Math 2.0

Integral transforms, operational calculus Definition

Integral transform

Builds on FunctionLebesgue integral

Definition

Let II and JJ be intervals of ℝ\R, with −∞≤t1<t2≤∞-\infty \le t_1 < t_2 \le \infty the endpoints of II. Let K ⁣:I×J→ℂK \colon I \times J \to \C be a measurable function. Let f ⁣:I→ℂf \colon I \to \C be a function. The integral transform of ff with kernel KK is the function TfTf given by (Tf)(u)=∫t1t2f(t) K(t,u) dt(Tf)(u) = \int_{t_1}^{t_2} f(t)\, K(t, u)\, dt at each u∈Ju \in J for which the integral converges. The integral is a Lebesgue integral or an improper Riemann integral, depending on the setting.

Remarks

Example: take I=[0,∞)I = [0, \infty), J=(0,∞)J = (0, \infty) and K(t,u)=e−utK(t, u) = e^{-ut}. This is the Laplace transform; for f=1f = 1 the integral ∫0∞e−ut dt=1/u\int_0^\infty e^{-ut}\, dt = 1/u converges for every u∈Ju \in J. Non-example: the map f↦f′f \mapsto f' is linear but is not an integral transform with a function kernel. The value f′(u)f'(u) depends only on the values of ff near uu, and no integrable kernel produces this for all smooth ff. The map f↦Tff \mapsto Tf is linear wherever both sides converge. The same definition works with II and JJ replaced by measure spaces, for example subsets of ℝn\R^n.

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Element, definition, no evidence levelElementdefinitionIntegrable function, concept, no evidence levelIntegrable functionconceptIntegral of a non-negative function, concept, no evidence levelIntegral of a non-neg…conceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionLebesgue integral, definition, no evidence levelLebesgue integraldefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionIntegral transform, definition, no evidence levelIntegral transformdefinition
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  1. Element definition
  2. Integrable function concept
  3. Integral of a non-negative function concept
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  5. Set definition
  6. Lebesgue integral definition
  7. Implication definition
  8. Predicate definition
  9. Ordered pair definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Function definition
  17. Integral transform definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Integral_transform
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DefinesIntegral transform
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