delta sequence
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11: 1.16 Distributions
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►A sequence
of test functions converges to a test function if the support of every is contained in a fixed compact set and as the sequence
converges uniformly on to for .
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►We say that a sequence of distributions
converges to a distribution in if
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§1.16(iii) Dirac Delta Distribution
… ►The Dirac delta distribution is singular. … ►A sequence of tempered distributions converges to in if …12: 2.1 Definitions and Elementary Properties
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►In (2.1.5) can be replaced by any fixed ray in the sector , or by the whole of the sector .
(Here and elsewhere in this chapter is an arbitrary small positive constant.)
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►Let , , be a sequence of functions defined in such that for each
…Then is an asymptotic sequence or scale.
Suppose also that and satisfy
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13: 25.11 Hurwitz Zeta Function
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25.11.40
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►As in the sector , with and fixed, we have the asymptotic expansion
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►Similarly, as in the sector ,
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14: 19.30 Lengths of Plane Curves
15: 18.2 General Orthogonal Polynomials
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►For such a system, functions and sequences
() satisfying can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2):
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►The Hankel determinant
of order is defined by and
…Also define determinants by , and
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►The operator is a delta operator, i.
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16: 18.33 Polynomials Orthogonal on the Unit Circle
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18.33.1
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18.33.17
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18.33.20
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►This states that for any sequence
with and the polynomials generated by the recurrence relations (18.33.23), (18.33.24) with satisfy the orthogonality relation (18.33.17) for a unique probability measure with infinite support on the unit circle.
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