Multivariate polynomial approximation and convex bodies

Transcription

1 Multivariate polynomial approximation and convex bodies

2 Outline 1 Part I: Approximation by holomorphic polynomials in C d ; Oka-Weil and Bernstein-Walsh 2 Part 2: What is degree of a polynomial in C d, d > 1? Bernstein-Walsh, revisited 3 Part 3: P extremal functions in C d (P a convex body in (R + ) d ) Mostly joint work with L. Bos; based on joint work with T. Bayraktar and T. Bloom.

3 Part I: Approximation by holomorphic polynomials Let P n denote the space of holomorphic polynomials of degree at most n in C d, d 1. Let f be a continuous complex-valued function on a compact set K C d. Let K := {z C d : p(z) p K = max ζ K p(ζ), all p np n }. Theorem (Oka-Weil) If K = K, then any f holomorphic on a neighborhood of K can be approximated uniformly on K by holomorphic polynomials. For d = 1, K = K if and only if K has connected complement (non-example: K = T := {z : z = 1}) and this is a version of Runge s theorem.

4 Sketch of Oka-Weil: Oka s proof Let f be holomorphic on a neighborhood U of K C d (i.e., f : U C is holomorphic in each variable separately). 1 If K = K d := {z : z j < 1, j = 1,..., d}, can approximate K by polynomial polyhedron: Π = {z : z j 1, j = 1,..., d, p k (z) 1, k = 1,..., l}, p k polynomials; i.e., K Π U. 2 Lift problem to d+l C d+l : we consider z (z, p 1 (z),..., p l (z)), i.e., the graph of (p 1,..., p l ). Using elementary theory on polynomial polyhedra, get F holomorphic in neighborhood of d+l with F (z, p 1 (z),..., p l (z)) = f (z), z Π (Oka extension theorem). 3 Expand F in Taylor series and truncate. Note: Runge and Oka-Weil are not quantitative.

5 Quantitative Runge and Oka-Weil: Bernstein-Walsh A result quantifying the Runge and Oka-Weil theorems is the Bernstein-Walsh theorem. The key tool is an extremal function: V K (z) := sup{u(z) : u L(C d ), u 0 on K} 1 = max[0, sup{ deg(p) log p(z) : p K := max p(ζ) 1}] ζ K where L(C d ) := {u PSH(C d ) : u(z) log z = 0(1), z } and p n P n. Here K C d is compact. For p n P n, p n (z) p n K exp(nv K (z)), z C d. (BW ) 1 K = {z : z a r}: V K (z) = log + z a r := max[0, log z a r ] et p n (z) p n K z a n pour z a > r. 2 K = [ 1, 1] C: V K (z) = log z + z 2 1.

6 Bernstein-Walsh For a continuous complex-valued function f on K, let D n (f, K) := inf{ f p n K : p n P n }. Theorem Let K C d (d 1) be compact with V K continuous. Let R > 1, and let Ω R := {z : V K (z) < log R}. Let f be continuous on K. Then lim sup D n (f, K) 1/n 1/R n if and only if f is the restriction to K of F holomorphic in Ω R.

7 Easy direction: lim sup n D 1/n n 1/R for some R > 1 To show f extends hol. to Ω R, take p n with D n = f p n K. Claim: p (p n p n 1 ) converges locally uniformly on Ω R to a holomorphic function F which agrees with f on K. Proof: Fix R with 1 < R < R; by hypothesis f p n K M R for n some M > 0. Fix 1 < ρ < R, and apply (BW) to the polynomial p n p n 1 P n on Ω ρ to obtain sup p n (z) p n 1 (z) ρ n p n p n 1 K Ω ρ ρ n ( p n f K + f p n 1 K ) ρ n M(1 + R ) R n. Since ρ and R were arbitrary with 1 < ρ < R < R, p (p n p n 1 ) converges locally uniformly on Ω R to a holomorphic function F. Converse requires (pluri-)potential theory.

8 Converse proof of BW in C using polynomial interpolation Let Γ C be a rectifiable Jordan curve with z 1,..., z n inside Γ, and let f be holomorphic inside and on Γ. Let L n 1 (z) be the Lagrange interpolating polynomial (LIP) associated to f, z 1,..., z n : L n 1 P n 1 and L n 1 (z j ) = f (z j ), j = 1,..., n. Proposition (Hermite Remainder Formula) For any z inside Γ, f (z) L n 1 (z) = 1 ω n (z) f (t) 2πi Γ ω n (t) (t z) dt, where ω n (z) = n k=1 (z z k). Take z (n) 1,..., z(n) n Fekete points of order n 1 for K: ω n satisfies 1 n log( ω n (z) ω n K ) VK (z) loc. unif. in C \ K. For f holomorphic in Ω R take LIP s for f, z (n) 1,..., z(n) n ; Γ Ω R.

9 Remarks on a converse proof in C d, d > 1 (Siciak/Bloom) Step 1: Let b n =dimp n and Q 1,..., Q bn be a basis for P n. Define D R := {z C d : Q j (z) < R n, j = 1,..., b n }. Proposition Let f be holomorphic in a neighborhood of D R. Then for each positive integer m, there exists G m P m such that for all ρ R, f G m Dρ B(ρ/R) m where B is a constant independent of m. Proof: Use a version of lifting result of Oka. Let S : C d C bn via S(z) := (Q 1 (z),..., Q bn (z)). Then S(C d ) is a subvariety of C bn and S(D R ) is a subvariety of the polydisk R := {ζ C bn : ζ j < R n, j = 1,..., b n }. Oka: Get F holomorphic in R C bn with F S = f on D R.

10 Remarks, cont d Step 2: Construct polynomials Q 1,..., Q bn so that for n large the sets D R approximate the sublevel sets Ω R of V K (for 0 < R 1 < R, there exists n 0 such that for all n n 0, D R1 Ω R D R ). We use Fekete points {z (n) j } of order n for K to construct fundamental Lagrange interpolating polynomials Q j := l (n) j P n ; so l (n) j (z (n) k ) = δ jk and l (n) j K = 1. More later... Step 3: For any R < R the Lagrange interpolating polynomials L n (f ) for f associated with a Fekete array for K satisfy f L n (f ) K B/(R ) n where B is a constant independent of n.

11 Part 2: What is degree of a polynomial in C d, d > 1? Above, the degree of a polynomial is its total degree; i.e., if p(z) = c α z α, c α C, z α = z α 1 1 zα d d, the total degree of p is its maximal l 1 degree: max{ α := α α d : c α 0}. Trefethen (PAMS, 2017) considered D n (f, K) for l 1 degree (total degree), l 2 degree (Euclidean degree) or l -degree (max degree). He compared these three (numerically) for K = [ 1, 1] d and f a multivariate Runge-type function, e.g., f (z) := 1 r 2 + d j=1 z2 j, r > 0. This f is holomorphic in a neighborhood of K in C d. We explain exactly his numerical conclusions.

12 The right framework Fix a convex body P (R + ) d ; i.e., P is compact, convex and P o (e.g., P is a non-degenerate convex polytope, i.e., the convex hull of a finite subset of (Z + ) d in (R + ) d with nonempty interior). We consider the finite-dimensional polynomial spaces Poly(nP) := {p(z) = c J z J : c J C} J np (Z + ) d for n = 1, 2,... where z J = z j 1 1 zj d d for J = (j 1,..., j d ). We let b n be the dimension of Poly(nP). For P = Σ where Σ := {(x 1,..., x d ) R d : 0 x i 1, d x i 1}, j=1 we have Poly(nΣ) = P n, the usual space of holomorphic polynomials of degree at most n in C d. We assume that Σ kp for some k Z + : note 0 P so Poly(nP) are linear spaces.

13 Convexity and remark on degree Convexity of P implies that p n Poly(nP), p m Poly(mP) p n p m Poly((n + m)p). We may define an associated norm for x = (x 1,..., x d ) R d + via x P := inf {x λp} λ>0 and define a general degree of a polynomial p(z) = α Z d + a αz α associated to the convex body P as deg P (p) := max a α 0 α P. Then Poly(nP) = {p : deg P (p) n}.

14 Trefethen degree: PAMS, 2017 For q 1, if we let P q := {(x 1,..., x d ) : x 1,..., x d 0, x q x q d 1} be the (R + ) d portion of an l q ball then we have, in Trefethen s notation (on the left), d T (p) := deg P1 (p) d E (p) := deg P2 (p) d max (p) := deg P (p) (total degree); (Euclidean degree); (max degree). Example p(z 1, z 2 ) = z 2 1 z3 2 has deg P 1 (p) = 5; deg P2 (p) = 13; deg P (p) = 3 so p Poly(5P 1 ) Poly(4P 2 ) Poly(3P ). We fix a convex body P (R + ) d with Σ kp for some k Z +.

15 Bernstein-Walsh, revisited For K C d compact and nonpluripolar and f continuous on K, [ 1 V P,K (z) := lim sup{ n n log p n(z) : p n Poly(nP), p n K 1} ] Theorem and D n (f, P, K) := inf{ f p n K : p n Poly(nP)}. (Bos-L.) Let K be compact with V P,K continuous. Let R > 1 and Ω R := Ω R (P, K) = {z : V P,K (z) < log R}. Let f be continuous on K. Then lim sup D n (f, P, K) 1/n 1/R. n if and only if f is the restriction to K of F holomorphic in Ω R.

16 The proof that lim sup n D n (f, P, K) 1/n 1/R implies f is the restriction to K of a function holomorphic in Ω R works exactly as before. For the converse direction, the proof is a modification of the case P = Σ: 1 use a version of the lifting result of Oka on D R sets 2 construct basis {Q j } for Poly(nP) so D R approximate Ω R 3 Use Lagrange interpolating polynomials L n (f ) for f at P Fekete points of K. Remark: The true definition of V P,K will be given later. We will give some examples first and discuss P Fekete points (and more general P pluripotential theory) later.

17 Examples: V P,K for product sets For P R d a convex body, the indicator function is φ P (x 1,..., x d ) := sup (x 1 y 1 + x d y d ). (y 1,...,y d ) P Proposition Let E 1,..., E d C be compact with V Ej continuous. Then V P,E1 E d (z 1,..., z d ) = φ P (V E1 (z 1 ),..., V Ed (z d )). We use a global domination principle (TBD later). In particular, for K = T d = {(z 1,..., z d ) : z 1 = = z d = 1}, the unit d torus in C d, V P,T d (z) = H P (z) := max J P log zj = φ P (log + z 1,..., log + z d ) (logarithmic indicator function of P). Here z J := z 1 j1 z d j d.

18 Trefethen Runge-type example Let 1/q + 1/q = 1 so φ Pq (x) = x lq. If E 1,..., E d C, V Pq,E 1 E d (z 1,..., z d ) = [V E1 (z 1 ), V E2 (z 2 ), V Ed (z d )] lq = [V E1 (z 1 ) q + + V Ed (z d ) q ] 1/q. For the particular product set, K := [ 1, 1] d where E j = [ 1, 1] for j = 1,..., d, that Trefethen considers, V Ej (z j ) = log z j + zj 2 1 and hence we have for q > 1 d ( V Pq,[ 1,1] d (z 1,..., z d ) = log z j + zj 2 1 j=1 1 For f (z) := r 2 +z1 2+ +z2 d except on its singular set S = {z C d algebraic variety having no real points. ) q and K = [ 1, 1] d, f is holomorphic 1/q. : d j=1 z2 j = r 2 }, an

19 Trefethen Runge-type example, cont d Note V P1,[ 1,1] d (z 1,..., z d ) = max j=1,...,d log z j + zj 2 1. By the theorem, the D n (f, P, K) decay like 1 R n where R = R(P, K) := sup{r > 0 : Ω R S = }. Clearly log(r(p, K)) = min z S V P,K (z). We show (Bos.-L): R(P 1, K) = r/ d r 2 /d < r r 2 = R(P 2, K) = R(P, K) (indep. of d!). Thus the approximation order of the Euclidean degree is considerably higher than for the total degree, while the use of max degree provides no additional advantage, as reported by Trefethen.

20 Unfair! This is not fair: the dimensions of {p : deg P (p) n} are proportional (asymptotically) to the volume vol d (P): dim({p : deg P (p) n}) = dim(poly(np)) vol d (P) n d. To equalize their dimensions we scale P q by c = c(q) = ( ) vold (P 1 ) 1/d vol d (P q ) and observe that R(cP, K) = (R(P, K)) c. For d = 2 we compare R(P 1, K) = r/ r 2 /2, R(P 2, K) c(2) = ( r r 2) 2/π. We have R(P 2, K) c(2) > R(P 1, K) for small r (r < ); so Euclidean degree, even normalized, has a better approximation order than the total degree case, but now with a lesser advantage.

21 Conclusion and questions The bottom line is: the Bernstein-Walsh type theorem shows that the geometry of the singularities of the approximated function f provided f is, indeed, holomorphic on a neighborhood of K! relative to the sublevel sets of the P extremal function V P,K govern the asymptotics of the sequence {D n (f, P, K)} and hence the number R(P, K) (which we should write as R(P, K, f )). Given K, P, f, let Ω R(P,K) := {z : V P,K (z) < log R(P, K)} where lim sup n D n (f, P, K) 1/n = 1/R(P, K). Questions: 1 Given K, f find the best P (with equalized dim(poly(np))) so that R(P, K) is largest. 2 If lim sup n D n (f, K) 1/n = lim sup n D n (g, K) 1/n = 1/R, then f, g are holomorphic in Ω R = {z : V K (z) < log R} (classical Bernstein-Walsh). Which is better?

22 Example Example In C, let K = {z : z 1/R}. Then f (z) = 1 z 1 is better than g(z) = z n! as holomorphic functions in the unit disk Ω R. We can t detect this, but in C d, d > 1, given K, f, we can compute Ω R(P,K) for various convex bodies P to get a better picture of the true region of holomorphicity of f : it contains P Ω R(P,K). For real theory, we can look, e.g., at traces Ω R(P,K) R d for f real-analytic on a neighborhood of K R d. We return to the proof of the hard direction of the theorem: to show that f holomorphic in Ω R = {z : V P,K (z) < log R} implies lim sup D n (f, P, K) 1/n 1/R. n

23 Sketch of proof Step 1: Let b n =dimpoly(np). Let Q 1,..., Q bn Poly(nP). Define be a basis for D R := {z C d : Q j (z) < R n, j = 1,..., b n }. Proposition Let f be holomorphic in a neighborhood of D R. Then for each positive integer m, there exists G m Poly(mP) such that for all ρ R, f G m Dρ B(ρ/R) m where B is a constant independent of m. Proof follows P = Σ case and requires S : C d C bn via S(z) := (Q 1 (z),..., Q bn (z)) be one-to-one on D R (follows for n k where Σ kp).

24 Sketch of proof, continued Step 2: Construct polynomials Q 1,..., Q bn so that for n large the sets D R approximate the sublevel sets Ω R of V P,K ; i.e., given 0 < R 1 < R, there exists n 0 such that for all n n 0, D R1 Ω R. Use P Fekete points and fundamental Lagrange interpolating polynomials to construct Q j. Here 1 V P,K (z) = lim n n log Φ P,n(z) where Φ P,n (z) := sup{ p n (z) : p n Poly(nP), p n K 1} and if V P,K (z) is continuous, the convergence is locally uniform on C d. Step 3: For any R < R the Lagrange interpolating polynomials L n (f ) for f associated with a P Fekete array for K satisfy f L n (f ) K B/(R ) n where B is a constant independent of n.

25 Part 3: P extremal functions in C d We briefly describe some basics of the P pluripotential theory. Associated to P (R + ) d a convex body, recall we have the logarithmic indicator function on C d H P (z) := sup J P log z J := sup log[ z 1 j1 z d j d ]; J P L P = L P (C d ) := {u PSH(C d ) : u(z) H P (z) = 0(1), z }. For p Poly(nP), n 1 we have 1 n log p L P. Given K C d, the P extremal function of K is V P,K (z) where the a priori definition of V P,K is V P,K (z) := sup{u(z) : u L P (C d ), u 0 on K}. This is a Perron-Bremmerman family:(dd c V P,K ) d = 0 outside K. Example: K = T d, the unit d torus in C d. Then V P,T d (z) = H P (z) = sup log z J. J P

26 Example: V P,T d(z) = H P (z) = sup J P log z J Here, (dd c V P,T d ) d = d!vol(p) dθ (2π) d 1 dθ d is the Monge-Ampère measure of V P,T d. For a C 2 function u on C d, (dd c u) d = dd c u dd c 2 u u = c d det[ ] j,k dv z j z k where dv = ( i 2 )d d j=1 dz j d z j is the volume form on C d and c d is a dimensional constant. Here, dd c u = 2i n j,k=1 2 u z j z k dz j d z k = u dv in C. If u locally bounded psh (dd c u) d well-defined as positive measure.

27 P extremal functions and P = Σ case Using Hörmander L 2 theory, T. Bayraktar showed: Theorem (T. Bayraktar) We have [ 1 V P,K (z) = lim sup{ n n log p n(z) : p n Poly(nP), p n K 1} ]. This is the starting point to develop a P pluripotential theory. For P = Σ = {(x 1,..., x d ) R d : 0 x i 1, d x i 1}, j=1 Poly(nΣ) = P n, and we recover classical pluripotential theory: H Σ (z) = max[0, log z 1,..., log z d ] = max[log + z 1,..., log + z d ] and L Σ = L; V Σ,K = V K.

28 Remark: Global Domination Principle We have L + P := {u L P : u(z) H P (z) + c u }. We call u P PSH(C d ) a strictly psh P potential if 1 u P L + P is strictly psh and 2 there exists C such that u P (z) H P (z) C for all z C d. We have existence of u P which may replace H P : and L P = {u PSH(C d ) : u(z) u P (z) = 0(1), z } L + P = {u L P : u(z) u P (z) + c u }. Using u P we can prove a global domination principle: Theorem Let u L P and v L + P with u v a.e. (dd c v) d. Then u v in C d.

29 Discretization Recall b n is the dimension of Poly(nP). We can write Poly(nP) = span{e 1,..., e bn } where {e j (z) := z α(j) } j=1,...,bn are the standard basis monomials. The ordering is unimportant. For points ζ 1,..., ζ bn C d, let VDMn P (ζ 1,..., ζ bn ) := det[e i (ζ j )] i,j=1,...,bn (1) e 1 (ζ 1 ) e 1 (ζ 2 )... e 1 (ζ bn ) = det e bn (ζ 1 ) e bn (ζ 2 )... e bn (ζ bn ) For K C d compact, P Fekete points of order n maximize VDMn P (ζ 1,..., ζ bn ) over ζ 1,..., ζ bn K. Let l n := b n j=1 deg(e j). Then (nontrivial!) the limit δ(k, P) := lim n max ζ 1,...,ζ bn K VDMP n (ζ 1,..., ζ bn ) 1/ln exists.

30 Asymptotic P Fekete arrays and (dd c V P,K ) d Theorem Let K C d be compact. For each n, take points z (n) 1, z(n) 2,, z(n) b n K for which lim n VDMP n (z (n) 1,, z(n) b n ) 1 ln = δ(k, P) (asymptotic P Fekete arrays) and let µ n := 1 b n bn µ n 1 d!vol(p) (dd c V P,K ) d weak. j=1 δ z (n) j. Then The proof of these results rely on techniques from Berman and Boucksom, Invent. Math., (2010) and involve weighted notions of V P,K and δ(k, P). See Bayraktar, Bloom, L., Pluripotential Theory and Convex Bodies. Remark: (dd c V P,K ) d is the target for zeros of random polynomial mappings and our motivation.

31 Questions: product sets Recall V P,E1 E d (z 1,..., z d ) = φ P (V E1 (z 1 ),..., V Ed (z d )). For T d = {(z 1,..., z d ) : z j = 1, j = 1,..., d} and 1 q, (dd c V Pq,T d )d is a multiple of Haar measure on T d and for [ 1, 1] d, d ( V Pq,[ 1,1] d (z 1,..., z d ) = log z j + zj 2 1 j=1 ) q 1/q. 1 Is supp (dd c V Pq,[ 1,1] d )d = [ 1, 1] d always? 2 More generally, what can one say about supp(dd c V Pq,E 1 E d ) d for 1 < q <?

32 Other sets: the complex Euclidean ball in C 2 For K = B 2 = {(z 1, z 2 ) C 2 : z z 2 2 1} and P = P, we have shown: { 1 2 log( z2 2 ) log(1 z 1 2 ) } z 1 2 1/2, z 2 2 1/2 { V P,B2 (z) = 1 2 log( z1 2 ) log(1 z 2 2 ) } z 1 2 1/2, z 2 2 1/2 log z 1 + log z 2 + log(2) z 1 2 1/2, z 2 2 1/2. Thus the measure (dd c V P,B 2 ) 2 is Haar measure on the torus { z 1 = 1/ 2, z 2 = 1/ 2} (with total mass 2). On the other hand, it is well known that (dd c V P 1,B 2 ) 2 is normalized surface measure on B 2. What can we say about supp(dd c V P q,b 2 ) 2 for 1 < q <? What is V P q,b 2?

33 A final question... from numerical analysis What can one say if P is not convex? For example, let P q := {(x 1,..., x d ) : x 1,..., x d 0, x q x q d 1}, for 0 < q < 1. Is there a Bernstein-Walsh theorem? Even for q = 0 : P 0 := d j=1{(x 1,..., x d ) : 0 x j 1, x k = 0, k j}. Let K = [ 1, 1] 2 R 2 C 2 and f (z, w) = g(z) + h(w), g, h holomorphic in a neighborhood of [ 1, 1]. Here one should only use polynomials of the form p(z) + q(w) and thus D n (f, P q, K) = D n (f, P 1, K), 0 q 1. What is the true extremal function V P,K?

34 CONGRATULATIONS to TOM!!! THANK YOU ALL!!!