On the best approximation of function classes from values on a uniform grid in the real line

Transcription

1 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, On te best approximation of function classes from values on a uniform grid in te real line GENNADI VAINIKKO Institute of Applied Matematics Tartu University J Liivi, Tartu ESTONIA gennadivainikko@utee Abstract: Given te values of function f on a uniform grid of a step size in te real line, we construct te spline interpolant of order m, defect 1 using te B-spline basis obtained by wavelet-type dilation and sifts from te fater B-spline wic can be considered as a scaling function for a special (nonortogonal) spline-wavelet system We establis an unimprovable error estimate on te class of functions wit bounded mt derivative, and we sow tat in te sense of worst case, oter approximations to f using te same information about f are more coarse tan te spline interpolant Some furter results of tis type are establised Key Words: splines, nonortogonal wavelets, interpolation, error estimates, exact constants 1 Introduction Tis paper is devoted to unimprovable error estimates for spline interpolation and to te optimality of spline interpolants compared wit oter approximations Te spline interpolation as been widely examined during te last years, see in particular te monograps [1, 3, 5, 6, 10] and te references tere Neverteless, tere are open problems concerning exact (unimprovable) constants in error estimates for interpolants and quasi-interpolants Suc results are of great general interest; our special interest is caused by examining fast solvers for integral equations (see, eg, [7, 9]) In Section of te present paper we discuss te construction of te spline interpolants of functions on te real line Given te values of a function f on a uniform grid R of a step size, we construct te spline interpolant Q,m f of order m, defect 1 using te B-spline basis obtained by wavelet-type dilation and sifts from te fater B-spline B m wic can be considered as a scaling function for a special (nonortogonal) wavelet system, see [] Tecnically, our approac is equivalent to tat in [6] but we use anoter start idea Our main results are presented in Section 3 We establis an unimprovable estimate for f Q,m f on te class V m, (R) of functions wit bounded mt derivative Tis estimate essentially extends a result of [3] concerning 1-periodic functions f and te step size of te form = 1/n wit even n N Furter, we sow tat in te sense of te worst case in V m, (R), oter approximations to f using te same information about f as Q,m f are more coarse tan Q,m f We also present some furter estimates for Q,m f, in particular, error estimates for derivatives and error estimates in te case of modestly smoot f We use te following standard notations: R = (, ), N = {1,, }, Z = {,, 1, 0, 1,, } Let us caracterise more precisely te spaces of functions on R used in te sequel As usual, C(R) is te space of continuous functions on R, and C m (R) is te space of functions on R tat ave continuous derivatives up to te order m By BC(R) we mean te Banac space of bounded continuous functions f on R equipped wit te norm f = sup f(x) ; x R BUC(R) is te (closed) subspace of BC(R) consisting of bounded uniformly continuous functions on R Te Sobolev space W m, (R), m N, consists of functions f suc tat f itself and its derivatives up to te order m are measurable, bounded functions on R (actually ten f, f,, f (m 1) are continuous in R; te derivatives are understood in te sense of distributions) Finally, te Sobolev space V m, (R) consists of functions f suc tat f (m) is measurable and bounded in R; ten f, f,, f (m 1) are continuous

2 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, but not necessarily bounded in R Wit te elp of te Taylor formula m 1 f (l) (0) f(x)= x l 1 + l! (m 1)! l=0 x 0 (x t) m 1 f (m) (t)dt we observe tat for f V m, (R), x it olds f(x) 1 m! f (m) x m + O(x m 1 ) Tere is an equivalent way to define V m, (R) as te space of functions f C (m 1) (R) suc tat f (m 1) is uniformly Lipscitz continuous: f (m 1) (x 1 ) f (m 1) (x ) L f x 1 x, x 1, x R A Lipscitz continuous function, f (m 1) in our case, is differentiable almost everywere as well as differentiable in te sense of distributions, and inf L f = f (m) = vraisup x R f (m) (x) were te infimum is taken over all Lipscitz constants L f for f (m 1) We do not need norms in W m, (R) and V m, (R) Clearly, W m, (R) + P m V m, (R); tis inclusion is strict Construction of te interpolant Te formula for te fater B-spline B m is given by ( ) m B m (x) = 1 m (m 1)! i=0( 1) i (x i) m 1 i +, x R, m N, were, as usual, 0! = 1, 0 0 := lim x 0 x x = 1, { (x i) m 1 (x i) + := m 1, x i 0 0, x i < 0 Some properties of B m, m, are as follows: } B m [i,i+1] P m 1 (polynomials of degree m 1), i Z, B m C (m ) (R), ie, B m is a spline of degree m 1, defect 1 on te cardinal knot set Z, supp B m = [0, m], B m (x) > 0 for 0 < x < m, B m (x) = B m (m x), 0 x m, ( B m ) m = max B m(x), x R B m (x)dx = 1, B m (x j) = 1, x R R Introduce in R te uniform grid Z = {i: i Z} of a step size > 0 Denote by S,m, m N, te space of splines of order m (or, of degree m 1) and defect 1 wit te knot set Z Clearly te dilated and sifted B-splines B m ( 1 x j), j Z, belong to S,m, and te same is true for d j B m ( 1 x j) wit arbitrary coefficients d j ; tere are no problems wit te convergence of te series since it is locally finite: for x [i, (i + 1))), i Z, it olds d j B m ( 1 x j) = i j=i m+1 d j B m ( 1 x j) Given a function f C(R) of possibly polynomial growt as x, we look for te interpolant Q,m f S,m in te form (Q,m f)(x) = d j B m ( 1 x j), x R, (1) and determine te coefficients d j from te interpolation conditions (Q,m f)((k + m )) = f((k + m )), k Z () Tis leads to te bi-infinite system of linear equations B m (k + m j)d j = f((k + m )), k Z, or b k j d j = f k, k Z, (3) were for k Z b k =b k,m =B m (k+ m ), f k =f k,,m =f((k+ m )), (4) b k = b k > 0 for k µ, b k = 0 for k > µ, b k = 1, (5) µ := int ((m 1)/) = k µ { (m )/, m even (m 1)/, m odd } Tus (3) is a bi-infinite system wit te symmetric Toeplitz band matrix B = (b k j ) k, of te band widt µ + 1 For m =, system (3) reduces to relations d k = f((k + 1))), k Z, and (Q, f)(x) = f((j+1)))b (nx j) is te usual piecewise linear interpolant wic can be constructed on every subinterval [i, (i + 1)] independently from oter subintervals All is clear in te cases m = 1,

3 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, and we focuse our attention to te case m 3 A delicate problem appears tat te solution of system (3) always exists but is nonunique for m 3 if we allow an exponential growt of d j as j Only one of te solutions of system (3) is reasonable We call it te Wiener solution since it is related to te Wiener teorem (see [11]) about trigonometric (or equivalent Laurent) series Te following construction of te Wiener interpolant is in more details elaborated in [8] and it is equivalent to tat in [6] Wit b k = b k,m = B m (k + m ) defined in (4), introduce te following functions: b(z) = b m (z) := µ b k z k = b 0 + b k (z k + z k ), k µ k=1 (6) P µ (z) = P m µ(z) = z µ b(z) (7) (te caracteristic polynomial of B m ) Denote by z ν = z ν,m, ν = 1,, µ, te roots of te caracteristic polynomial Pµ m P µ (we call tem te caracteristic roots) From (6) we observe tat togeter wit z ν also 1/z ν is a caracteristic root Te polynomials (7) were introduced in [6] starting from different considerations As proved in [6], all caracteristic roots are real and simple; ten clearly z ν < 0, ν = 1,, µ, and z ν 1, ν = 1,, µ, tus tere are exactly µ caracteristic roots z ν, ν = 1,, µ, in te interval ( 1, 0) and µ caracteristic roots z µ+ν = 1/z ν, ν = 1,, µ in te interval (, 1) It is not complicated to sow tat te function a(z) = a m (z) := 1/b m (z) = z µ /P m µ(z) (8) as te Laurent expansion a m (z)= a k,m z k wit a k,m = µ ν=1 z µ 1 ν,m P µ (z ν,m) z k ν,m = a k,m, k Z, (9) and te Wiener solution of system (3) is given by d k = a k j,m f((j + m )), k Z (10) An important observation from (9) is tat a k,m decays exponentially as k, tus te series in (10) converges provided tat f is bounded or of a polynomial growt as x Tus we ave te following result Furter properties of a k,m is tat a k,m = 1, a k,m = ( 1)µ P µ ( 1), a k,m = ( 1) k a k,m 0, k Z In te vector space X of all bisequences (d j ), te null space N (B) of te matrix B = (b k j ) k, of system (3) is of dimension µ being spanned by bisequences (zν) j, ν = 1,, µ Hence for any nontrivial (d (0) j ) N (B), d (0) j grows exponentially as j or as j Clearly, Q,m BC(R) BC(R) a k,m but tis estimate is coarse To present an exact formula for Q,m BC(R) BC(R), introduce te fundamental spline F m (x) := a k,m B m (x k); it decays exponentially as x and satisfies F m (j + m ) = δ j,0, j Z, were δ j,k is te Kronecker symbol Clearly, (Q,m f)(x) = f((j + m ))F m( 1 x + j), and tis implies Q,m BC(R) BC(R) = sup F m (x + j) x R = max F m (x + j) x [ m, m+1 ] Numerical values of q m := Q,m BC(R) BC(R) and α m := a k,m for some m are presented in te following table taken from [4]: m q m α m For 4 m 0, te computed values of q m fit into te model q m e 4 + π logm, and it seems tat q m ( e 4 + π logm) 0 as m ; for m = 0 tis difference is of order 0001 We can also observe tat α m+1 /α m π/ = as m ; for m = 0 tis ratio is , already very close to π/ It is callenging to confirm tese empiric guesses analytically Finally, it is easily seen tat for any bisequence d j, j Z, wit sup j d j <, it olds Teorem 1 For a bounded or polynomially growing f C(R), te Wiener interpolant Q,m f is well defined by te formulae (1), (9), (10) sup j d j α m sup d j B( 1 x j) sup d j x R j

4 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, Error estimates of te spline interpolant To formulate te main results of te paper, we need some information concerning te Euler splines [3] A spline E S,m satisfying E (m 1) (x) = ( 1) i for i < x < (i+1), i Z, is called perfect If E S,m is perfect ten so is E + g wit any g P m For m = 1, te Euler perfect spline E,1 S,1 is defined by te formula E,1 (x)=sign sin( 1 πx)= 4 sin(k+1) 1 πx π k + 1 (11) For m, te Euler perfect spline E,m S,m is determined recursively as a special integral function of E,m 1, namely, E,m (x) = x / E,m 1 (y)dy, m = l x E,m 1 (y)dy, m = l ; te lower bounds of integration are cosen so tat te -periodicity and te zero mean value of E,m 1 over a period is inerited to E,m Starting from (11) we recursively find tat = 4 ( 1) l m 1 π π m 1 4 ( 1) l m 1 π π m 1 E,m (x) cos(k+1) 1 πx (k+1), m = l m sin(k+1) 1 πx (k+1) m, m = l + 1 (1) By construction, E,m = E,m 1 for m Furter, we observe tat x = (i + 1 ), i Z, are te zeroes of E,m for even m, and x = i, i Z, are te zeroes of E,m for odd m A unified formulation is tat x = (i + m 1 ), i Z, are te zeroes of E,m and x = (i + m ), i Z, are te local extrema of E,m (te zeroes of E,m = E,m 1) Tere are no oter zeroes and extrema of E,m tis can be easily seen recursively, since by Rolle s teorem an additional zero of E,m involves an additional zero of E,m = E,m 1 It is clear also tat te zeroes of E,m are simple Furter, for m = l, E,m = E,m (0) = 4 π m 1 π m 1 1 (k + 1) m (te absolute value of E,m at oter local extremum points x = i is same) Similarly, for m = l + 1, E,m = E,m ( ) = 4 m 1 ( 1) k π π m 1 (k + 1) m Unifying tese two formulae, we can write wit E,m = Φ m π (m 1) m 1, m N, (13) Φ m = 4 π 1 (k+1) m, m = l ( 1) k (k+1) m, m = l + 1 known as te Favard constant In particular,, m N, (14) Φ 1 = 1, Φ = π/, Φ 3 = π /8, Φ 4 = π 3 /4, and it olds lim m Φ m = 4 π, Φ 1 < Φ 3 < Φ 5 < < 4 π < < Φ 6 < Φ 4 < Φ We are ready to formulate our main result Teorem For f V m, (R), m N, tere old te pointwise estimate f(x) (Q,m f)(x) f (m) E,m+1 (x), x R, (15) and te uniform estimate f Q,m f Φ m+1 π m m f (m) (16) For f = E,m+1 W m, (R) V m, (R), inequalities (15) and (16) turn into equalities Tis teorem is known [3] in te case of 1- periodic f and = 1/n wit an even n N (ten also Q,m f and E,m+1 are 1-periodic) Actually, only (15) is a complicated assertion for te proof, wereas (16) is an obvious consequence of (15), (13), and te assertion about te sarpness of estimates for f = E,m+1 is elementary We step after step extend te teorem first for functions f wit compact support, ten for f V m, (R) of slow growt as x, and finally for arbitrary f V m, (R) Tecnically, te extensions are based on te following lemma; for details, see [8] Lemma 3 Suppose tat functions g δ C(R), δ > 0, satisfy g δ (x) 0 as δ 0 x R, g δ (x) c(1 + x r ) x R were r 0 Ten (Q,m g δ )(x) 0 as δ 0 x R

5 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, Remark 4 Using Banac Steinaus teorem and Teorem, it is easily seen tat for f BUC(R), f Q,m f 0 as 0 Let us discuss optimality properties of te spline interpolation compared wit oter metods tat use te same information about te values of f on te uniform grid = {(j + m ) : j Z} Suc a metod can be identified wit a mapping M : C( ) C(R) were C( ) is te vector space of grid functions defined on and aving values in R or C Remark 5 For given γ > 0, we ave in accordance to Teorem sup f Q,m f =Φ m+1 π m m γ f V m, (R), f (m) γ wereas for any mapping M : C( ) C(R) (linear or nonlinear, continuous or discontinuous), it olds sup f M (f ) f V m, (R), f (m) γ Φ m+1 π m m γ (17) Indeed, (17) is trivially fulfilled if M (0) / BC(R), so we may assume tat M (0) BC(R) Consider two functions f ± = ±γe,m+1 Clearly, f ± W m, (R) V m, (R), f (m) ± = γ, and since E,m+1 = 0, we obtain (17) by te following argument: sup f M (f ) f V m, (R), f (m) γ max{ f + M (f + ), f M (f ) } = max{ f + M (0), f M (0) } 1 ( f + M (0) + f M (0) ) 1 f + f = E,m+1 γ =Φ m+1 π m m γ Remark 6 Let = 1/n wit an even n N Consider te subspace C per (R) of C(R) consisting of 1-periodic continuous functions on R, and denote Wper m, (R) = C per (R) W m, (R); denote by C per ( ) te space of 1-periodic (grid) functions on te grid, ie, f (i) = f (1 + i), i Z, for f C per ( ) Ten for any mapping M : C per ( ) C per (R), it olds f W m, per sup (R), f (m) γ Φ m+1 π m m γ f M (f ) Te proof is same as in te case of Remark 5, we only need to observe tat E,m+1 Wper m, (R) for = 1/n wit an even n N Remark 7 For functions wit compact supports, similar result as Remark 5 olds asymptotically as 0 Denote by W m, 0 (R) te subspace of W m, (R) consisting of functions f W m, (R) wit support in [0, 1] For any mapping M : C( ) C(R), it olds lim inf 0 sup f W m, 0 (R), f (m) γ f M (f ) Φ m+1 π m m 1 γ Tis follows by a sligt modification of te argument in Remark 5 Namely, instead of f ± = ±γe,m+1, use f ± = ±γee,m+1 were e C m (R) is supported in (0, 1), 0 e(x) 1 for all x R and e(x) = 1 for 1 3 x 3 Ten for sufficiently small, it still olds ee,m+1 = E,m+1 = Φ m+1 π m m, and te Leibniz differentiation rule yields (ee,m+1 ) (m) 1 as 0 Next we formulate some furter error estimates Teorem 8 For f V m, (R), l = 1,, m 1, it olds f (l) (Q,m f) (l) Φ m l+1 π (m l) m l (1 + α m ) f (m), (18) f (l) (Q,m f) (l) Φ m l+1 π (m l) m l (1 + q m l α m,l ) f (m) (19) were α m = a k,m, α m,l = a k j,m b j,m l j int{(m l 1)/} < α m, (0) q m l = Q,m l BC(R) BC(R), b j,m l = B m l (j + m l ) (cf (4)), and a k,m are defined in (9) Remark 9 For f V l, (R) wit f (l) BUC(R), 0 < l < m, it olds f (l) (Q,m f) (l) 0 as 0 Teorem 10 For f V l, (R), 0 < l < m, it olds f Q,m f Φ l+1 π l l (1 + α m ) f (l), (1) f Q,m f Φ l+1 π l l (1 + q m l α m,l ) f (l) ()

6 Proceedings of te 5t WSEAS Int Conf on System Science and Simulation in Engineering, Tenerife, Canary Islands, Spain, December 16-18, wit constants α m and α m,l defined in (0) If, in addition, f (l) BUC(R) ten f Q,m f = o( l ) as 0 Te proof of Teorems 8 and 10 is based on te following lemma Lemma 11 For f V l, (R), N l <m, it olds (Q,m f) (l) α m f (l) (Q,m f) (l) q m l α m,l f (l) wit constants defined in Teorem 8 Introduce te space V m, (R), m, of functions g C m (R) suc tat te derivatives g (m 1), g (m) exist on every interval (i, (i + 1)) and g (m 1) (i,(i+1)) C((i, (i + 1))), g (m) (i,(i+1)) L ((i, (i + 1))), i Z, σ,m (g) := sup i Z sup g (m) (x) < i<x<(i+1)) Clearly, V m, (R) V m, (R) and f (m) = σ,m (f) for f V m, (R) Lemma 1 For m, it olds V m, (R) = V m, (R) + S,m, ie, every g V m, (R) as a representation g = f + g, f V m, (R), g S,m (3) In particular, (3) olds wit f(x) = 1 (m 1)! x 0 (x t) m 1 G m (t)dt, x R, (4) were G m L (R) is defined by G m (x) = g (m) (x) for x (i, (i + 1)), i Z (and oter f in (3) differ from (4) by an additive polynomial of degree m 1) As te consequence of Teorem and Lemma 1 we obtain te following result Teorem 13 Let m Assume tat g V m, (R) satisfies te inequality g(x) c(1 + x r ), x R, (5) were r 0 and c 0 Ten g(x) (Q,m g)(x) σ,m (g) E,m+1 (x), x R, (6) g Q,m g Φ m+1 π m m σ,m (g) (7) References: [1] C de Boor, A practical Guide to Splines, Springer, New York 001 [] C K Cui, An Introduction to Wavelets, Academic Press, Boston 199 [3] N P Korneycuk, Splines in te Approximation Teory, Nauka, Moscow 1984 (in Russian) [4] E Leetma and G Vainikko, Quasi-interpolation by splines on te uniform knot sets, Matematical Modelling and Analysis (submitted) [5] L L Scumaker, Spline Functions: Basic Teory, Krieger Publ, Malabar, Florida 1993 [6] S B Steckin and Yu N Subbotin, Splines in Numerical Matematics, Nauka, Moscow 1976 (in Russian) [7] G Vainikko, Fast solvers of integral equations of te second kind: quadrature metods, J Integral Equations and Applications 17, 005, pp [8] G Vainikko, Error estimates for te spline interpolation on te real line (in progress) [9] G Vainikko, A Kivinukk and J Lippus, Fast solvers of integral equations of te second kind: wavelet metods, J Complexity 1, 005, pp [10] Yu S Zav yalov and B I Kvasov, V L Mirosnicenko, Metods of Spline Functions, Nauka, Moscow 1980 (in Russian) [11] A Zygmund, Trigonometric Series, 1, Cambridge Univ Press 1959 Acknowledgements: Tis work was partly supported by Estonian Science Foundation (researc grant No 5859)