Inner products

We define the inner product measurement of the tensor field with respect to "probe d... 

If A is invertible for all m and oj then the complete Radon transform of the tensor field is easily obtained from the inner product measurements using... 

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

As for the Fourier Transform (FT), the Continuous Wavelet Transform (CWT) is expressed by the mean of an inner product between the signal to analyze s(t) and a set of analyzing function ... 

This construction requires one matrix-vector multiplication with S and two inner products in each recursive step. Therefore, it is not necessary to store S explicitly as a matrix. The Lanczos process yields the approximation [21, 7, 12]... 

The work assumption may be reduced by taking the inner product of (5.32) with V... 

Substituting (5.11) into this equation, taking the inner product of each side with d jdk, and using the consistency condition (5.14)... 

Using (5.26) and the permutation properties of the inner product, this becomes, in component form... 

In this case, may be eliminated and y may be evaluated by substituting (5.77) into the left-hand side of the normality condition, taking the inner product of each side with the term in chain brackets in (5.79), and using the consistency condition (5.79), which results in... 

Because the scalar inner product of a symmetric and an antisymmetric tensor vanishes, from (A.l 1)... 

We ll consider each of the relations in Equation 39 in turn. In the first case, by forming the inner product of each side with we obtain the following expression... 

Since the y s are orthonormal, the inner product of any with itself is one, and the inner product of any distinct two of them is 0. 

The expression for E also follows easily from simple linear algebra. We begin by again forming the inner product of both sides of the second relation from Equation 39 with... 

We will form the inner product of both sides of Equation 49 with an arbitrary substituted wavefunction y, and then solve for a, ... 

This set forms a Hilbert space with an inner product defined by... 

Linear bounded operators in a real Hilbert space. Let H he a real Hilbert space equipped with an inner product x,y) and associated norm II X II = (x, x). We consider bounded linear operators defined on the space... 

Let A be a positive self-adjoint linear operator. By introducing on the space H the inner product x,y) = Ax,y) and the associated norm x) we obtain a Hilbert space Ha, which is usually called the energetic space Ha- It is easy to show that the inner product... 

Linear operators in finite-dimensional spaces. It is supposed that an n-dimensional vector space R is equipped with an inner product (, ) and associated norm a = / x x). By the definition of finite-dimensional space, any vector x G i n can uniquely be represented as a linear combina-... 

In what follows we will use, as a rule, several norms associated with inner products in the space Hh (the grid analogs of the L2 and IT2-norms are available in Chapter 1). 

In what follows we will frequently employ a non-equidistant grid denoting it by LOf in contrast to an equidistant introduced in Section 1. On any such grid the formulae of the inner product and the difference summation by parts formulae are somewhat different ... 

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