loc C 1 , 1 convex extensions of jets, and some applications Daniel - PowerPoint PPT Presentation

C 1 , 1 and C 1 , 1 loc C 1 , 1 convex extensions of jets, and some applications Daniel Azagra Based on joint works with E. Le Gruyer, P. Hajłasz and C. Mudarra Fitting Smooth Functions to Data Austin, Texas, August 2019 C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 1 / 77

Two related problems: C 1 , 1 extension and C 1 , 1 convex extensions of 1-jets Two related problems: C 1 , 1 extension and C 1 , 1 convex extensions of 1-jets C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 2 / 77

Two related problems: C 1 , 1 extension and C 1 , 1 convex extensions of 1-jets Problem ( C 1 , 1 convex extension of 1-jets) Given E a subset of a Hilbert space X , and a 1-jet ( f , G ) on E (meaning a pair of functions f : E → R and G : E → X ), how can we tell whether there is a C 1 , 1 convex function F : X → R which extends this jet (meaning that F ( x ) = f ( x ) and ∇ F ( x ) = G ( x ) for all x ∈ E )? Problem ( C 1 , 1 extension of 1-jets) Given E a subset of a Hilbert space X , and a 1-jet ( f , G ) on E , how can we tell whether there is a C 1 , 1 function F : X → R which extends ( f , G ) ? C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 3 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets Previous solutions to the C 1 , 1 extension problem for 1-jets C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 4 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets The C 1 , 1 version of the classical Whitney extension theorem theorem tells us that there exists a function F ∈ C 1 , 1 ( R n ) with F = f on C and ∇ F = G on E if and only there exists a constant M > 0 such that | f ( x ) − f ( y ) − � G ( y ) , x − y �| ≤ M | x − y | 2 , and | G ( x ) − G ( y ) | ≤ M | x − y | for all x , y ∈ E . We can trivially extend ( f , G ) to the closure E of E so that the inequalities hold on C with the same constant M . The function F can be explicitly defined by � f ( x ) if x ∈ C F ( x ) = � if x ∈ R n \ C , Q ∈Q ( f ( x Q ) + � G ( x Q ) , x − x Q � ) ϕ Q ( x ) where Q is a family of Whitney cubes that cover the complement of the closure C of C , { ϕ Q } Q ∈Q is the usual Whitney partition of unity associated to Q , and x Q is a point of C which minimizes the distance of C to the cube Q . Recall also that Lip ( ∇ F ) ≤ k ( n ) M , where k ( n ) is a constant depending only on n (but with lim n →∞ k ( n ) = ∞ ). C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 5 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets In 1973 J.C. Wells improved this result and extended it to Hilbert spaces. Theorem (Wells, 1973) Let E be an arbitrary subset of a Hilbert space X, and f : E → R , G : E → X. There exists F ∈ C 1 , 1 ( X ) such that F | E = f and ( ∇ F ) | E = G if and only if there exists M > 0 so that f ( y ) ≤ f ( x ) + 1 2 � G ( x ) + G ( y ) , y − x � + M 4 � x − y � 2 − 1 4 M � G ( x ) − G ( y ) � 2 ( W 1 , 1 ) for all x , y ∈ E. In such case one can find F with Lip ( F ) ≤ M. We will say jet ( f , G ) on E ⊂ X satisfies condition ( W 1 , 1 ) if it satisfies the inequality of the theorem. It can be checked that ( W 1 , 1 ) is absolutely equivalent to the condition in the C 1 , 1 version of Whitney’s extension theorem, which we will denote ( � W 1 , 1 ) . C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 6 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets Well’s proof was quite complicated, and didn’t provide any explicit formula for the extension when E is infinite. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 7 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets Well’s proof was quite complicated, and didn’t provide any explicit formula for the extension when E is infinite. In 2009 Erwan Le Gruyer showed, by very different means, another version of Well’s result. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 7 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets Theorem (Erwan Le Gruyer, 2009) Given a Hilbert space X, a subset E of X, and functions f : E → R , G : E → X, a necessary and sufficient condition for the 1 -jet ( f , G ) to have a C 1 , 1 extension ( F , ∇ F ) to the whole space X is that �� � Γ( f , G , E ) := sup A 2 x , y + B 2 x , y + | A x , y | < ∞ , (2.1) x , y ∈ E where A x , y = 2 ( f ( x ) − f ( y )) + � G ( x ) + G ( y ) , y − x � and � x − y � 2 B x , y = � G ( x ) − G ( y ) � x , y ∈ E , x � = y . for all � x − y � Moreover, Γ( F , ∇ F , X ) = Γ( f , G , E ) = � ( f , G ) � E , where � ( f , G ) � E := inf { Lip ( ∇ H ) : H ∈ C 1 , 1 ( X ) and ( H , ∇ H ) = ( f , G ) on E } is the trace seminorm of the jet ( f , G ) on E. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 8 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets The number Γ( f , G , E ) is the smallest M > 0 for which ( f , G ) satisfies Well’s condition ( W 1 , 1 ) with constant M > 0. In particular Le Gruyer’s condition is also absolutely equivalent to the condition in the C 1 , 1 version of Whitney’s extension theorem. Le Gruyer’s theorem didn’t provide any explicit formula for the extension either (it uses Zorn’s lemma). C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 9 / 77

Previous solutions to the C 1 , 1 extension problem for 1-jets What about the convex case? Theorem (Azagra-Mudarra, 2016) Let E be a subset of R n , and f : E → R , G : E → R n be functions. There exists a convex function F ∈ C 1 ,ω ( R n ) if and only if there exists M > 0 such that, for all x , y ∈ E, 1 2 M | G ( x ) − G ( y ) | 2 . f ( x ) − f ( y ) − � G ( y ) , x − y � ≥ |∇ F ( x ) −∇ F ( y ) | ≤ k ( n ) M. Moreover, sup x � = y | x − y | Here, as in Whitney’s theorem, k ( n ) only depends on n , but goes to ∞ as n → ∞ (not surprising, as Whitney’s extension techniques were used in the proof, which was constructive). We say that ( f , G ) satisfies condition ( CW 1 , 1 ) if it satisfies the inequality of the theorem for some M > 0. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 10 / 77

A constructive optimal solution to these C 1 , 1 extension problems. A constructive optimal solution to these C 1 , 1 extension problems. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 11 / 77

A constructive optimal solution to these C 1 , 1 extension problems. Theorem (Azagra-Le Gruyer-Mudarra, 2017) Let ( f , G ) be a 1 -jet defined on an arbitrary subset E of a Hilbert space X. There exists F ∈ C 1 , 1 conv ( X ) such that ( F , ∇ F ) extends ( f , G ) if and only if f ( x ) ≥ f ( y ) + � G ( y ) , x − y � + 1 2 M | G ( x ) − G ( y ) | 2 x , y ∈ E , for all where | G ( x ) − G ( y ) | M = M ( G , E ) := . sup | x − y | x , y ∈ E , x � = y The function � � y ∈ E { f ( y ) + � G ( y ) , x − y � + M 2 | x − y | 2 } F ( x ) = conv inf defines such an extension, with the property that Lip ( ∇ F ) ≤ M. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 12 / 77

A constructive optimal solution to these C 1 , 1 extension problems. Recall that conv ( g )( x ) = sup { h ( x ) : h is convex and continuous, h ≤ g } . Other useful expressions for conv ( g ) are given by    n + 1 n + 1 n + 1  � � � λ j x j , n ∈ N conv ( g )( x ) = inf λ j g ( x j ) : λ j ≥ 0 , λ j = 1 , x =   j = 1 j = 1 j = 1 (or with n fixed as the dimension of R n in the case X = R n ), and by the Fenchel biconjugate of g , that is, conv ( g ) = g ∗∗ , where h ∗ ( x ) := sup v ∈ R n {� v , x � − h ( v ) } . C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 13 / 77

A constructive optimal solution to these C 1 , 1 extension problems. Corollary (Wells 1973, Le Gruyer 2009, Azagra-Le Gruyer-Mudarra 2017) Let E be an arbitrary subset of a Hilbert space X, and f : E → R , G : E → X. There exists F ∈ C 1 , 1 ( X ) such that F | E = f and ( ∇ F ) | E = G if and only if there exists M > 0 so that f ( y ) ≤ f ( x )+ 1 2 � G ( x )+ G ( y ) , y − x � + M 4 | x − y | 2 − 1 4 M | G ( x ) − G ( y ) | 2 ( W 1 , 1 ) for all x , y ∈ E. Moreover, F = conv ( g ) − M 2 | · | 2 , where 2 | x − y | 2 } + M 2 | x | 2 , y ∈ E { f ( y ) + � G ( y ) , x − y � + M g ( x ) = inf x ∈ X , defines such an extension, with the additional property that Lip ( ∇ F ) ≤ M. C 1 , 1 and C 1 , 1 Daniel Azagra loc convex extensions of jets Fitting Smooth Functions to Data 14 / 77