Harmonic analysis

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Harmonic analysis

In music, if a note has frequency , integer multiples of that frequency, and so on, are known as harmonics. As a result, the mathematical study of overlapping waves is called harmonic analysis.Harmonic analysis is a diverse field including such branches as Fourier series, isospectral manifolds (hearing the shape of a drum), and topological groups. Signal processing, medical imaging, and quantum mechanics are three of the fields that use harmonic analysis extensively.

Haar measure

Any locally compact Hausdorff topological group has a unique (up to scalars) nonzero left invariant measure which is finite on compact sets. If the group is Abelian or compact, then this measure is also right invariant and is known as the Haar measure.More formally, let be a locally compact group. Then a left invariant Haar measure on is a Borel measure satisfying the following conditions: 1. for every and every measurable . 2. for every nonempty open set . 3. for every compact set . For example, the Lebesgue measure is an invariantHaar measure on real numbers.In addition, if is an (algebraic) group, then with the discrete topology is a locally compact group. A left invariant Haar measure on is the counting measure on .

Subharmonic function

Let be an open set and a real-valued continuous function on . Suppose that for each closed disk and every real-valued harmonic function defined on a neighborhood of which satisfies on , it holds that on the open disk . Then is said to be subharmonic on (Krantz 1999, p. 99).1. If are subharmonic on , then so is . 2. If is subharmonic on and is a constant, than is subharmonic on . 3. If are subharmonic on , then is also subharmonic on .

Kelvin transformation

Let be a domain in for . Then the transformationonto a domain , whereis called a Kelvin transformation. If is a harmonic function on , then is also harmonic on .

Harnack's principle

Let be harmonic functions on a connected open set . Then either uniformly on compact sets or there is a finite-values harmonic function on such that uniformly on compact sets.

Dirichlet problem

The problem of finding the connection between a continuous function on the boundary of a region with a harmonic function taking on the value on . In general, the problem asks if such a solution exists and, if so, if it is unique. The Dirichlet problem is extremely important in mathematical physics (Courant and Hilbert 1989, pp. 179-180 and 240; Logan 1997; Krantz 1999b).If is a continuous function on the boundary of the open unit disk , then define(1)where is the boundary of . Then is continuous on the closed unit disk and harmonic on (Krantz 1999a, p. 93).For the case of rational boundary data without poles, the resulting solution of the Dirichlet problem is also rational (Ebenfelt and Viscardi 2005), the proof of which led to Viscardi winning the 2005-2006 Siemens-Westinghouse competition (Siemens Foundation 2005; Mathematical Association of America 2006)...

Harnack's inequality

Let be an open disk, and let be a harmonic function on such that for all . Then for all , we have

Dirichlet energy

Let be a real-valued harmonic function on a bounded domain , then the Dirichlet energy is defined as , where is the gradient.

Harmonic function

Any real function with continuous second partial derivatives which satisfies Laplace's equation,(1)is called a harmonic function. Harmonic functions are called potential functions in physics and engineering. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component vector field to a 1-component scalar function. A scalar harmonic function is called a scalar potential, and a vector harmonic function is called a vector potential.To find a class of such functions in the plane, write the Laplace's equation in polar coordinates(2)and consider only radial solutions(3)This is integrable by quadrature, so define ,(4)(5)(6)(7)(8)(9)so the solution is(10)Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes(11)(12)Other solutions may be obtained by differentiation, such as(13)(14)(15)(16)and(17)Harmonic functions..

Poisson kernel

The integral kernel in the Poissonintegral, given by(1)for the open unit disk . Writing and taking gives(2)(3)(4)(5)(6)(Krantz 1999, p. 93).In three dimensions,(7)where and(8)The Poisson kernel for the -ball is(9)where is the outward normal derivative at point on a unit -sphere and(10)Let be harmonic on a neighborhood of the closed unit disk , then the reproducing property of the Poisson kernel states that for ,(11)(Krantz 1999, p. 94).

Harmonic conjugate function

The harmonic conjugate to a given function is a function such thatis complex differentiable (i.e., satisfies the Cauchy-Riemann equations). It is given bywhere , , and is a constant of integration.Note that is a closed form since is harmonic, . The line integral is well-defined on a simply connected domain because it is closed. However, on a domain which is not simply connected (such as the punctured disk), the harmonic conjugate may not exist.

Scalar potential

A conservative vector field (for which the curl ) may be assigned a scalar potentialwhere is a line integral.