Bump function

which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.The set of all bump functions with domainforms a vector space, denotedNote that the support of this function is the closed interval, where the closure is taken with respect the Euclidean topology of the real line.The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article.scaled to fit into the unit disc: the substitutionvariables is obtained by taking the product ofThis function is supported on the unit ball centered at the origin.The function has a strictly positive denominator everywhere on the real line, hence g is also smooth.To have the smooth transition in the real interval [a, b] with a < b, consider the function For real numbers a < b < c < d, the smooth function equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function., leads to: which is not an infinitely differentiable function (so, is not "smooth"), so the constraints a < b < c < d must be strictly fulfilled.make smooth transition curves with "almost" constant slope edges (a bump function with true straight slopes is portrayed this Another example).Bump functions defined in terms of convolution The construction proceeds as follows.The latter is just a bump function with a very small support and whose integral isSuch a mollifier can be obtained, for example, by taking the bump functionfrom the previous section and performing appropriate scalings.An alternative construction that does not involve convolution is now detailed.and denote the usual Euclidean norm byis endowed with the usual Euclidean metric).the partial derivatives all vanish (equalthe values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number.As a corollary, given two disjoint closed subsetsthe above construction guarantees the existence of smooth non-negative functionsBump functions are often used as mollifiers, as smooth cutoff functions, and to form smooth partitions of unity.They are the most common class of test functions used in analysis.The space of bump functions is closed under many operations.to fulfill the requirement of "smoothness", it has to preserve the continuity of all its derivatives, which leads to the following requirement at the boundaries of its domain:Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power offrom above can be analyzed by a saddle-point method, and decays asymptotically as