Space of main functions

The space of basic functions is the structure by which the space of generalized functions is constructed (the space of linear functionals on the space of basic functions).

Generalized functions are of great importance in mathematical physics , and the space of basic functions is used as the basis for the construction of generalized functions (formally this is the domain of definition of the corresponding generalized functions). Differential equations are considered in the so-called. in the weak sense , that is, it does not consider pointwise equality, but the equality of the corresponding regular linear functionals on an appropriate space of basic functions. See Sobolev spaces .

Usually as a space of basic functions ${\mathcal {D}}(\Omega )$ ${\ displaystyle {\ mathcal {D}} (\ Omega)}$ ${\ displaystyle {\ mathcal {D}} (\ Omega)}$ the space of infinitely differentiable functions with compact support is chosen (the so-called compactly supported functions) $C_{0}^{\infty }(\Omega )$ ${\ displaystyle C_ {0} ^ {\ infty} (\ Omega)}$ ${\ displaystyle C_ {0} ^ {\ infty} (\ Omega)}$ on which the following convergence is introduced (and therefore the topology ):

Sequence $\left\{u_{j}\right\}_{j=1}^{\infty }\subset {\mathcal {D}}(\Omega )$ {\ displaystyle \ left \ {u_ {j} \ right \} _ {j = 1} ^ {\ infty} \ subset {\ mathcal {D}} (\ Omega)} ${\ displaystyle \ left \ {u_ {j} \ right \} _ {j = 1} ^ {\ infty} \ subset {\ mathcal {D}} (\ Omega)}$ converges to $u\in {\mathcal {D}}(\Omega )$ ${\ displaystyle u \ in {\ mathcal {D}} (\ Omega)}$ ${\ displaystyle u \ in {\ mathcal {D}} (\ Omega)}$ , if a:

Functions $u_{j}$ ${\ displaystyle u_ {j}}$ $u_ {j}$ are uniformly finite , i.e. $\exists K$ ${\ displaystyle \ exists K}$ ${\ displaystyle \ exists K}$ - compact in $\Omega$ ${\ displaystyle \ Omega}$ $\ Omega$ including $\forall j\;\mathrm {supp} \,u_{j}\subset K$ ${\ displaystyle \ forall j \; \ mathrm {supp} \, u_ {j} \ subset K}$ ${\ displaystyle \ forall j \; \ mathrm {supp} \, u_ {j} \ subset K}$ .
$\forall \alpha \;D^{\alpha }u_{j}(x)\to D^{\alpha }u(x)$ ${\ displaystyle \ forall \ alpha \; D ^ {\ alpha} u_ {j} (x) \ to D ^ {\ alpha} u (x)}$ ${\ displaystyle \ forall \ alpha \; D ^ {\ alpha} u_ {j} (x) \ to D ^ {\ alpha} u (x)}$ uniformly across $x$ ${\ displaystyle x}$ $x$ .

Here $\Omega$ ${\ displaystyle \ Omega}$ $\ Omega$ - limited area in $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ $\ mathbb {R} ^ {n}$ .

For questions of the Fourier transform , generalized functions of slow growth are used. For them, the Schwartz class is chosen as the main ${\mathcal {S}}={\mathcal {S}}(\mathbb {R} ^{n})$ ${\ displaystyle {\ mathcal {S}} = {\ mathcal {S}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle {\ mathcal {S}} = {\ mathcal {S}} (\ mathbb {R} ^ {n})}$ - infinitely smooth on $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ $\ mathbb {R} ^ {n}$ functions decreasing when $|x|\to \infty$ ${\ displaystyle | x | \ to \ infty}$ $| x | \ to \ infty$ faster than any degree $|x|^{-1}$ ${\ displaystyle | x | ^ {- 1}}$ ${\ displaystyle | x | ^ {- 1}}$ along with all its derivatives. Convergence on it is defined as follows: a sequence of functions $\phi _{1},\phi _{2},\dots$ ${\ displaystyle \ phi _ {1}, \ phi _ {2}, \ dots}$ ${\ displaystyle \ phi _ {1}, \ phi _ {2}, \ dots}$ converges to $\phi ^{*}$ ${\ displaystyle \ phi ^ {*}}$ ${\ displaystyle \ phi ^ {*}}$ , if a

\forall \alpha ,\beta \in \mathbb {N} \ |x|^{\alpha }D^{\beta }\phi _{j}(x)\to |x|^{\alpha }D^{\beta }\phi ^{*}(x)

{\ displaystyle \ forall \ alpha, \ beta \ in \ mathbb {N} \ | x | ^ {\ alpha} D ^ {\ beta} \ phi _ {j} (x) \ to | x | ^ {\ alpha } D ^ {\ beta} \ phi ^ {*} (x)}

{\ displaystyle \ forall \ alpha, \ beta \ in \ mathbb {N} \ | x | ^ {\ alpha} D ^ {\ beta} \ phi _ {j} (x) \ to | x | ^ {\ alpha } D ^ {\ beta} \ phi ^ {*} (x)}

uniformly across

x

{\ displaystyle x}

x

.

The choice of the Schwartz class for constructing the Fourier transform on the space of generalized functions is determined by the fact that the Fourier transform is an automorphism on the Schwartz class.

Literature

V.S. Vladimirov . Generalized functions in mathematical physics. - ed. 2nd. - M .: Science , 1979 . - 320 p.

Space of main functions

Literature

See also

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