤ E . Similarly, p 2 ( x i ) − f ( x i ) ≤ E . By the above, we have that both are actually equal to E and so p 1 ( x i ) = p 2 ( x i ). Since p 1 and p 2 coincide on at least n + 1 points, they must be equal.

Approximating the proof Let us now see how one approximates the proof on the previous slide. First, for trivial reasons, the polynomials can be assumed to be in the closed ball Z of radius 5 2 � f � (which is compact, as it lies inside the finite dimensional space P n ).

Approximating the proof Let us now see how one approximates the proof on the previous slide. First, for trivial reasons, the polynomials can be assumed to be in the closed ball Z of radius 5 2 � f � (which is compact, as it lies inside the finite dimensional space P n ). 1 for all p 1 , p 2 ∈ Z and all ε > 0, if � f − p 1 � , � � � � � f − p 1 + p 2 � f − p 2 � ≤ E + Φ 1 ( ε ), then � ≤ E + ε . 2

Approximating the proof Let us now see how one approximates the proof on the previous slide. First, for trivial reasons, the polynomials can be assumed to be in the closed ball Z of radius 5 2 � f � (which is compact, as it lies inside the finite dimensional space P n ). 1 for all p 1 , p 2 ∈ Z and all ε > 0, if � f − p 1 � , � � � � � f − p 1 + p 2 � f − p 2 � ≤ E + Φ 1 ( ε ), then � ≤ E + ε . 2 2 (the “ ε -alternation theorem”) for all p ∈ Z and all ε > 0 with � f − p � ≤ E + Φ 2 ( ε ) there is a j ∈ { 0 , 1 } and x 1 < . . . < x n +1 in [0 , 1] such that for every i ∈ { 1 , . . . , n + 1 } , | ( p − f )( x i ) − ( − 1) i + j E | ≤ ε.

Last steps I shall omit steps 3 and 4, as I am not going to focus on them.