Oscillating integral

An oscillating integral is an object from the mathematical sub-area of functional analysis or from micro-local analysis . It is a generalized integral term, which is used especially in the field of distribution theory . Since the phase function causes the integrand to oscillate , the integral was called the oscillating integral. This term was introduced by Lars Hörmander .

Phase function

definition

A function is called a phase function, if for all ${\displaystyle \phi \in C^{\infty }(X\times \mathbb {R} ^{n}\backslash \{0\})}$ ${\displaystyle (x,\xi )\in X\times \mathbb {R} ^{n}\backslash \{0\}}$

• the imaginary part is nonnegative, that is

${\displaystyle \operatorname {Im} \phi (x,\xi )\geq 0}$.

• the function is homogeneous , that is

${\displaystyle \phi (x,\lambda \xi )=\lambda \phi (x,\xi )}$for everyone . ${\displaystyle \lambda >0}$

• the differential is non-zero, that is

${\displaystyle {\frac {\mathrm {d} \phi (x,\xi )}{\mathrm {d} (x,\xi )}}\neq 0}$.

example

• The images , where the standard scalar product denotes, are phase functions which occur during the Fourier transformation and its inverse transformation. ${\displaystyle (x,\xi )\mapsto \pm \langle x,\xi \rangle }$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

motivation

Be a phase function like for example and be a symbol with . Keep defining ${\displaystyle \phi }$ ${\displaystyle \phi (x,\xi )=\langle x,\xi \rangle }$ ${\displaystyle a\in S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{n})}$ ${\displaystyle m+k<-n,\ k\in \mathbb {N} }$

${\displaystyle I_{\phi }(a)(x):=\int _{\mathbb {R} ^{n}}e^{i\phi (x,\xi )}a(x,\xi )\mathrm {d} \xi \in C^{k}(X)}$.

The image

${\displaystyle S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{n})\to C^{k}(X),\quad a\mapsto I_{\phi }(a)}$

is steady . These types of parameter integrals are common in the field of functional analysis. For example, the Fourier transform and the bilateral Laplace transform have this shape. Or the solution of Bessel's differential equation

${\displaystyle J_{k}(\lambda )=(2\pi )^{-1}\int _{0}^{2\pi }e^{i\lambda \sin(\xi )}e^{ik\xi }\mathrm {d} \xi }$

can be noted like this.

Continuation sentences

Fourier transform on L ²

The Fourier transformation can be carried out on the Schwartz space by the integral operator ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$

${\displaystyle I_{-tx}(f)(t)={\mathcal {F}}(f)(t)={\frac {1}{\left(2\pi \right)^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} tx}f(x)\,\mathrm {d} x}$

To be defined. This operator can be continued using a density argument , but the Fourier integral does not converge for every function. The operator must therefore be represented differently. ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$

Space of symbol classes

With the space of the distributions on and with the space of the symbol classes . Be a phase function and is , . Then there is exactly one possibility of a mapping ${\displaystyle {\mathcal {D}}'(X)}$ ${\displaystyle X}$ ${\displaystyle S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{n})}$ ${\displaystyle \phi }$ ${\displaystyle 0<\rho \leq 1}$ ${\displaystyle 0\leq \delta <1}$

${\displaystyle I_{\phi }:S_{\rho ,\delta }^{\infty }(X\times \mathbb {R} ^{n})=\bigcup _{m\in \mathbb {R} }S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{n})\to {\mathcal {D}}'(X)}$

to define so that for the integral ${\displaystyle a\in S_{\rho ,\delta }^{m}(X\times \mathbb {R} ^{n}),\ m<-n}$

${\displaystyle I_{\phi }(a)(x)=\int _{\mathbb {R} ^{n}}e^{i\phi (x,\xi )}a(x,\xi )\mathrm {d} \xi }$

exists and the mapping is continuous. ${\displaystyle I_{\phi }:S_{\rho ,\delta }^{\infty }(X\times \mathbb {R} ^{n})\to {\mathcal {D}}'(X)}$

definition

The two continuation clauses mentioned above show that it is desirable to have an integral term so that the continuations can also be expressed in the integral notation. The oscillating integral defined below can be used for this.

Oscillating integral

Be a clipping function with for and for . In addition, let it be a phase function and a symbol class . Now you bet ${\displaystyle \chi \in C_{c}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle \chi (\xi )=1}$ ${\displaystyle |\xi |\leq 1}$ ${\displaystyle \chi (\xi )=0}$ ${\displaystyle |\xi |\geq 2}$ ${\displaystyle \phi \colon \mathbb {R} ^{n}\times \mathbb {R} ^{N}\to \mathbb {R} }$ ${\displaystyle a\in S^{m}(\mathbb {R} ^{n}\times \mathbb {R} ^{N})}$

${\displaystyle I_{\phi }(a)(x):=\int _{\mathbb {R} ^{N}}e^{i\phi (x,\xi )}a(x,\xi )\mathrm {d} \xi :=\lim _{j\to \infty }\int _{\mathbb {R} ^{N}}\chi \left({\tfrac {\xi }{j}}\right)e^{i\phi (x,\xi )}a(x,\xi )\mathrm {d} \xi \,}$

where the limit value is to be understood in terms of distributions . That means the limit is through

${\displaystyle \langle I_{\phi }(a),u\rangle =\lim _{j\to \infty }\int _{\mathbb {R} ^{N}}\int _{\mathbb {R} ^{n}}\chi \left({\tfrac {\xi }{j}}\right)e^{i\phi (x,\xi )}a(x,\xi )u(x)\mathrm {d} x\mathrm {d} \xi }$

explained for all test functions . The integral term is called the oscillating integral. ${\displaystyle u\in {\mathcal {D}}(\mathbb {R} ^{n})\cong C_{c}^{\infty }(\mathbb {R} ^{n})}$ ${\displaystyle I_{\phi }}$

Oscillating integral operator

Be again a phase function and a symbol of class . The image ${\displaystyle \phi \colon \mathbb {R} ^{n}\times \mathbb {R} ^{N}\to \mathbb {R} }$ ${\displaystyle a\in S^{m}(\mathbb {R} ^{n}\times \mathbb {R} ^{N})}$

${\displaystyle u\mapsto T_{\lambda }(u)(x):=I_{\phi }(au)(x)=\int _{\mathbb {R} ^{N}}e^{i\lambda \phi (x,\xi )}a(x,\xi )u(\xi )\mathrm {d} \xi }$

is an oscillating integral operator .

Boundedness to L ²

Lars Hörmander showed that oscillating integral operators are bounded operators on the space of square integrable functions under certain conditions . ${\displaystyle L^{2}(\mathbb {R} ^{n})}$

Let be a phase function and let the symbol class be a smooth function with compact support. Then there is a constant such that ${\displaystyle \phi }$ ${\displaystyle a\colon \mathbb {R} ^{n}\times \mathbb {R} ^{N}\to \mathbb {R} }$ ${\displaystyle C}$

${\displaystyle \|T_{\lambda }(u)\|_{L^{2}(\mathbb {R} ^{n})}\leq C\lambda ^{-{\frac {n}{2}}}\|u\|_{L^{2}(\mathbb {R} ^{n})}}$

holds, which means that the linear operator is bounded, i.e. continuous. In addition, it follows from the Banach-Steinhaus theorem that the family of operators is uniformly bounded . ${\displaystyle T_{\lambda }}$ ${\displaystyle L^{2}}$ ${\displaystyle (T_{\lambda })_{\lambda }}$

Examples

Bessel function

→ Main article : Bessel function

The Bessel function

${\displaystyle {\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{\mathrm {i} \,x\sin \varphi }e^{-\nu \varphi }\,\mathrm {d} \varphi \,.}$

is an oscillating integral with the phase function and the symbol . ${\displaystyle \sin \varphi }$ ${\displaystyle e^{-\nu \varphi }}$

Fourier transform

→ Main article : Fourier transform

Let be a smooth function with compact support and with and be the phase function. By rescaling, one can find the oscillating integral operator ${\displaystyle a\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle a(0,0)={\tfrac {1}{\left(2\pi \right)^{\frac {n}{2}}}}}$ ${\displaystyle \phi (x,\xi )=-\langle x,\xi \rangle }$

${\displaystyle T_{\lambda }(u)(x)=I_{\phi }(au)(x)=\int _{\mathbb {R} ^{n}}e^{-i\lambda \langle x,\xi \rangle }a(x,\xi )u(\xi )\mathrm {d} \xi }$

in

${\displaystyle {\tilde {T}}_{\lambda }(u)(x)=\int _{\mathbb {R} ^{n}}e^{-i\langle x,\xi \rangle }a\left({\frac {x}{\sqrt {\lambda }}},{\frac {\xi }{\sqrt {\lambda }}}\right)u(\xi )\mathrm {d} \xi }$

transform. This family of operators is equally restricted to and for one obtains the Fourier transform ${\displaystyle L^{2}}$ ${\displaystyle \lambda \to \infty }$

${\displaystyle {\tilde {T}}_{\infty }(u)(x)={\mathcal {F}}(u)(x)={\frac {1}{\left(2\pi \right)^{\frac {n}{2}}}}\int _{\mathbb {R} ^{n}}e^{-\mathrm {i} x\xi }u(\xi )\,\mathrm {d} \xi }$.

Pseudo differential operator

→ Main article : Pseudodifferential operator

With the help of the oscillating integral one defines a special continuous and linear operator

${\displaystyle T\colon {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})}$

on the Schwartz room , which through

${\displaystyle {\begin{aligned}T(u)(x)=I_{\langle x,\xi \rangle }(a\,{\mathcal {F}}(u))(x)&=\int _{\mathbb {R} ^{n}}e^{i\langle x,\xi \rangle }a(x,\xi ){\mathcal {F}}(u)(\xi )\mathrm {d} \xi \\&=\int _{\mathbb {R} ^{n}}e^{i\langle x,\xi \rangle }a(x,\xi )\int _{\mathbb {R} ^{n}}e^{-i\langle y,\xi \rangle }u(y)\mathrm {d} y\,\mathrm {d} \xi \\&=\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{i\langle x-y,\xi \rangle }a(x,\xi )u(y)\mathrm {d} y\,\mathrm {d} \xi \end{aligned}}}$

given is. The function is a symbolic function and the operator is called the pseudo differential operator. It is a generalization of a differential operator . The integral kernel of this operator is ${\displaystyle a\in S^{m}(\mathbb {R} ^{n}\times \mathbb {R} ^{n})}$ ${\displaystyle T}$

${\displaystyle K(x,y):=\int _{\mathbb {R} ^{n}}e^{i\langle x-y,\xi \rangle }a(x,\xi )\mathrm {d} \xi }$

and is a typical Schwartz core .

literature

• Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).
• Elias M. Stein : Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton NJ 1993, ISBN 0-691-03216-5 ( Princeton mathematical Series 43 = Monographs in harmonic Analysis 3).
• Alain Grigis, Johannes Sjöstrand: Microlocal analysis for differential operators. An introduction. Cambridge University Press, Cambridge et al. 1994, ISBN 0-521-44986-3 ( London Mathematical Society lecture note series 196).

Individual evidence

1. ↑ L. Hörmander: Fourier integral operators , Acta Math. 127 (1971), 79-183. doi : 10.1007 / BF02392052
2. ^ Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals . Princeton University Press, 1993. ISBN 0-691-03216-5 , p. 377.
3. ↑ Christopher D. Sogge: Fourier integrals in classical analysis . Cambridge University Press, 1993, ISBN 0-521-06097-4 , pp. 41 .