Smoothness beyond dierentiability

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Smoothness beyond dierentiability Peter Mathé Tartu, October 15, 014 1 Brief Motivation Fundamental Theorem of Analysis Theorem 1. Suppose that f H 1 (0, 1).

1 Smoothness beyond dierentiability Peter Mathé Tartu, October 15, Brief Motivation Fundamental Theorem of Analysis Theorem 1. Suppose that f H 1 (0, 1). Then there exists a g L (0, 1) such that f(x) f(0) = x g(t) dt, 0 < x < 1. 0 We dene the linear mapping (Sg)(x) := x g(t) dt. 0 Corollary 1. If f H 1 (0, 1) then f f(0) R(S). Thus, smoothness means that f is in the range of a smoothing operator. Theorem. Suppose that f H 1 (0, 1) (periodic). For the Fourier coecients ˆf(n) of f it holds n=1 n ˆf(n) <. Corollary. Smoothness implies a certain decay of the Fourier coecients. Goal 1. Make a general theory from these observations! Singular value decomposition Erhard Schmidt (13 January 1876 December 1959) The singular value decomposition for linear operators in Hilbert space was rst developed by E. Schmidt: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener, Math. Ann.,

2 Erhard Schmidt was born in Tartu (German: Dorpat), also studied here. He graduated under D. Hilbert. He was Prof. for Mathematics at the University of Berlin. In 1946 he founded the Mathematical Institute of the Academy of Sciences, nowadays: Weierstrass Institute. svd for operators in Hilbert space I Let A: X Y be a compact operator. Then its symmetrization A A: X X is compact and self-adjoint. Using the Riesz-Schauder theory of compact operators we nd that the spectrum of A A consists entirely of eigenvalues, it is at most countable, and the only accumulation points is 0. The corresponding eigen-spaces, corresponding to non-zero eigenvalues are nite dimensional. Gluing nite-dimensional orthonormal bases of these eigen-space together we end up by 1. an ordered sequence µ 1 > µ > > 0 of non-zero eigenvalues;. We agree to arrange these taking the multiplicities into account to obtain an ordered sequence λ 1 λ 0; 3. We assign the corresponding orthonormal system u 1, u,..., such that A Au j = λ j u j, j = 1,,... Thus we arrive at the representation A Ax = λ j >0 λ j x, u j u j, x X. svd for operators in Hilbert space II In a second step we use the polar decomposition of A as A = U A with A x := λ j >0 λj x, u j u j, x X and a partial isometry, which maps the orthonormal system u j, j = 1,,... onto an orthonormal system v j := Uu j, j = 1,,.... Denition 1 (singular value decomposition (svd)). Let A: X Y be a compact operator between Hilbert spaces. There exist orthonormal families u j X and v j Y, j = 1,,... as well as a non-negative sequence of numbers s j, j = 1,,... such that Ax = s j >0 s j x, u j v j, x X. The (non-increasing) numbers s j, j = 1,,... are called singular numbers of the operator A.

3 We add the dening equations, which are easily seen from the denition. Au j = s j v j, and A v j = s j u j, j N. Example The integration operator Sg(x) = x g(t) dt has the following svd: 0 1 s-numbers: s j = (n+1/)π orth. system: u j (t) = cos ((n + 1/)πt) orth. system: v j (t) = sin ((n + 1/)πt) We can revisit the decay of the Fourier coecients from before, and rewrite f H 1 (0, 1) if ˆf(j) <. j=1 s j Theorem 3 (Equivalency). f R(S) = R ( (SS ) 1/) if and only if ˆf(j) j=1 s j. If we let ϕ(t) = t then this means ˆf(j) f R(ϕ(SS )) if and only if ϕ (s j ). < Functions of operators Suppose that an operator A has svd (s j, u j, v j ). We assign H := A A. For a bounded continuous function ϕ we let ϕ(h)v := ϕ(s j) v, u j u j, v X. j=1 If the function ϕ is non-decreasing then the linear operator ϕ(h) has singular numbers s j (ϕ(h)) = ϕ(s j). Denition (Index function). Let A be a linear operator, with companion H = A A. A continuous increasing (non-decreasing) function ϕ: (0, ) (0, ) is called index function if ϕ(0+) = 0. Theorem 4 (Equivalency). We have x R(ϕ(H)) if and only if x,u j j=1 < ϕ (s j. ) 3 j=1 <

4 3 Smoothness relative to an operator: Variable Hilbert scales Variable Hilbert scales We x the injective self-adjoint positive operator H := A A: X X. If ϕ is an index function then ϕ(h) is also injective self-adjoint positive. For x = ϕ(h)v, y = ϕ(h)w we assign the scalar product x, y ϕ := v, w. If x R(ϕ(H)) then x D(ϕ(H) 1 ), and we have the corresponding norm x ϕ := v X = ϕ(h) 1 x X, x R(ϕ(H)). Theorem 5. Let ϕ be any index function. The space X H ϕ := (R(ϕ(H)), ϕ ) is a Hilbert space. Denition 3 (Variable Hilbert scale, Source set). The collection of spaces { X H ϕ, ϕ index function } constitutes a variable Hilbert scale. The set H ϕ = {x, x = ϕ(h)v, v X 1} is the unit ball (source set). Calculus I Proposition 1. Let ϕ, ψ be index functions. 1. There is a dense natural embedding Xϕ H Xψ H, if and only if there is a constant C <, such that ϕ(t) Cψ(t), 0 t a.. The above embedding is compact if and only if 3. The adjoint space ( X H ϕ lim ϕ(t)/ψ(t) = 0. (1) t 0 ) to X H ϕ is isometric to X H 1/ϕ. Sketch. We only mention the following obvious fact: We have that ( ) ϕ R(ϕ(H)) = R(ψ(H)) (H). ψ 4

5 Picture, Examples Draw a picture to highlight the situation. ( ) The triple Xϕ H, X, X1/ϕ H constitutes a Gelfand triple. Example 1. In most text on inverse problems the following setup is used. Solution smoothness is measured in terms of source sets: x = H p v, v E. This may be rewritten as x = ϕ p,e (H)ṽ with ϕ p,e (t) = 1 E tp and ṽ = 1 E v, and hence ṽ 1. We thus have that x H ϕp,e. Example. In severely ill-posed problems we nd that X = log µ (H 1 ) with ( H < 1). The function t log 1 (1/t) is no index function, but it may be replaced by one with same asymptotics as t 0. Smoothness is everywhere Why shall we consider general smoothness in terms of general source sets, and not just power type smoothness? In the previous examples we saw that logarithmic smoothness is important. However, one may nd elements x which do not belong to the range R(ϕ(H)) for any logarithmic type function ϕ(t) = log µ (1/t). The concept of general smoothness is general in the following sense. Theorem 6 ([5]). Let H : X X be a compact non-negative and injective operator. For every x X there is an index function ϕ such that x H ϕ. As a consequence, by iterating this argument, given any x X there is no maximal smoothness ϕ for which x H ϕ. Remark 1. Although smoothness is everywhere the latter consequence says that this notion of smoothness is never sharp! 5

6 4 Element-wise interpolation Problem of interpolation Remember the following situation from the previous proposition: If ϕ, ψ are index functions with ϕ(t) Cψ(t) then Xϕ H Xψ H X. Problem 1. If we know the norm sizes of some element x Xϕ H can we say something about its norm in Xψ H? and in X Example 3 (HardyLittlewoodPolya). Let x W n (R). For 0 k n we have that W n (R) W k (R) L (R), and x (k) x n k n x (n) k n. Problem. What is the position of X H ψ between (XH ϕ, X)? How does the above inequality generalize? First Interpolation inequality Theorem 7 (see [7, 6]). Suppose that ϕ, ψ and κ are functions such that the function ϕ/κ is an index function, and ϕ/ψ is an increasing index function. If the composition ( ( ϕ ) (ϕ ) ) 1 t (t), 0 < t < κ ψ is concave, then ( ϕ ) ( ) 1 x κ κ x ϕ ( ϕ ψ ) 1 ( x ψ x ϕ Proof. Exercise on convexity on blackboard! Intermediate spaces ), 0 x X H ϕ. Denition 4 (intermediate space). Suppose that the spaces Xϕ H Xψ H are ordered by inclusion. A space Xκ H is called intermediate between ( ) Xϕ H, Xψ H if the function ϕ/κ is non-decreasing, and ( ( ϕ ) (ϕ ) ) 1 t (t), 0 < t < is concave. κ ψ 6

7 Example 4. Let r s t and ϑ be such that s = ϑr + (1 ϑ)t. Then, with ϕ(α) := α t, κ(α) := α s and ψ(α) := α r we see, that the above function is α α (t s)/(t r), and this is concave, because 0 t s t r. Applying the interpolation inequality we arrive at ( ) 1/(t s) ( ) 1/(t r) x s x r, x t x t thus x s x ϑ r x 1 ϑ t Polya! Application: Modulus of continuity Consider the following problem: with ϑ = (t s)/(t r). Cf. HardyLittlewood Let A: X Y be an injective compact linear operator between Hilbert spaces. Let M X be a convex centrally-symmetric subset. Denition 5 (modulus of continuity). Let ω(a 1, M, δ) := sup { x, x M X, Ax δ} denote the modulus of continuity of A 1 on A(M). Remark. This is a very important subject in approximation theory and in inverse problems! Problem 3. Can be bound the modulus of continuity for source sets M := H ϕ? Bounding the modulus of continuity Recall that ω(a 1, M, δ) := sup { x, We rst rewrite with H := A A x M X, Ax δ}. Ax = H 1/ x = x 1/ t. { } Thus we have that ω(a 1, H ϕ, δ) := sup x, x ϕ 1, x 1/ t δ An application of the interpolation inequality gives where Θ(t) := tϕ(t). ω(a 1, H ϕ, δ) ϕ ( Θ 1 (δ) ), δ > 0, Remark 3. There is an explicit representation for the modulus of continuity in terms of piece-wise linear interpolation, which was rst established in [3]. For many index functions ϕ this is sharp! 7

8 G :Xρ G S H :X H r J ρ G ϕ S J r H f J ϕ S J f Figure 1: The setup of interpolation. The position of Xϕ G between Xρ G and X is given by the function t ϕ ((ρ ) 1 (t)), and f is determined in such a way that Xf H has the appropriate position in the scale on bottom. PeierlsBogolyubov Inequality Theorem 8 (see [1]: Matrix analysis). Let A: X Y be a bounded operator, and suppose that the function f : (0, ) (0, ) is convex. If x X with x = 1 then f( x, A Ax ) x, f(a A)x. Proof. This is a consequence of the interpolation inequality with θ = 1, ϕ = 1/ t and ψ(t) = 1/ f(t) 5 Operator interpolation Interpolation problem We are given two variable Hilbert scales, generated by the operators G and H. Suppose that an operator S acts boundedly: S : X X and S : X G ρ X H r. When is it bounded from S : X G ϕ X H f? We depict the interpolation setup: Operator concavity We use the following partial ordering for self-adjoint operators. Denition 6 (parial ordering). Let G and H be self-adjoint operators in some Hilbert spaces X. We say that G H if for all x D(H) the inequality Gx, x Hx, x holds true. 8

9 Denition 7 (operator concave function). Let f : [0, ) R + be a continuous function. It is called operator concave if for any pair G, H 0 of self-adjoint operators we have f( G + H f(g) + f(h) ). Remark 4. Operator monotonicity, concavity and related operator theoretic concepts were intensively studied. The rst monograph we by [4], and it was then completed in []. Examples There is a complete description of operator monotone (concave) functions. Linear functions are operator monotone; higher powers are not! Fractional powers f(t) = t θ with 0 < θ 1 are operator monotone. The function t 1/t is operator monotone. Compositions of operator monotone functions are operator monotone. The complete characterization uses the theory of complex functions. Remark 5. For further reading we advertise the monographs Donoghue, Jr., William F., Monotone matrix functions and analytic continuation, Springer, 1974 and Bhatia, Rajendra, Matrix analysis, Graduate Texts in Mathematics, Springer, Interpolation Theorem Theorem 9 (Interpolation theorem, [8]). Let G, H 0 be self-adjoint operators, respectively. Furthermore, let ϕ, ρ and r be index functions (ρ strictly increasing). If t ϕ ((ρ ) 1 (t)) is operator concave then yield Sx C 1 x and ρ(g)sx C r(h)x, x X, ϕ(g)sx max {C 1, C } f(h)x, x X. G :Xρ G S J ρ J ϕ G ϕ S S H :X H r J r H f J f 9

10 Application: L"ownerHeinz Inequality, 1951 The following famous result is often used in the theory of inverse problems. Theorem 10. Let G, H be positive self-adjoint operators in Hilbert space. If 0 θ 1 then the inequality G H yields that G θ H θ. Proof. We rewrite the assumption as follows: G H means G 1/ x = Gx, x Hx, x = H 1/ x We apply the interpolation theorem to the scales generated by G 1/ and H 1/, respectively. The operator S is the identity. The functions ρ(t) = r(t) = t are linear, and ϕ(t) = t θ. function ϕ ((ρ ) 1 (t) = t θ ; it is operator concave. Thus the We obtain that G θ x, x = G θ/ x H θ/ x = H θ x, x. 6 Summary We showed that 'classical' smoothness may be regarded as a special case of the more general concept: element belongs to the range of a smoothing (compact) operator. The range may be stretched or squeezed by index functions ϕ. This leads to the concept of a variable Hilbert scale. We analyzed these scales and found intermediate spaces. This lead to the notion of interpolation. The following hierarchy between functions ϕ, ψ proved important: monotonicity: ϕ ψ yields that X H ϕ X H ψ. concavity: point-wise interpolation. op. concavity: operator interpolation. In several examples we highlighted applications, in particular within the theory of inverse problems. 10

11 Summary One of my teachers, A. Pietsch, made the following remark, which I quote here: If a mathematical theory has not yet proved useful, then it should at least be beautiful. A. Pietsch, Eigenvalues and s-numbers, Akadem. Verlagsgesellschaft Geest & Portig, Leipzig, 1987 [Chapt (Practical applications)]. Thank you for the attention. References [1] Rajendra Bhatia. Matrix analysis. Springer-Verlag, New York, [] William F. Donoghue, Jr. Monotone matrix functions and analytic continuation. Springer-Verlag, New York, [3] V. K. Ivanov and T. I. Koroljuk. The estimation of errors in the solution of linear ill-posed problems. š. Vy isl. Mat. i Mat. Fiz., 9:3041, [4] K. Löwner. Über monotone Matrixfunktionen. Math. Z., 38:17716, [5] Peter Mathé and Bernd Hofmann. How general are general source conditions? Inverse Problems, 4(1):015009, 5, 008. [6] Peter Mathé and Sergei V. Pereverzev. Discretization strategy for linear ill-posed problems in variable Hilbert scales. Inverse Problems, 19(6):163177, 003. [7] Peter Mathé and Sergei V. Pereverzev. Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Problems, 19(3):789803, 003. [8] Peter Mathé and Ulrich Tautenhahn. Interpolation in variable Hilbert scales with application to inverse problems. Inverse Problems, (6):7197,

A Spectral Characterization of Closed Range Operators 1

A Spectral Characterization of Closed Range Operators 1 M.THAMBAN NAIR (IIT Madras) 1 Closed Range Operators Operator equations of the form T x = y, where T : X Y is a linear operator between normed linear