equation is taken to mean that, for any test function φ, The result follows from the formula for the Newtonian potential (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the Radon transform because it recovers the value of φ(x) from its integrals over hyperplanes. For instance, if n is odd and , then the integral on the right hand side is where is the Radon transform of φ: An alternative equivalent expression of the plane wave decomposition, from , is for n even, and for n odd. Fourier kernels In the study of Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a periodic function converges to the function. The nth partial sum of the Fourier series of a function f of period 2 is defined by convolution (on the interval ) with the Dirichlet kernel: Thus, where A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval tends to a multiple of the delta function as . This is interpreted in the distribution sense, that for every compactly supported smooth function f. Thus, formally one has on the interval . Despite this, the result does not hold for all compactly supported continuous functions: that is DN does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods to produce convergence. The method of Cesàro summation leads to the Fejér kernel The Fejér kernels tend to the delta function in a stronger sense that for every compactly supported continuous function f. The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point. Hilbert space theory The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L2 of square-integrable functions. Indeed, smooth compactly supported functions are dense in L2, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L2 and to give a stronger topology on which the delta function defines a bounded linear functional. Sobolev spaces The Sobolev embedding theorem for Sobolev spaces on the real line R implies that any square-integrable function f such that is automatically continuous, and