Generalized trigonometric functions

Transcription

1 7/211, Santiago de Compostela (Etwas allgemein machen heisst, es denken. G.W.F. Hegel)

2 Based on: * David E. Edmunds, Petr Gurka, J.L., Properties of generalized Trigonometric functions, Preprint * David E. Edmunds, J.L., Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics 216, Springer

3 Let 1 < p, q < and define a (differentiable) function F p,q : [, 1] R by F p,q (x) = x 1 p dt, x 1. (1) 1 t q Since F p,q is strictly increasing it is a one-to-one function on [, 1] with range [, π p,q /2], where 1 π p,q = 2 1 p dt, x 1. (2) 1 t q The inverse of F p,q on [, π p,q /2] we denote by sin p,q and extend as in the case of sin (p=q=2) to [, π p,q ] by defining sin p,q (x) = sin p,q (π p,q x) for x [π p,q /2, π p,q ]; further extension is achieved by oddness and 2π p,q -periodicity on the whole of R. By this means we obtain a differentiable function on R which coincides with sin when p = q = 2.

4 Corresponding to this we define a function cos p,q by the prescription cos p,q (x) = d dx sin p,q(x), x R. (3) Clearly cos p,q is even, 2π p,q -periodic and odd about π p,q ; and cos 2,2 = cos. If x [, π p,q /2], then from the definition it follows that cos p,q (x) = (1 (sin p,q (x)) q ) 1/p. (4) Moreover, the asymmetry and periodicity show that sin p,q (x) q + cos p,q (x) p = 1, x R. (5) We will use: π p := π p,p, sin p := sin p,p and cos p := cos p,p.

5 Figure: sin 6,6, cos 6,6 and sin 1.2,1.2, cos 1.2,1.2

6 π p 2 = p 1 1 (1 s p ) 1/p s 1/p 1 ds = p 1 B(1 1/p, 1/p) = p 1 Γ(1 1/p)Γ(1/p), where B is the Beta function, Γ is the Gamma function and π p = Clearly π 2 = π and, with p = p/(p 1), 2π p sin(π/p). (6) pπ p = 2Γ(1/p )Γ(1/p) = p π p. (7) Using (6) and (7) we see that π p decreases as p increases, with lim π p =, p 1 lim π p = 2, p lim (p 1)π p = lim π p = 2. (8) p 1 p 1

7 Properties of π p,q Lemma Let p, q (1, ). Then p π p,q q π p,q is decreasing on (1, ) for any fixed q (1, ), is decreasing on (1, ) for any fixed q (1, ), qπ p,q = p π q,p π p,q 2 = 1 (1 t q ) 1/p dt = p q = p q 1 1 (1 y p ) 1/q dy = p q y p /p (1 y p ) 1/q y p /p dy π q,p 2.

8 Lemma Let p, q (1, ). (i) If p q then π p,q π q q. (ii) If p > q then π p,q p q π p,p. (iii) 2 π p,p 4. π p,q = 2 B(1/p,1/q) q = 2 Γ(1/p ) Γ(1/q) q Γ(1/p +1/q).

9 An integral operator and generalized trig. functions On the interval I = [, 1] let Tf (x) := x f (t)dt. (9) At first we consider T as a map from L 2 (, 1) into L 2 (, 1). It is obvious that T is compact and that there exists a function in L 2 (, 1) at which the norm of T is attained. In this case it is quite simple to show that T L 2 (, 1) L 2 (, 1) = 2/π and that the norm is attained when: ( πx ) π f (t) = cos 2 2 so that Tf (t) = sin ( πx ). 2

10 When 1 < p, q < then again T is a compact map from L p (, 1) into L q (, 1) and there exists a function at which the norm is attained. In [Levin, Schmidt] it was proved that T L p (, 1) L q (, 1) = = (p + p) 1 1 p + 1 q (p ) 1/q q 1/p B( 1 p, 1 q ) and that the norm is attained when: f (t) = cos p,q ( πp,q x 2 ) πp,q 2 and ( πp,q x ) Tf (t) = sin p,q. 2

11 Eigenfunctions for the p-laplacian We recall the definition of the p-laplacian which is a natural extension of the Laplacian: p u = ( u p 2 u ). Evidently 2 u = u. Then the Dirichlet eigenvalue problem on (, 1) is p u + λ u p 2 } u = on (, 1), (1) u() =, u(1) =. In [Elbert, Lindqvist, Drabek] it is shown that all eigenvalues of this problem are of the form λ n = (nπ p ) p p p with corresponding eigenfunctions u n (t) = sin p (nπ p t).

12 Consider the pq-laplacian and the corresponding eigenvalue problem p u + λ u q 2 } u = on (, 1), u() =, u(1) =. (11) Then all eigenvalues of this problem are of the form λ n,α = (nπ p,q ) q α p q q p, α R \ {} with corresponding eigenfunctions u nα (t) = αt nπ p,q sin p,q (nπ p,q t).

13 The approximation theory Let 1 < p < and < a < b <. Consider the Sobolev embedding on I = [a, b], E : W 1,p (I ) L p (I ) (12) where W 1,p (I ) is the Sobolev space of functions on the interval I with zero trace equipped with the following norm: ( b 1/p u W 1,p (I ) := u (t) dt) p. a It is well known that (12) is a compact map and that more detailed information about its compactness plays an important role in different branches of mathematics. The properties of compact maps can be well described by using the Kolmogorov, Bernstein and Gel fand n-widths together with the approximation numbers.

14 Definition Let T L(X, Y ). Then the n-th approximation and isomorphism of T are defined by a n (T ) = inf{ T K : K L(X, Y ), rank(t ) < n}, i n (T ) = sup{ A 1 B 1 }, where the supremum is taken over all possible Banach spaces G and maps A : Y G and B : G X such that I G = ATB is an identity on G and dim(g) n. Note that other important strict s numbers as Gelfand, Bernstein, Kolmogorov and Mityagin numbers lies between the approximation and isomorphism numbers. a n (T ) d n (T ), b n (T ), d n (T ), m n (T ) i n (t)

15 Theorem Let s n stands for any strict s-number (i.e. a n, d n, d n, m n, b n, i n ). Let n be an integer, then s n (E ) = I nπ p ( p p ) 1/p and s n (E ) = (E R n )g L p (I ), ( ) x a where g(x) = sin p b a π pn. Here where n 1 R n f = P i f i=1 P i f (x) = χ Ii (x)f ( a + i I ). n

16 We can see from the previous theorem that the largest element in BW 1,p (I ) := {f ; f W 1,p 1} in the L p (I ) norm is ( ) sin x a p b a π p f 1 (x) := ( ). sin x a p b a π p W 1,p (I ) Let us approximate BW 1,p (I ) by a one-dimensional subspace in L p (I ). The most distant element from the optimal one-dimensional approximation is ) f 2 (x) := ( sin x a p b a 2π p sin p ( x a b a 2π p ) W 1,p (I ).

17 We present below figures which show an image of BW 1,p (I ) restricted to a linear subspace span{f 1, f 2, f 3 } in L p (I ). In the case p = 2 we obtain an ellipsoid (here the x, y, z axes correspond to f 1, f 2, f 3 ). Figure 5: p = 2

18 When p = 1 and p = 1.1 we have the images below: Figure 6: p = 1 p = 1.1 We can see that the main difference between Fig. 5 and Fig 6 is that the pictures in Fig. 6 are not convex. This suggests that possibly the functions f 1, f 2, f 3 are not orthogonal in the James sense.

19 Basis Theorem Let p, q (1, ) and let p q < 4 π (13) 8 Then the sequence ( sin p,q (n π p,q t) ) n N L r (, 1) for any r (1, ). is a a Schauder basis in The functions f n,p (x) := sin p (nπ p x) form a basis in L q (, 1) for every q (1, ) if p < p <, where p is defined by the equation π p = 2π2 π 2 8. (14) (i.e. p 1.5)

20 1 p' q =2.14 1/q 1 p, 1 p' 1/p 1

21 Identities Lemma For all x [, π p,q /2), d dx cos p,q x = p q (cos p,q x) 2 p (sin p,q x) q 1, d dx tan p,q x = 1 + p (sin p,q x) q q (cos p,q x) p, d dx (cos p,q x) p 1 p(p 1) = (sin p,q x) q 1, q d dx (sin p,q x) p 1 = (p 1) (sin p,q x) p 1 cos p,q x.

22 Lemma For all y [, 1], cos 1 p,q y = sin 1 p,q ( (1 y p ) 1/q), sin 1 p,q y = cos 1 p,q ( (1 y q ) 1/p), 2 π p,q sin 1 p,q (y 1/q ) + 2 ( ) π sin 1 q,p q,p (1 y) 1/p = 1, ( cosp,q (π p,q y/2) ) p ( = sinq,p (π q,p (1 y)/2)) p. Lemma Let x [, π 4/3,4 /4). Then sin 4/3,4 (2x) = 2 sin 4/3,4 x (cos 4/3,4 x) 1/3 ( (sin4/3,4 x) 4 (cos 4/3,4 x) 4/3) 1/2. (15)