Difference between revisions of "Convolution of functions"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Gel'fand, G.E. Shilov, "Generalized functions" , '''1–5''' , Acad. Press (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{ZBL|0954.35001}} {{ZBL|0652.35002}} {{ZBL|0695.35001}} {{ZBL|0699.35005}} {{ZBL|0607.35001}} {{ZBL|0506.35001}} {{ZBL|0223.35002}} {{ZBL|0231.35002}} {{ZBL|0207.09101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Gel'fand, G.E. Shilov, "Generalized functions" , '''1–5''' , Acad. Press (1964) (Translated from Russian) {{MR|435831}} {{ZBL|0115.33101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) {{MR|0942661}} {{ZBL|0017.40404}} {{ZBL|63.0367.05}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Kecs, "The convolution product and some applications" , Reidel & Ed. Academici (1982)</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Kecs, "The convolution product and some applications" , Reidel & Ed. Academici (1982) {{MR|0690953}} {{ZBL|0512.46041}} </TD></TR></table> |
Revision as of 11:58, 27 September 2012
and
belonging to
The function defined by
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it is denoted by the symbol . The function
is defined almost everywhere and also belongs to
. The convolution has the basic properties of multiplication, namely,
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for any three functions in . Therefore,
with the usual operations of addition and of multiplication by a scalar, with the operation of convolution as the multiplication of elements, and with the norm
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is a Banach algebra (for this norm ). If
denotes the Fourier transform of
, then
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and this is used in solving a number of applied problems.
Thus, if a problem has been reduced to an integral equation of the form
![]() | (*) |
where
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then, under the assumption that , by applying the Fourier transformation to (*) one obtains
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hence
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and the inverse Fourier transformation yields the solution to (*) as
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The properties of a convolution of functions have important applications in probability theory. If and
are the probability densities of independent random variables
and
, respectively, then
is the probability density of the random variable
.
The convolution operation can be extended to generalized functions (cf. Generalized function). If and
are generalized functions such that at least one of them has compact support, and if
is a test function, then
is defined by
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where is the direct product of
and
, that is, the functional on the space of test functions of two independent variables given by
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for every infinitely-differentiable function of compact support.
The convolution of generalized functions also has the commutativity property and is linear in each argument; it is associative if at least two of the three generalized functions have compact supports. The following equalities hold:
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where is the differentiation operator and
is any multi-index,
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in particular, , where
denotes the delta-function. Also, if
,
are generalized functions such that
, and if there is a compact set
such that
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then
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Finally, if is a generalized function of compact support and
is a generalized function of slow growth, then the Fourier transformation can be applied to
, and again
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The convolution of generalized functions is widely used in solving boundary value problems for partial differential equations. Thus, the Poisson integral, written in the form
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is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature can be not only an ordinary function but also a generalized one.
Both for ordinary and generalized functions the concept of a convolution carries over in a natural way to functions of several variables; then in the above and
must be regarded as vectors from
and not as real numbers.
References
[1] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101 |
[2] | I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1–5 , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101 |
[3] | E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) MR0942661 Zbl 0017.40404 Zbl 63.0367.05 |
Comments
References
[a1] | W. Kecs, "The convolution product and some applications" , Reidel & Ed. Academici (1982) MR0690953 Zbl 0512.46041 |
Convolution of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convolution_of_functions&oldid=17031