Top 10 similar words or synonyms for poisson_kernel
scalar_curvature 0.827424
poincaré_metric 0.821784
laplace_operator 0.821343
laplace_beltrami_operator 0.816276
sectional_curvature 0.812080
hilbert_transform 0.810969
dirac_delta_function 0.810574
dimensional_hausdorff_measure 0.804678
bivector 0.802449
dirac_operator 0.802097
Top 30 analogous words or synonyms for poisson_kernel
Article | Example |
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Poisson kernel | If formula_2 is the open unit disc in C, T is the boundary of the disc, and "f" a function on T that lies in "L"(T), then the function "u" given by |
Poisson kernel | is harmonic in D and has a radial limit that agrees with "f" almost everywhere on the boundary T of the disc. |
Poisson kernel | Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in "L"(T) . Let "f" ∈ "L"(T) have Fourier series {"f"}. After the Fourier transform, convolution with "P"("θ") becomes multiplication by the sequence {"r"} ∈ "l"(Z). Taking the inverse Fourier transform of the resulting product {"rf"} gives the Abel means "Af" of "f": |
Poisson kernel | Rearranging this absolutely convergent series shows that "f" is the boundary value of "g" + "h", where "g" (resp. "h") is a holomorphic (resp. antiholomorphic) function on "D". |
Poisson kernel | Thus, again, the Hardy space "H" on the upper half-plane is a Banach space, and, in particular, its restriction to the real axis is a closed subspace of formula_11. The situation is only analogous to the case for the unit disk; the Lebesgue measure for the unit circle is finite, whereas that for the real line is not. |
Poisson kernel | The Poisson kernel for the upper half-space appears naturally as the Fourier transform of the Abel kernel |
Poisson kernel | in which "t" assumes the role of an auxiliary parameter. To wit, |
Poisson kernel | In particular, it is clear from the properties of the Fourier transform that, at least formally, the convolution |
Poisson kernel | When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space. This is true when the negative Fourier coefficients of "f" all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle. |
Poisson kernel | That the boundary value of "u" is "f" can be argued using the fact that as "r" → 1, the functions "P"("θ") form an approximate unit in the convolution algebra "L"(T). As linear operators, they tend to the Dirac delta function pointwise on "L"(T). By the maximum principle, "u" is the only such harmonic function on "D". |