The Extended Abstracts of The 4th Seminar on Functional Analysis and its Applications 2-3rd March 2016, Ferdowsi University of Mashhad, Iran 6 1 0 2 n a ON MULTIPLIERS OF REPRODUCING KERNEL J BANACH AND HILBERT SPACES 5 ] A ALI EBADIAN1,∗SAEED HASHEMI SABABE2 AND MAYSAM ZALLAGHI3 F . 1,2 Department of Mathematics, Payame Noor University (PNU), Iran; h [email protected], Hashemi [email protected] t a m 3 Department of Mathematics, University of Isfahan, Isfahan, Iran; [ [email protected] 1 v Abstract. Thispaperisdevotedtothestudyofreproducingker- 3 9 nel Hilbert spaces. We focus on multipliers of reproducing kernel 2 Banach and Hilbert spaces. In particular we tried to extend this 1 concept and prove some theorems. 0 . 1 0 6 1 1. Introduction : v i In functional analysis, a reproducing kernel Hilbert space (RKHS) X is a Hilbert space associated with a kernel that reproduces every func- r a tion in the space or, equivalently, where every evaluation functional is bounded. The reproducing kernel was first introduced in the 1907 work of Stanisaw Zaremba concerning boundary value problems for harmonic and biharmonic functions. James Mercer simultaneously ex- amined functions which satisfy the reproducing property in the theory of integral equations. These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics. Repro- ducing kernel Hilbert spaces are particularly important in the field of 2010 Mathematics Subject Classification. Primary 47B32;Secondary 47A70. Key words and phrases. Hilbert space, reproducing kernels, representation. ∗ Speaker. 1 2 EBADIAN,HASHEMISABABE, ZALLAGHI statistical learning theory because of the celebrated Representer the- orem which states that every function in an RKHS can be written as a linear combination of the kernel function evaluated at the training points. More details can be found in [1, 2, 3]. Given a set X, if we equip the set of all functions from X to F, F(X,F)with theusual operationsofaddition, (f+g)(x) = f(x)+g(x), andscalar multiplication, (λ.f)(x) = λ.(f(x)), thenF(X,F)isavector space over F. Given a set X, we will say that H is a reproducing kernel Hilbert space(RKHS) on X over F, provided that: (1) H is a vector subspace of F(X,F), (2) H is endowed with an inner product, h.,.i, making it into a Hilbert space, (3) for every y ∈ X, the linear evaluation functional, E : H → F, y defined by E (f) = f(y), is bounded. y If H is a RKHS on X, then since every bounded linear functional is given by the inner product with a unique vector in H, we have that for every y ∈ X, there exists a unique vector, k ∈ H, such that y f(y) = hf,k i ∀f ∈ H (1.1) y The function k is called the reproducing kernel for the point y. The y 2-variablefunctiondefinedby K(x,y) = k (x)iscalled thereproducing y kernel for H. Note that we have, K(x,y) = k (x) = hk ,k i (1.2) y y x kE k2 = kk k2 = hk ,k i = K(y,y). (1.3) y y y y Definition 1.1. Let H be a RKHS on X with kernel function, K. A function f : X → C is called a multiplier of H provided that fH = {fh : h ∈ H} ⊆ H. We let M(H) or M(K) denote the set of multipliers of H. More generally, if H , i = 1,2 are RKHSs on X with i reproducing kernels, K , i = 1,2 then a function, f : X → C, such that i fH1 ⊆ H2, is called a multiplier of H1 into H2 and we let M(H1,H2) denote the set of multipliers of H1 into H2, so that M(H,H) = M(H). Given a multiplier, f ∈ M(H1,H2), we let Mf : H1 → H2, de- note the linear map, M (h) = fh. Clearly, the set of multipliers, f M(H1,H2) is a vector space and the set of multipliers, M(H), is an algebra. Definition 1.2. A reproducing kernel Banach space (RKBS) on X is a reflexive Banach space of functions on X such that its topological ON MULTIPLIERS OF REPRODUCING KERNEL BANACH AND HILBERT SPACES3 dual B′ is isometric to a Banach space of functions on X and the point evaluations are continuous linear functionals on both B and B′. In this case, There is a kernel function K : X ×X → C such that [f,K(.,x)] = f(x) ∀f ∈ B ∀x ∈ X, (1.4) B and B = span{K(.,x); x ∈ X} Definition 1.3. Let X be a set. We call a uniformly convex and uniformly Frechet differentiable RKBS on X an s.i.p. reproducing kernel Banach space (s.i.p.RKBS). Theorem 1.4. (Riesz representation theorem) [5] For each g ∈ B′, there exists a unique h ∈ B such that g = h∗, i.e., g(f) = [f,h] ,f ∈ B B and kgkB′ = khkB where [.,.]B denotes the semi-inner product on B. Definition 1.5. (The adjoint operator in a semi-inner product space) Suppose B1 and B2 are tow s.i.p. Banach spaces. The adjoint operator T∗ for a map T : B1 → B2 is defined such that the domain of T∗ is D(T∗) = {g∗ ∈ B2∗ : g∗T is continuous on B1}, (1.5) and T∗ : D(T∗) → BC is defined by T∗g∗ = g∗T where BC is the space 1 1 of all continuous functionals on B1 Definition 1.6. A normed vector space V of functions on X satisfies the Norm Consistency Property if for every Cauchy sequence {f : n ∈ n N} in V, lim f (x) = 0 x ∈ X =⇒ lim kf k = 0. (1.6) n n V n→∞ n→∞ Suppose X be a set and B be a s.i.p. RKBS on X with K as its kernel. let B♯ = span{K(x,.); x ∈ X} (1.7) We can define a new norm as follows |[f,g] | kgk = sup B g ∈ B♯ (1.8) B♯ f∈B,f6=0 kfkB Theorem 1.7. [4] The norm k.k is well-defined and point evalua- B♯ tion functionals are continuous on B♯ if and only if point evaluation functionals are continuous on B. 2. Main Results Theorem 2.1. Let X be a set and H) be a reproducing kernel Hilbert space on X. Then function π : M(H) × H → H with π (f,h) = H H M (h) = fh is a representation. f 4 EBADIAN,HASHEMISABABE, ZALLAGHI Definition 2.2. Suppose X be a set and B be a s.i.p. RKBS on X. A function f : X → C is called a multiplier of B provided that fB = {fg : g ∈ B} ⊆ B. We let M(B) denote the set of multipliers of B. Suppose M be the set of multipliers of a s.i.p. RKBS. It is endowed B with a semi inner product inherited of B. So it can be embedded in an inner product space. We denote H a Hilbert space spanned by MB M . B Theorem 2.3. Let X be a set and B be a reproducing kernel Banach space on X. Then function π : M(B) × B → H with π (f,g) = B MB B M (g) = fg is a representation. f Theorem 2.4. Let B , i = 1,2 be s.i.p. RKBS’s on X with reproducing i kernels, Ki(x,y) = kyi(x), i = 1,2. If f ∈ M(B1,B2), then for every y ∈ X, M∗(k2) = f(y)k1. f y y Proof. For any h ∈ B1, we have that [h,f(y)ky1]1 = f(y)h(y) = [Mf(h),ky2]2 = [h,M∗f(ky2)], (2.1) and hence, f(y)k1 = M∗(k2). (cid:3) y f y Theorem 2.5. Suppose B and B♯ defined as above. then M ∼= M B B♯ Theorem 2.6. The space of B0 = {g ∈ B♯; kgkB♯ = 1} is a subspace of B as a s.i.p. RKBS. Acknowledgement Research supported in part by Kavosh Alborz Institute of Mathe- matics and Applied Sciences. References 1. A.Ebadian,S.HashemiSababeandSh.Najafzadeh,On generalized reproducing kernel Hilbert spaces , Submited in Operators and Matrices, arXiv:1504.04668. 2. A. Ebadian, S. Hashemi Sababe and Sh. Najafzadeh, Generalized reproducing kernel Banach spaces in machine learning, Submited in Bulletin of the Korean Mathematical Society. 3. G. Lumer, Semi-inner-product spaces, AMS journals, 1960. 4. Guohui Song, Haizhang Zhang and Fred J. Hickernell, Reproducing Kernel Ba- nach Spaces with the 1 Norm,arXiv:1101.4388v3 5. Haizhang Zhang, Yuesheng Xu and Jun Zhang Reproducing Kernel Banach Spaces for Machine Learning, Journal of Machine Learning Research 10 (2009) 2741-2775.