I would have thought to find this question here on MSE, but I couldn't find it. I know from https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space#Common_examples that the Laplacian kernel is the reproducing kernel of $H^1(\mathbb R)$ (why is there no $\sigma$ in the norm, btw?), but I couldn't find any reference for a compact interval $[a,b]$. Can anybody help me out?
2026-03-26 06:12:50.1774505570
What's the reproducing kernel of $H^1(a,b)$?
86 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- how is my proof on equinumerous sets
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Optimization - If the sum of objective functions are similar, will sum of argmax's be similar
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Is this relating to continuous functions conjecture correct?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Absolutely continuous functions are dense in $L^1$
- A particular exercise on convergence of recursive sequence
Related Questions in FUNCTIONAL-ANALYSIS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- Prove or disprove the following inequality
- Unbounded linear operator, projection from graph not open
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Elementary question on continuity and locally square integrability of a function
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?
Related Questions in REPRODUCING-KERNEL-HILBERT-SPACES
- Prove strictly positive-definite kernel
- Example of an infinite dimensional Hilbert space that is not an RKHS
- Is a Reproducing Kernel Hilbert Space just a Hilbert space equipped with an "indexed basis"?
- RKHS rough definition
- Does the optimal function in kernel method have a sparse representation?
- A biliographic inquiry into Fredholm's kernel
- Orthonormal system $\{ e_n(t)\}_{n=1}^{\infty}$ is complete $\Leftrightarrow$ $k(t,t) = \sum_{n=0}^{\infty}{|e_n(t)|^2}, \forall t \in \Omega$
- Invertibility of Grammian in Reproducing Kernel Hilbert Space
- Linear regression with feature representation confusion - is design matrix column space the feature space?
- Is there a positive-semidefinite convolution kernel, that is continuous at $0$ but discontinuous elsewhere?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
The kernel is: $$ K(x,y)= \begin{cases} \sigma\frac{\cosh(\sigma(x - b))\cosh(\sigma (t - a))}{\sinh(\sigma(b-a))}\text{ for }a\leq t\leq x\leq b\\ \sigma\frac{\cosh(\sigma(x - a))\cosh(\sigma(t - b))}{\sinh(\sigma(b-a))}\text{ for }a\leq x\leq t\leq b.\\ \end{cases}$$ This can be checked by direct verification of the reproducing property using integration by parts from the definition of the inner product $\langle f,g \rangle=\int_a^bf(u)g(u)\,du + \frac 1 {\sigma^2}\int_a^bf'(u)g'(u)\,du$.
Complete derivation with $\sigma=1$ can be found in Section 2.11 of the script by Schaback or in Chapter 6 Section 1.6.1 of Berlinet/Thomas-Agnan.